# Minkowski Decomposition of Associahedra and Related Combinatorics

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## Abstract

Realisations of associahedra with linear non-isomorphic normal fans can be obtained by alteration of the right-hand sides of the facet-defining inequalities from a classical permutahedron. These polytopes can be expressed as Minkowski sums and differences of dilated faces of a standard simplex as described by Ardila et al. (Discret Comput Geom, 43:841–854, 2010). The coefficients \(y_I\) of such a Minkowski decomposition can be computed by Möbius inversion if tight right-hand sides \(z_I\) are known not just for the facet-defining inequalities of the associahedron but also for all inequalities of the permutahedron that are redundant for the associahedron. We show for certain families of these associahedra: (1) How to compute the tight value \(z_I\) for any inequality that is redundant for an associahedron but facet-defining for the classical permutahedron. More precisely, each value \(z_I\) is described in terms of tight values \(z_J\) of facet-defining inequalities of the corresponding associahedron determined by combinatorial properties of \(I\). (2) The computation of the values \(y_I\) of Ardila, Benedetti & Doker can be significantly simplified and depends on at most four values \(z_{a(I)},\,z_{b(I)},\,z_{c(I)}\) and \(z_{d(I)}\). (3) The four indices \(a(I),\,b(I),\,c(I)\) and \(d(I)\) are determined by the geometry of the normal fan of the associahedron and are described combinatorially. (4) A combinatorial interpretation of the values \(y_I\) using a labeled \(n\)-gon. This result is inspired from similar interpretations for vertex coordinates originally described by Loday and well-known interpretations for the \(z_I\)-values of facet-defining inequalities.

## Keywords

Realizations of Polytopes Associahedra Minkowski sums and differences Möbius inversion## 1 Introduction

*past any vertices*, compare Postnikov et al. [19]. This fine distinction and additional condition is easily overlooked but essential. For example, Proposition 1.2 does not hold for arbitrary polytopes \(P_n(\{ z_I\})\), we illustrate this by a simple example in Sect. 5. Fundamental examples of generalised permutahedra are dilations of the standard simplex \(\Delta _n=\mathrm{conv }\{ e_1, e_2, \ldots , e_n\}\) where \(e_i\) denotes the \(i\mathrm{{th}}\) standard basis vector of \({\mathbb {R}}^n\).

**Lemma 1.1**

([1, Lemma 2.1]). \(P_n(\{z_I\})+P_n(\{z^{\prime }_I\}) = P_n(\{z_I + z^{\prime }_I\})\).

If we consider the function \(I\longmapsto z_I\) that assigns every subset of \([n]\) the corresponding tight value \(z_I\) of \(P_n(\{z_I\})\), then the Möbius inverse of this function assigns to \(I\) the coefficient \(y_I\) of a Minkowski decomposition of \(P_n(\{z_I\})\) into faces of the standard simplex:

**Proposition 1.2**

*Every generalised permutahedron*\(P_n(\{z_I\})\)

*can be written uniquely as a Minkowski sum and difference of faces of*\(\Delta _n\):

In particular, we also have \(z_I = \sum _{J\subseteq I} y_J\). A basic example is the classical permutahdron: it is known to be a zonotope and it is the Minkowski sum of the edges and vertices of \(\Delta _n\). The reader is invited to check that the corresponding \(z_I\)-values obtained by this formula yield precisely the right-hand sides mentioned earlier.

\(I\) | \(\{1\}\) | \(\{2\}\) | \(\{3\}\) | \(\{1,2\}\) | \(\{1,3\}\) | \(\{2,3\}\) | \(\{1,2,3\}\) |
---|---|---|---|---|---|---|---|

\(z^{c_1}_I\) | 1 | 1 | 1 | 3 | 2 | 3 | 6 |

\(z^{c_2}_I\) | 1 | 0 | 1 | 3 | 3 | 3 | 6 |

We could stop here and be fascinated how the Möbius inversion relates the description by half spaces to the Minkowski decomposition. But we go beyond this alternating sum description for \(y_I\) and significantly simplify the formula for each \(y_I\) in Therom 4.2. In fact, each \(y_I\) can be expressed as an alternating sum of at most four non-zero values \(z_{a(I)},\,z_{b(I)},\,z_{c(I)}\) and \(z_{d(I)}\) which are tight right-hand sides for certain facet-defining inequalities as specified in the theorem. In other words, we extract combinatorial core data for the Möbius inversion of the function \(z_I\) and answer the question which subsets \(J\) of \(I\) are essential to compute \(y_I\) if the associahedron’s normal fan is the normal fan of \(\mathsf As _{n-1}^c\). Figure 9 illustrates how Theorem 4.2 can be used to compute the coefficients \(y_I\) for one of the two examples shown in Fig. 1. If the associahedron coincides with some \(\mathsf As ^c_{n-1}\) of Hohlweg and Lange [11], Theorem 4.3 states a purely combinatorial interpretation of the values \(y_I\). To illustrate this theorem, we recompute \(y_I\) for \(\mathsf As ^{c_1}_2\) and \(\mathsf As ^{c_2}_2\) in Examples 4.6 and 4.7.

The outline of the paper is as follows. Section 2 summarises necessary known facts about \(\mathsf As _{n-1}^c\) and indicates some occurrences of the realisations considered here in the mathematical literature. In Sect. 3 we introduce the notion of an up and down interval decomposition for subsets \(I\subseteq [n]\). This decomposition depends on the choice of a Coxeter element \(c\) (or equivalently on a partition of \([n]\) induced by \(c\)) and is essential to prove Proposition 3.8. This proposition gives a combinatorial characterisation of all tight values \(z_I\) for \(\mathsf As _{n-1}^c\) needed to evaluate \(y_I\) using Proposition 1.2. The main results, Theorems 4.2 and 4.3, are then stated in Sect. 4. The proof of Theorem 4.2 is long and convoluted and deferred to Sects. 6 and 7, while Theorem 4.3 is proved under the assumption of Theorem 4.2 in Sect. 4. To show that Proposition 1.2 and Theorem 4.2 do not hold for polytopes \(P_n(\{z_I\})\) that are not contained in the deformation cone of the classical permutahedron, we briefly study a realisation of a 2-dimensional cyclohedron in Sect. 5.

About the same time as some of these results were achieved, Pilaud and Santos showed that the associahedra \(\mathsf As _{n-1}^c\) are examples of brick polytopes [16, 17]. One of their results is that any brick polytope can be expressed as a Minkowski sum of other brick polytopes. As a consequence, we have two Minkowski decompositions of \(\mathsf As _{n-1}^c\) that are extremal in the following sense. The first decomposition of \(\mathsf As _{n-1}^c\) has a relatively complicated structure with respect to the coefficients \(y_I\) (possibly negative numbers) but is very simple with respect to the polytopes used (faces of a standard simplex). On the other hand, the second decomposition of \(\mathsf As _{n-1}^c\) has a simple structure in terms its coefficients (they are either 0 or 1) but is more complicated with respect to the polytopes used (brick polytopes). At the time of writing, the exact relationship of these two decompositions is not properly understood and remains a joint project of Pilaud with the author.

## 2 Associahedra as Generalised Permutahedra

Associahedra form a family of combinatorially equivalent polytopes and can be realised as generalised permutahedra. Since the combinatorics of a polytope is encoded in its face lattice, we define an associahedron as a polytope with a face lattice that is isomorphic to the lattice of sets of non-crossing proper diagonals of a convex and plane \((n+2)\)-gon \(Q\) ordered by reversed inclusion.^{1} This description immediately tells us that the set of \(k\)-faces is in bijection to the set of triangulations of \(Q\) with \(k\) proper diagonals removed. In particular, vertices correspond to triangulations and facets correspond to proper diagonals. Since associahedra turn out to be simple polytopes, a result of Blind and Mani-Levitska with an elegant proof due to Kalai, [3, 13], guarantees that the face lattice is already determined by the 1-skeleton, so it suffices to specify the vertex-edge graph to determine the combinatorics of the face-lattice. This graph is known as the flip graph of triangulations of \(Q\). In 2004, Loday published a beautiful combinatorial description for the vertex coordinates of associahedra constructed earlier by Shnider and Sternberg, Shnider and Stasheff [14, 24, 25]. Loday’s description is in terms of labeled binary trees dual to the triangulations of \(Q\). The construction of Shnider, Sternberg and Stasheff as well as Loday’s vertex description was subsequently generalised by Hohlweg and Lange [11]. The latter construction explicitly describes realisations \(\mathsf As _{n-1}^c\) of \((n-1)\)-dimensional associahedra and exhibits them as generalised permutahedra. The construction depends on the choice of a Coxeter element \(c\) of the symmetric group \(\Sigma _n\) on \(n\) elements.

*down set*and to \(\mathsf U _c\) as

*up set*. The partitions satisfy

\(R_\delta \) and \(\tilde{z}^c_{I}\) associated to the proper diagonals \(\delta \) of the labelled hexagon for the associahedron on the left of Fig. 1 (\(\mathsf D_c = {1,3,4}\) and \(\mathsf U_c = {2}\))

\(\delta \) | \(\{0,3\}\) | \(\{0,4\}\) | \(\{0,5\}\) | \(\{1,2\}\) | \(\{1,4\}\) | \(\{1,5\}\) | \(\{2,3\}\) | \(\{2,4\}\) | \(\{3,5\}\) |

\(R_\delta \) | \(\{1\}\) | \(\{1,3\}\) | \(\{1,3,4\}\) | \(\{2,3,4\}\) | \(\{3\}\) | \(\{3,4\}\) | \(\{1,2\}\) | \(\{1,2,3\}\) | \(\{4\}\) |

\(\tilde{z}^c_{R_\delta }\) | 1 | 3 | 6 | 6 | 1 | 3 | 3 | 6 | 1 |

\(R_\delta \) and \(\tilde{z}^c_{I}\) associated to the proper diagonals \(\delta \) of the labelled hexagon for the associahedron on the right of Fig. 1 (\(\mathsf D_c = {1,4}\) and \(\mathsf U_c = {2,3}\))

\(\delta \) | \(\{0,4\}\) | \(\{2,4\}\) | \(\{3,4\}\) | \(\{0,5\}\) | \(\{0,3\}\) | \(\{1,2\}\) | \(\{2,5\}\) | \(\{1,3\}\) | \(\{1,5\}\) |

\(R_\delta \) | \(\{1\}\) | \(\{1,2\}\) | \(\{1,2,3\}\) | \(\{1,4\}\) | \(\{1,3,4\}\) | \(\{2,3,4\}\) | \(\{1,2,4\}\) | \(\{3,4\}\) | \(\{4\}\) |

\(\tilde{z}^c_{R_\delta }\) | 1 | 3 | 6 | 3 | 6 | 6 | 6 | 3 | 1 |

We end this section relating \(\mathsf As ^c_n\) to earlier work. Firstly, we indicate a connection to cambrian fans, generalised associahedra and cluster algebras and secondly to convex rank texts and semigraphoids in statistics. Thirdly, we mention some earlier appearances of specific instances of \(\mathsf As ^c_{n-1}\) in the literature.

Fomin and Zelevinsky introduced generalised associahedra in the context of cluster algebras of finite type, [8], and it is well-known that associahedra are generalised associahedra associated to cluster algebras of type \(A\). The construction of [11] was subsequently generalised by Hohlweg, Lange, and Thomas to generalised associahedra, [12], and depends also on a Coxeter element \(c\). The geometry of the normal fans of these realisations is determined by combinatorial properties of \(c\) and the normal fans are \(c\)-cambrian fans (introduced by Reading and Speyer in [20]). Reading and Speyer conjectured the existence of a linear isomorphism between \(c\)-cambrian fans and \(g\)-vector fans associated to cluster algebras of finite type with acyclic initial seed (the notion of a \(g\)-vector fan for cluster algebras was introduced by Fomin and Zelevinsky [9]). In [21], Reading and Speyer describe and relate cambrian and \(g\)-vector fans in more detail and prove their conjecture up to an assumption of another conjecture of [9]. Yang and Zelevinsky gave an alternative proof of the conjecture of Reading and Speyer in [28]. Stella recently recovered the realizations of generalized associahedra for finite type of [12] and describes the relationship to cluster algebras in detail [27].

Generalised permutahedra and therefore the associahedra \(\mathsf As ^c_{n-1}\) are closely related to the framework of convex rank tests and semigraphoids from statistics as discussed by Morton et al. [15]. The semigraphoid axiom characterises the collection of edges of a permutahedron that can be contracted simultaneously to obtain a generalised permutahedron. The authors also study submodular rank tests, its subclass of Minkowski sum of simplices tests and graphical rank tests. The latter one relates to graph associahedra of Carr and Devadoss [5]. Among the associahedra studied in this manuscript, Loday’s realisation fits to Minkowski sum of simplices and graphical rank tests.

Some instances of \(\mathsf As ^c_{n-1}\) have been studied earlier. For example, the realisations of Loday, [14], and of Rote et al. [22], related to one-dimensional point configurations, are affine equivalent to \(\mathsf As _{n-1}^c\) if \(\mathsf U _c=\varnothing \) or \(\mathsf U _c=[n]{\setminus }\{1,n\}\). For \(\mathsf U _c = \varnothing \), the Minkowski decomposition into faces of a standard simplex is described by Postnikov in [18]. Moreover, Rote, Santos, and Streinu point out in Sect. 5.3 that their realisation is not affine equivalent to the realisation of Chapoton et al. [7]. It is not difficult to show that the realisation described in [7] is affine equivalent to \(\mathsf As _{3}^c\) if \(\mathsf U _c=\{2\}\) or \(\mathsf U _c=\{3\}\).

## 3 Tight Values for all \(z_I^c\) for \(\mathsf As _{n-1}^c\)

Since the facet-defining inequalities for \(\mathsf As ^c_{n-1}\) correspond to proper diagonals of \(Q_c\), we know precisely the irredundant inequalities for the generalised permutahedron \(P_n(\{\tilde{z}_I^c\})= \mathsf As ^{c}_{n-1}\). In this section, we determine tight values \(\tilde{z}^c_I\) for all \(I\subseteq [n]\) corresponding to redundant inequalities in order to be able to compute the coefficients \(y_I\) of the Minkowski decomposition of \(\mathsf As _{n-1}^c\) as described by Proposition 1.2. The concept of an up and down interval decomposition induced by the partitioning \(\mathsf D _c\sqcup \mathsf U _c\) (or, equivalently, induced by \(c\)) of a given interval \(I\subset [n]\) is a key concept that we introduce first, it allows us to describe any \(I\subseteq [n]\) in terms of unions and intersections of sets \(R_\delta \) for certain proper diagonals determined by this decomposition (or, equivalently, as unions of set differences of certain sets \(R_\delta \) and their complements).

**Definition 3.1**

*Up and down intervals*). Let \(\mathsf D _c=\{d_1=1<d_2<\cdots <d_\ell =n\}\) and \(\mathsf U _c=\{u_1<u_2<\cdots <u_m\}\) be the partition of \([n]\) induced by a Coxeter element \(c\).

- (a)
A set \(S \subseteq [n]\) is a non-empty interval of \([n]\) if \(S = \{ r, r+1, \ldots , s\}\) for some \(0 < r \le s < n\). We write \(S\) as closed interval \([r,s]\) (end-points included) or as open interval \((r-1,s+1)\) (end-points excluded). An empty interval is an open interval \((k,k+1)\) for some \(1\le k < n\).

- (b)
A non-empty open down interval is a set \(S = \{ d_r<d_{r+1}<\cdots <d_s\}\) \( \subseteq \mathsf D _c\) for some \(1\le r \le s \le \ell \). We write \(S\) as open down interval \((d_{r-1},d_{s+1})_\mathsf{D _c}\) where we allow \(d_{r-1}=0\) and \(d_{s+1}=n+1\), i.e. \(d_{r-1},d_{s+1}\in \overline{\mathsf{D }}_c\). For \(1\le r\le \ell -1\), we also have the empty down interval \((d_r,d_{r+1})_\mathsf{D _c}\).

- (c)
A closed up interval is a non-empty set \(S = \{ u_r<u_{r+1}<\cdots <u_s\}\) \( \subseteq \mathsf{U }_c\) for some \(1\le r \le s \le \ell \). We write \([u_r,u_s]_\mathsf{U _c}\).

We often omit the words *open* and *closed* when we consider down and up intervals. There will be no ambiguity, because we are not going to deal with closed down intervals or open up intervals. Up intervals are always non-empty, while down intervals may be empty. It will be useful to distinguish the empty down intervals \((d_r,d_{r+1})_\mathsf{D _c}\) and \((d_s,d_{s+1})_\mathsf{D _c}\) if \(r\ne s\) although they are equal as sets.

It might be helpful to read the following definition of the up and down interval decomposition in combination with Examples 3.3 and 3.5.

**Definition 3.2**

*Up and down interval decomposition*). Let \(\mathsf D _c \sqcup \mathsf U _c\) be the partition of \([n]\) induced by a Coxeter element \(c\) and \(I\subset [n]\) be non-empty. The up and down interval decomposition of type \((v,w)\) of \(I\) is a partition of \(I\) into disjoint up and down intervals \(I^\mathsf U _1, \ldots , I^\mathsf U _w\) and \(I^\mathsf D _1,\ldots ,I^\mathsf D _v\) obtained by the following procedure.

- 1.
Suppose there are \(\tilde{v}\) non-empty inclusion maximal down intervals of \(I\) denoted by \(\tilde{I}^\mathsf D _k=(\tilde{a}_k,\tilde{b}_k)_\mathsf{D _c},\,1\le k\le \tilde{v}\), with \(\tilde{b}_k \le \tilde{a}_{k+1}\) for \(1\le k < \tilde{v}\). Consider also all empty down intervals \(E^\mathsf{D }_i=(d_{r_i},d_{r_i+1})_\mathsf{D _c}\) with \(\tilde{b}_k\le d_{r_i}<d_{r_i+1}\le \tilde{a}_{k+1}\) for \(0\le k \le \tilde{v}\) where \(\tilde{b}_0=1\) and \(\tilde{a}_{\tilde{v}+1}=n\). Denote the open intervals \((\tilde{a}_i,\tilde{b}_i)\) and \((d_{r_i},d_{r_i+1})\) of \([n]\) by \(\tilde{I}_i\) and \(E_i\) respectively.

- 2.Consider all inclusion maximal up intervals of \(I\) contained in some interval \(\tilde{I}_i\) or \(E_i\) obtained in Step 1 and denote these up intervals byWithout loss of generality, we assume \(\alpha _i \le \beta _i < \alpha _{i+1}\).$$\begin{aligned} I^\mathsf U _1=[\alpha _1,\beta _1]_\mathsf{U _c},\ldots ,I^\mathsf U _w=[\alpha _w,\beta _w]_\mathsf{U _c}. \end{aligned}$$
- 3.
A down interval \(I^\mathsf D _i = (a_i,b_i)_{\mathsf{{D} }_c},\,1\le i \le v\), is a down interval obtained in Step 1 that is either a non-empty down interval \(\tilde{I}^\mathsf D _k\) or an empty down interval \(E^\mathsf{D }_k\) with the additional property that there is some up interval \(I^\mathsf U _j\) obtained in Step 2 such that \(I^\mathsf U _j \subseteq E_k\). Without loss of generality, we assume \(b_i \le a_{i+1}\) for \(1\le i < v\).

*Example 3.3*

- (i)
\(J_1=\{2,3\}\). The only non-empty inclusion maximal down interval of \(J_1\) is \(\tilde{I}^\mathsf D _1{=}(1,4)_{\mathsf{{D} }_c}{=}\{3\}\); there are no empty down intervals \(E^\mathsf{D }_i\) to be considered. As inclusion maximal up intervals of \(J_1\) contained in \(\tilde{I}_1=(1,4)=\{2,3\}\), we identify \(I^\mathsf U _1{=}[2,2]_{\mathsf{{U} }_c}{=}\{2\}\). The up and down interval decomposition of \(J_1\) is \((1,4)_{\mathsf{{D} }_c} \sqcup [2,2]_{\mathsf{{U} }_c}\). Its type is \((1,1)\).

- (ii)
\(J_2=\{2\}\). There is no non-empty inclusion maximal down interval of \(J_2\) to be considered, but there is one empty down interval \(E^\mathsf{D }_1=(1,3)_{\mathsf{{D} }_c}\) such that \(E_1=(1,3)=\{2\}\) contains one inclusion maximal up interval \(I^\mathsf U _1=[2,2]_{\mathsf{{U} }_c}=\{2\}\) of \(J_2\). It follows that the up and down interval decomposition of \(J_2\) is \((1,3)_{\mathsf{{D} }_c}\sqcup [2,2]_{\mathsf{{U} }_c}\). Its type is \((1,1)\).

- (iii)
Consider \(J_3=\{2,4\}\). The only non-empty inclusion maximal down interval of \(J_3\) is \(\tilde{I}^\mathsf D _1{=}(3,5)_{\mathsf{{D} }_c}{=}\{4\}\); there is one empty down interval \(E^\mathsf{D }_1=(1,3)_{\mathsf{{D} }_c}\) such that \(E_1\) contains an inclusion maximal up interval of \(J_3\), this is the up interval \(I^\mathsf U _1=[2,2]_{\mathsf{{U} }_c}=\{2\}\). There is no non-empty inclusion maximal up interval contained in \(\tilde{I}^\mathsf D _1\). It follows that the up and down interval decomposition of \(J_3\) is \(\left( (1,3)_{\mathsf{{D} }_c}\sqcup [2,2]_{\mathsf{{U} }_c}\right) \sqcup \left( (3,5)_{\mathsf{{D} }_c}\right) \). Its type is \((2,1)\).

**Definition 3.4**

(*Nested up and down interval decomposition, nested components*)

- (a)
The up and down interval decomposition of \(I\) is nested if its type is \((1,w)\).

- (b)
A nested component of \(I\) is an inclusion-maximal subset \(J\) of \(I\) such that the up and down decomposition of \(J\) is nested.

*Example 3.5*

- (i)
\(R_\delta =(a,b)_{\mathsf{{D} }_c}\) iff \(R_\delta \) has an up and down decomposition of type \((1,0)\).

- (ii)
\(R_\delta =(0,b)_{\mathsf{{D} }_c}\cup [u_1,a]_{\mathsf{{U} }_c}\) or \(R_\delta =(a,n+1)_{\mathsf{{D} }_c}\cup [b,u_m]_{\mathsf{{U} }_c}\) iff \(R_\delta \) has a decomposition of type \((1,1)\).

- (iii)
\(R_\delta =(0,n+1)_{\mathsf{{D} }_c}\cup [u_1,a]_{\mathsf{{U} }_c}\cup [b,u_m]_{Up_c}\) iff \(R_\delta \) has an up and down decomposition of type \((1,2)\).

**Lemma 3.6**

Given the partition \([n]=\mathsf D _c \sqcup \mathsf U _c\) induced by a Coxeter element \(c\). Let \(I\) be a non-empty proper subset of \([n]\) with up and down interval decomposition of type \((v,w)\) and nested components of type \((1,w_1), \cdots , (1,w_v)\). For \(1\le i \le v\) and \(1\le j \le w_i\), denote by \([\alpha _{i,j},\beta _{i,j}]_{\mathsf{{U} }_c}\) the inclusion maximal up intervals contained in the down interval \((a_i,b_i)_{\mathsf{{D} }_c}\) where \(\beta _{i,j}<\alpha _{i,j+1}\) and \(b_i\le a_{i+1}\).

*Proof*

It follows from the definition of nested components that \(\delta _{i,j}\) and \(\delta _{i^\prime ,j^\prime }\) are non-crossing if \(i\ne i^\prime \). That \(\delta _{i,j}\) and \(\delta _{i,j^\prime }\) are non-crossing within a nested component is implied by \(\beta _{i,j}<\alpha _{i,j+1}\).

To see the identities on \(I\), we first remark that \(I = \bigcap _{j\in [w_1+1]}R_{\delta _{1,j}}\) follows directly from the the up and down interval decomposition of \(I\) and the definition of \(R_\delta \) if \(I\) has only one nested component. If \(I\) consists of more than one nested component, we obtain the claim since it holds for each nested component separately. The second identity is a simple reformulation of the first. This is easily seen in case of just one nested component: instead of intersecting the sets \(R_\delta \), we choose \(\delta =\delta _{1,w_1+1}\) and remove the complements \([n]{\setminus } R_{\delta _{1,j}},\,1\le j\le w_1\) from \(R_\delta \). This yields \(\bigcap _{j\in [w_1+1]}R_{\delta _{i,j}}\). \(\square \)

*Example 3.7*

- (i)
\(J_1 =\) \((1,4)_{\mathsf{{D} }_c} \sqcup [2,2]_{\mathsf{{U} }_c}\) and the associated diagonals are \(\delta _{1,1}=\{1,2\}\) and \(\delta _{1,2}=\{2,4\}\).

- (ii)
\(J_2 =\) \((1,3)_{\mathsf{{D} }_c} \sqcup [2,2]_{\mathsf{{U} }_c}\) and the associated diagonals are \(\delta _{1,1}=\{1,2\}\) and \(\delta _{1,2}=\{2,3\}\).

- (iii)
\(J_3 =\) \(\left( (1,3)_{\mathsf{{D} }_c}\sqcup [2,2]_{\mathsf{{U} }_c}\right) \sqcup \left( (3,5)_{\mathsf{{D} }_c}\right) \) and the associated diagonals are \(\delta _{1,1}=\{1,2\},\,\delta _{1,2}=\{2,3\}\) and \(\delta _{2,1}=\{3,5\}\).

The final proposition of this section resolves the quest for tight values \(z_{I}^c\) of all redundant inequalities of an associahedron that has the normal fan of \(\mathsf As _{n-1}^c\). If we denote this associahedron by \(P_n(\{\tilde{z}^c_I\})\), then the inequalities that correspond to an index set \(I=R_\delta \) for some proper diagonal of \(Q_c\) are precisely the facet defining inequalities and all other inequalities are redundant.

**Proposition 3.8**

*Proof*

**Definition 3.9**

Let \(I\) be a non-empty proper subset of \([n]\) with up and down interval decomposition of type \((v,w)\) and nested components of type \((1,w_1), \cdots , (1,w_v)\). As in Lemma 3.6, we associate diagonals \(\delta _{i,j}\) for \(1\le i \le v\) and \(1\le j \le w_i\). The subset \({\mathcal {D}}_I\) of proper diagonals of \(\{\delta _{i,j}\vert 1\le i \le v\,\text {and}\,1\le j\}\) is called set of proper diagonals associated to \(I\). Similarly, we say that \(\delta \in {\mathcal {D}}_I\) is a proper diagonal associated to \(I\).

We end this section with some remarks. First, if a non-proper diagonal \(\delta =\{0,u_1\}\) or \(\delta {=}\{u_m,n+1\}\) occurs as a diagonal associated to the first or last nested component, the formula for \(z_I^c\) in Proposition 3.8 can be simplified by cancelation of the corresponding terms \(\tilde{z}^c_{[n]}\). Second, for any proper diagonal \(\delta \) of \(Q_c\), we obtain \(z^c_{R_\delta }=\tilde{z}^c_{R_\delta }\). And finally, we can characterise the face of \(P(\{\tilde{z}^c_I\})\) that minimises the linear functional \(\sum _{i\in I} x_i\) for a given non-empty and proper subset \(I\subset [n]\).

**Corollary 3.10**

Associate the linear functional \(\varphi _I(x)=\sum _{i\in I} x_i\) to a non-empty proper subset \(I\subset [n]\) and denote the facet of \(P(\{\tilde{z}^c_I\})\) that is supported by \(\sum _{i\in R_\delta } x_i=\tilde{z}^c_{R_\delta }\) for the proper diagonal \(\delta \) by \(F_{R_\delta }\). Then the intersection \(\bigcap _{\delta \in {\mathcal {D}}_I}F_{R_\delta }\) is the minimizing face of \(P(\{\tilde{z}^c_I\})\) for \(\varphi _I\).

## 4 Main Results and Examples

*Example 4.1*

- (i)Since \(J_1=\{2,3\} = (1,4)_{\mathsf{{D} }_c} \sqcup [2,2]_{\mathsf{{U} }_c}\) is nested, we have \(\gamma =2\) and \(\Gamma =3\). It follows thatIn this situation, all diagonals \(\delta _i\) except diagonal \(\delta _2=\{1,3\}\) are proper diagonals. Therefore, \({\fancyscript{D}}_I=\{\delta _1, \delta _3, \delta _4\}\)$$\begin{aligned} \delta _1=\{1,4\}, \qquad \delta _2=\{1,3\}, \qquad \delta _3=\{2,4\} \qquad \text {and}\qquad \delta _4=\{2,3\}. \end{aligned}$$
- (ii)Since \(J_2=\{2\} = \) \((1,3)_{\mathsf{{D} }_c}\sqcup [2,2]_{\mathsf{{U} }_c}\) is nested, we have \(\gamma =\Gamma =2\). This impliesIn this situation, the diagonals \(\delta _1\) and \(\delta _4\) are not proper while the diagonals \(\delta _2\) and \(\delta _3\) are proper. Hence, \({\fancyscript{D}}_I=\{\delta _2, \delta _3\}\).$$\begin{aligned} \delta _1=\{1,3\}, \qquad \delta _2=\{1,2\}, \qquad \delta _3=\{2,3\} \qquad \text {and}\qquad \delta _4=\{2,2\}. \end{aligned}$$
- (iii)
The set \(J_3=\{2,4\}\) is not nested since its up and down interval decomposition is of type \((2,1)\). We do not associate diagonals \(\delta _i\) to \(J_3\), the set \({\fancyscript{D}}_I\) is empty.

**Theorem 4.2**

We prove Theorem 4.2 in Sect. 6. An example illustrating the theorem for the left associahedron \(\mathsf As ^c_3\) of Fig. 1 (\(\mathsf D _c=\{1,3,4\}\) and \(\mathsf U _c=\{2\}\)) is given in Fig. 9 where we also explicitly compute the \(y_I\)-values for this realisation with \(z^c_I=\frac{|I|(|I|+1)}{2}\) for the facet-defining inequalities.

For the rest of this section, we specialise to realisations with this specific choice of \(z_I\)-values. We obtain a nice combinatorial interpretation of the coefficients \(y_I\) in Theorem 4.3 and characterise the vanishing \(y_I\)-values in Corollary 4.5.

*signed lengths*\(K_\gamma \) and \(K_\Gamma \) of \(I\) are integers defined as follows. Let \(|K_\Gamma |\) be the number of edges of the path in \(\partial Q\) connecting \(b\) and \(\Gamma \) that does not use the vertex labeled \(a\). The sign of \(K_\Gamma \) is negative if and only if \(\Gamma \in \mathsf D _c\). Similarly, \(|K_\gamma |\) is the length of path in \(\partial Q\) connecting \(a\) and \(\gamma \) not using label \(b\) and \(K_\gamma \) is negative if and only if \(\gamma \in \mathsf D _c\). Equivalently, we have that \(K_\gamma \) (respectively \(K_\Gamma \)) is a positive integer if and only if \(\gamma \in \mathsf U _c\) (respectively \(\Gamma \in \mathsf U _c\)) and that \(K_\gamma =-1\) (respectively \(K_\Gamma =-1\)) if and only if \(\gamma \in \mathsf D _c\) (respectively \(\Gamma \in \mathsf D _c\)). We can now express the coefficients \(y_I\) of \(\mathsf As _{n-1}^c\) in terms of \(K_\gamma \) and \(K_\Gamma \). The following theorem is an easy consequence of Theorem 4.2.

**Theorem 4.3**

*Proof*

**Corolloary 4.4**

For \(n\ge 2\) and any choice \(\mathsf D _c\sqcup \mathsf U _c\), we have \(y_{[n]}=(-1)^{|\mathsf U _c|}.\)

*Proof*

**Corollary 4.5**

Let \(n \ge 2\) and \(\mathsf D _c\sqcup \mathsf U _c\) be a partition induced by some Coxeter element \(c\). Then \(y_I=0\) if and only if \(I\) has an up and down decomposition of type \((v_I,w_I)\) with \(v_I>1\) or \(n=3\) and \(I=\mathsf U _c=\{2\}\).

*Proof*

Since \(K_\gamma \) and \(K_\Gamma \) are non-zero, Theorem 4.3 implies \(y_I\ne 0\) if \(I\ne \{u_s\}\). So we assume \(I=\{u_s\}\). It now suffices to prove that \(y_I= 0\) if and only if \(n=3\).

We now illustrate Theorem 4.3 by recomputing the \(y_I\)-values for \(\mathsf As ^{c_1}_2\) and \(\mathsf As ^{c_2}_2\) mentioned in the introduction. For \(n=3\), there are two possible partitions of \(\{1,2,3\}\) that correspond to the two Coxeter elements of \(\Sigma _3\): either \(\mathsf D _{c_1}=\{1,2,3\}\) and \(\mathsf U _{c_1}=\varnothing \) or \(\mathsf D _{c_2}=\{1,3\}\) and \(\mathsf U _{c_2}=\{2\}\).

*Example 4.6*

- (i)
We have \(y_I=1\) for \(I=\{ i\}\) and \(1\le i\le 3\). The up and down interval decomposition of \(\{ i\}\) is \((i-1,i+1)_{\mathsf{{D} }}\) and \(\gamma =\Gamma =i\). It follows that \(K_\gamma =K_\Gamma =-1\) and \(I{\setminus } (a,b)_{\mathsf{{D} }}=\varnothing \). Thus \(y_I=1\).

- (ii)
We have \(y_I=1\) for \(I=\{ i,i+1\}\) and \(1\le i\le 2\). Then \(I=(i-1,i+2)_{\mathsf{{D} }},\,\gamma =i\), and \(\Gamma =i+1\). It follows that \(K_\gamma =K_\Gamma =-1\) and \(I{\setminus } (a,b)_{\mathsf{{D} }}=\varnothing \). Thus \(y_I=1\).

- (iii)
We have \(y_I=0\) for \(I=\{1,3\}\). Then \(I=(0,2)_{\mathsf{{D} }}\sqcup (2,4)_{\mathsf{{D} }}\), so \(I\) is of type \((2,0)\) and \(y_I=0\) by Corollary 4.5.

- (iv)
We have \(y_I=1\) for \(I=\{1,2,3\}\). Then \(I=(0,4)_{\mathsf{{D} }},\,\gamma =1\) and \(\Gamma =3\) implies \(K_\gamma =K_\Gamma {=}-1\) and \(I{\setminus } (a,b)_{\mathsf{{D} }}{=}\varnothing \). Thus \(y_I=1\). Of course, we could also use Corollary 4.4 instead.

*Example 4.7*

- (i)
We have \(y_I=1\) for \(I=\{1\}\) and \(I=\{3\}\). The up and down interval decomposition is \((0,3)_{\mathsf{{D} }}\) and \((1,4)_{\mathsf{{D} }}\) respectively. Therefore we have \(\gamma =\Gamma =1\) and \(\gamma =\Gamma =3\) respectively. It follows \(K_\gamma =K_\Gamma =-1\) and \(I{\setminus } (a,b)_{\mathsf{{D} }}=\varnothing \).

- (ii)
We have \(y_I=0\) for \(I=\{2\}\). The up and down interval decomposition is \((1,3)_{\mathsf{{D} }} \sqcup [2,2]_{\mathsf{{U} }}\), so \(I\) is of type \((1,1)\). We have \(\gamma =\Gamma =2\) which implies \(K_\gamma =K_\Gamma =2\). Since \(n=3\), we conclude \(y_I=(3+1)-2\cdot 2=0\). Of course, we could have used Corollary 4.5 instead.

- (iii)
We have \(y_I=2\) for \(I=\{i,i+1\}\) and \(1\le i\le 2\). Then \(I=(i-1,i+2)_{\mathsf{{D} }},\,\gamma =i\), and \(\Gamma =i+1\), that is, \(K_\gamma {=}-1,\,K_\Gamma {=}2\). Moreover, \(I{\setminus } (a,b)_{\mathsf{{D} }}=\{2\}\) if \(I=\{1,2\}\) and \(K_\gamma =2,\,K_\Gamma =-1\), and \(I{\setminus } (a,b)_{\mathsf{{D} }}=\{2\}\) if \(I=\{2,3\}\).

- (iv)
We have \(y_I=1\) for \(I=\{1,3\}\). Then \(I=(0,4)_\mathsf{D },\,\gamma =1\), and \(\Gamma =3\). It follows that \(K_\gamma =K_\Gamma =-1\) and \(I{\setminus } (a,b)_{\mathsf{{D} }}=\varnothing \).

- (v)
We have \(y_I=-1\) for \(I=\{ 1,2,3\}\). Then \(I=(0,4)_{\mathsf{{D} }} \sqcup [2,2]_{\mathsf{{U} }}\) with \(\gamma =1\) and \(\Gamma =3\). It follows that \(K_\gamma =K_\Gamma =-1\) and \(I{\setminus } (a,b)_{\mathsf{{D} }}=\{ 2\}\). Again, we could have used Corollary 4.4 instead.

## 5 A Remark on Cyclohedra

We now show that Proposition 1.2 does not hold if we consider a polytope obtained by ‘moving some inequalities of the permutahedron past vertices’. The example is a cyclohedron which also known as Bott-Taubes polytope or type B generalised permutahedron [4, 7, 26]. A Minkowski decomposition of ‘generalised permutahedra of type \(B\)’ (similar to Proposition 1.2 for generalised permutahedra) is not known.

*symmetric*choices \(c\). To obtain these realisations of cyclohedra, we follow [11] and intersect \(\mathsf As _{2n-1}^c\) with ‘type \(B\) hyperplanes’ \(x_i+x_{2n+1-i}=2n-1\) for \(1\le i < n\). A 2-dimensional cyclohedron \(\mathsf Cy _{2}^c\) obtained from \(\mathsf As _{3}^c\) (with up set \(\mathsf U _c=\{2\}\)) by intersection with \(x_1+x_4=5\) is shown in Fig. 10 (the hyperplane \(x_2+x_3=5\) is implicitly used since \(\mathsf As _{3}^c\) is contained in \(x_1+x_2+x_3+x_4=10\)). A similar construction does not yield a cyclohedron if one starts with the other associahedron of Fig. 1 where \(\mathsf U _c=\{2,3\}\). The tight right-hand sides of this realisation of the cyclohedron are obviously the tight right-hand sides of \(\mathsf As _{3}^c\) except \(z^c_{\{1,4\}}=z^c_{\{2,3\}}=5\). The inequalities \(x_1+x_4\ge 2\) and \(x_2+x_3\ge 2\) are redundant for \(\mathsf As _{3}^c\) and altering the level sets for these inequalities from 2 (for \(\mathsf As _{3}^c\)) to 5 (for \(\mathsf Cy _2^c\)) means that we move past the four vertices \(A\), \(B\), \(C\), and \(D\), so the realisation of the cyclohedron is not in the deformation cone of the classical permutahedron. We now show by example that Proposition 1.2 does not hold in this situation. To this respect, we list the function \(z_I\) of tight right hand-sides for all inequalities of the permutahedron (that is, facet-defining or not for the cyclohedron) and its Möbius inverse \(y_I\), both defined on the boolean lattice: In other words, if Proposition 1.2 were true for ‘generalised permutahedra not in the deformation cone of the classical permutahedron’, then the following equation of polytopes has to hold:

## 6 A Proof of Theorem 4.2

This section is devoted to the proof of Theorem 4.2 under the assumption that Lemma 6.3 holds; Lemma 6.3 is proved in Sect. 7. We begin with an outline of the strategy to prove Theorem 4.2.

First, we prove Proposition 6.2 which weakens Theorem 4.2 in two senses: we restrict to \(I\subset [n]\) with a nested decomposition and we restrict to the situation where \({\fancyscript{D}}_I=\{\delta _1,\delta _2,\delta _3,\delta _4\}\), that is, where all four diagonals \(\delta _i\) are proper. That the statement of Proposition 6.2 is actually the statement of Theorem 4.2 weakened by these additional assumptions follows from Corollary 6.7.

Lemma 6.3 states precisely which subsets of \(\{\delta _1,\delta _2,\delta _3,\delta _4\}\) are sets \({\fancyscript{D}}_I\) for some \(I\subset [n]\) with a nested up and down interval decomposition. Lemma 6.4 then expresses the Minkowski coefficients \(y_I\) using these sets \({\fancyscript{D}}_I\) if \(I\subset [n]\) has a nested up and down interval decomposition and \(|{\fancyscript{D}}_I|<4\). Lemmas 6.5 and 6.6 then imply the claim of Theorem 4.2 when \(I\subset [n]\) has a nested decomposition and not all \(\delta _i\) are proper. Finally, Lemma 6.8 covers the cases \(I\subset [n]\) where \(I\) does not have a nested decomposition and Lemma 6.9 settles \(I=[n]\).

Suppose now that the proper diagonal \(\delta \) occurs on the right-hand side of this rewritten formula for \(y_I\), that is, \(\delta \) is one of the associated diagonals \(\delta ^J_{i,j}\) for some \(J\subseteq I\). We now distinguish whether \(\delta \) occurs as a single summand \(\tilde{z}_{R_{\delta ^J_{i,m_{J,i}}}}^c\) or as a compound summand \((\tilde{z}_{R_{\delta ^J_{i,j}}}^c - \tilde{z}^c_{[n]})\) and make the following definition.

**Definition 6.1**

- (a)
A proper diagonal \(\delta \) (associated to \(J\subseteq I\)) is of type \(\tilde{z}_{R_\delta }^c\) (in the expression for \(y_I\)), if there exists an index \(i\in [v_J]\) such that \(\delta =\delta ^J_{i,m_{J,i}}\).

- (b)
A proper diagonal \(\delta \) (associated to \(J\subseteq I\)) is of type \(\bigl (\tilde{z}_{R_\delta }^c - \tilde{z}^c_{[n]}\bigr )\) (in the expression for \(y_I\)), if there exist indices \(i\in [v_J]\) and \(j\in [m_{J,i}-1]\) such that \(\delta =\delta ^J_{i,j}\)

A geometric interpretation of these notions is the following. The proper diagonal \(\delta \) (associated to \(J\subseteq I\)) is of type \(\tilde{z}_{R_\delta }^c\) (in the expression for \(y_I\)), if \(\delta \) is the ‘rightmost’ proper diagonal associated to a nested component of \(J\). Similarly, the proper diagonal \(\delta \) (associated to \(J\subseteq I\)) is of type \(\bigl (\tilde{z}_{R_\delta }^c - \tilde{z}^c_{[n]}\bigr )\) (in the expression for \(y_I\)), if \(\delta \) is a proper diagonal associated to a nested component of \(J\), but it is not the rightmost one.

**Proposition 6.2**

*Proof*

- 1.\(R_\delta \) has up and down decomposition of type \((1,0)\), see Fig. 11. Then \(R_{\delta }=(\tilde{a}, \tilde{b})_{\mathsf{{D} }_c}\subseteq (a,b)_{\mathsf{{D} }_c}\) and we may consider \(J=R_\delta \subseteq I\) as witness for the occurrence of \(\delta \) in the right-hand side of (1). Let \(S\subseteq I\) be a set with \(\delta \in {\mathcal {D}}_S\). Then \(J=(\tilde{a},\tilde{b})_{\mathsf{{D} }_c}\) is necessarily a nested component of type \((1,0)\) of \(S\) and all other nested components are subsets of \((a,\tilde{a})\cap I\) and \((\tilde{b},b)\cap I\). It follows that \(S\subseteq I\) satisfies \(\delta \in {\mathcal {D}}_S\) if and only ifWe now collect all terms for \(\tilde{z}_{R_\delta }^c\) in the expression for \(y_I\). Since \(\delta \) is a proper diagonal, we have \(\tilde{z}_{R_\delta }^c\ne 0\) and the resulting alternating sum vanishes if and only if there is more than one term of this type, that is, if and only if$$\begin{aligned} R_{\delta }\subseteq S \subseteq R_{\delta }\cup \bigl ((a,\tilde{a})\cap I\bigr )\cup \bigl ((\tilde{b},b)\cap I\bigr ). \end{aligned}$$If \(\bigl ((a,\tilde{a})\cap I\bigr )\cup \bigl ((\tilde{b},b)\cap I\bigr )= \varnothing \), we obtain \((-1)^{|I{\setminus } R_\delta |}\tilde{z}_{R_\delta }^c\) as contribution for \(y_I\). For later use in this proof, we note that \(\bigl ((a,\tilde{a})\cap I\bigr )\cup \bigl ((\tilde{b},b)\cap I\bigr )= \varnothing \) guarantees \(\delta \in {\fancyscript{D}}_I\). Note that \(R_{\delta _1}\) is always of type \((1,0)\) if the up and down decomposition of \(R_\delta \) is of type \((1,0)\). Similarly, we have \(\delta _2\in {\fancyscript{D}}_I\) with \(R_{\delta _2}\) of type \((1,0)\) if additionally \(\Gamma \in \mathsf D _c,\,\delta _3\in {\fancyscript{D}}_I\) with \(R_{\delta _3}\) of type \((1,0)\) if additionally \(\gamma \in \mathsf D _c\), and \(\delta _4\in {\fancyscript{D}}_I\) with \(R_{\delta _4}\) of type \((1,0)\) if additionally \(\gamma ,\Gamma \in \mathsf D _c\).$$\begin{aligned} \bigl ((a,\tilde{a})\cap I\bigr )\cup \bigl ((\tilde{b},b)\cap I\bigr )\ne \varnothing . \end{aligned}$$
- 2.\(R_\delta \) has up and down decomposition of type \((1,1)\). In contrast to Case 1, \(R_\delta \subseteq I\) is not true in general any more. We distinguish two cases, either \(\delta =\{\tilde{\beta }, \tilde{b}\}\) with \(\tilde{\beta }<\tilde{b},\,\tilde{\beta }\in \mathsf U _c\) and \(\tilde{b}\in \mathsf D _c\) or \(\delta =\{\tilde{a}, \tilde{\alpha }\}\) with \(\tilde{a}<\tilde{\alpha },\,\tilde{a}\in \mathsf D _c\) and \(\tilde{\alpha }\in \mathsf U _c\).
- a.\(\delta =\{\tilde{\beta }, \tilde{b}\}\), see Fig. 12 Observe first that \(R_\delta =(0,\tilde{b})_{\mathsf{{D} }_c}\cup [u_1,\tilde{\beta }]_{\mathsf{{U} }_c}\) with \(\tilde{\beta }<\tilde{b}\le b\). Since we assume that \(\delta \) appears in the right-hand side of (1), we have \(\tilde{\beta }\in I\) and may consider \(J=R_\delta \cap I\). If \(S\subseteq I\) is a subset with \(\delta \in {\mathcal {D}}_S\) then \(\delta \) must be the ‘rightmost’ diagonal of one nested component for \(S\). This means that the diagonal \(\delta \) associated to \(S\) is never of type \((\tilde{z}_{R_{\delta }}^c-\tilde{z}^c_{[n]})\) in the expression for \(y_I\). Similarly to Case 1, we conclude that the terms \(\tilde{z}_{R_\delta }^c\) cancel if and only ifAgain, \(\tilde{z}_{R_\delta }^c\ne 0\) since \(\delta \) is a proper diagonal and the terms for \(\tilde{z}_{R_\delta }^c\) do not cancel if and only if there is only one subset \(S\subseteq I\) with \(\delta \in {\mathcal {D}}_S\), that is, if \(((a,\tilde{\beta })\cap I)\cup ((\tilde{b}, b)\cap I)= \varnothing \). For later use in this proof, we mention the two possible scenarios if$$\begin{aligned} \bigl ((a,\tilde{\beta })\cap I\bigr )\cup \bigl ((\tilde{b}, b)\cap I\bigr )\ne \varnothing \qquad \text {or}\qquad \tilde{z}_{R_\delta }^c=0. \end{aligned}$$Firstly, if \(\gamma \in \mathsf U _c\) and \(\Gamma \in \mathsf D _c\), then \(\delta \in \{\delta _3,\delta _4\}\) and the contribution of \(\delta _3\) and \(\delta _4\) to \(y_I\) is$$\begin{aligned} ((a,\tilde{\beta })\cap I)\cup ((\tilde{b}, b)\cap I)= \varnothing . \end{aligned}$$Secondly, if \(\gamma ,\Gamma \in \mathsf U _c\), then \(\delta =\delta _3\) and the contribution to \(y_I\) is \((-1)^{|I{\setminus } R_{\delta _3}|}\tilde{z}_{R_{\delta _3}}^c\).$$\begin{aligned} (-1)^{|I{\setminus } R_{\delta _3}|}\tilde{z}_{R_{\delta _3}}^c \qquad \text {and}\qquad (-1)^{|I{\setminus } R_{\delta _4}|}\tilde{z}_{R_{\delta _4}}^c. \end{aligned}$$
- b.\(\delta =\{\tilde{a}, \tilde{\alpha }\}\) Observe first that \(R_\delta =(\tilde{a},n+1)_{\mathsf{{D} }_c}\cup [\tilde{\alpha },u_m]_{\mathsf{{U} }_c}\) with \(a\le \tilde{a}<\tilde{\alpha }\). Since we assume that \(\delta \) appears in the right-hand side of (1), we have \(\tilde{\alpha }\in I\) and may consider \(J=R_\delta \cap I\). If \(S\subseteq I\) with \(\delta \in {\mathcal {D}}_S\), then \(\delta \) (associated to \(S\)) can be of type \(\tilde{z}_{R_{\delta }}^c\) or \((\tilde{z}_{R_{\delta }}^c-\tilde{z}^c_{[n]})\) in the expression for \(y_I\). The diagonal \(\delta \) is of type \(\tilde{z}_{R_{\delta }}^c\) if and only if \(R_\delta =R_\delta \cap I\) and \(S=R_\delta \cup M\) for some subset \(M\subseteq (a,\tilde{a})\cap I\). The diagonal \(\delta \) is of type \((\tilde{z}_{R_{\delta }}^c-\tilde{z}^c_{[n]})\) for all other subsets \(S\subseteq I\) with \(\delta \in {\mathcal {D}}_S\), in particular, we conclude \(R_\delta \supset R_\delta \cap S\). We distinguish two sub-cases: either \(\delta \) (associated to \(S\)) is of type \((\tilde{z}_{R_{\delta }}^c-\tilde{z}^c_{[n]})\) (in the expression for \(y_I\)) for all \(S\subseteq I\) with \(\delta \in {\mathcal {D}}_S\) or there is an \(S\subseteq I\) with \(\delta \in {\mathcal {D}}_S\) such that \(\delta \) (associated to \(S\)) is of type \(\tilde{z}_{R_{\delta }}^c\) (in the expression for \(y_I\)).
- i.\(\delta \) is of type \((\tilde{z}_{R_{\delta }}^c-\tilde{z}^c_{[n]})\) for all \(S\subseteq I\) with \(\delta \in {\mathcal {D}}_S\), see Fig. 13. As mentioned, we have \(R_\delta \supset R_\delta \cap S\) for all sets \(S\subseteq I\) with \(\delta \in \mathcal S\). Moreover, these sets are in bijection to the subsets of \(\bigl ((a,\tilde{a})\cap I\bigr )\cup \bigl ((\tilde{\alpha },b)\cap I\bigr )\):If there is more than one set \(S\subseteq I\) with \(\delta \in {\mathcal {D}}_S\), then collecting all summands \((\tilde{z}_{R_{\delta }}^c-\tilde{z}^c_{[n]})\) in the expression for \(y_I\) yields a vanishing alternating sum. If there is only one set \(S\subseteq I\) with \(\delta \in {\mathcal {D}}_S\) as associated diagonal then \(\bigl ((a,\tilde{a})\cap I\bigr )\cup \bigl ((\tilde{\alpha },b)\cap I\bigr )= \varnothing \) and it follows that \(\Gamma =\tilde{\alpha }\in \mathsf U _c\) and \(\tilde{a}\in \{a,\gamma \}\cap \mathsf D _c\). For later use in this proof, we note that \(\gamma \in \mathsf D _c\) implies \(\delta \in \{\delta _2,\delta _4\}\). The only possible contributions of \(\delta \) in the expression for \(y_I\) are therefore$$\begin{aligned} S=\bigl (R_\delta \cap (I{\setminus } B)\bigr )\cup A\qquad \text {for}\, A\subseteq (a,\tilde{a})\cap I\,\text { and }B \subseteq (\tilde{\alpha },b)\cap I. \end{aligned}$$But since the corresponding subsets \(R_{\delta _2}\cap I\) and \(R_{\delta _4}\cap I\) differ by \(\gamma \), the contribution to \(y_I\) can be simplified to$$\begin{aligned} (-1)^{|I{\setminus } R_{\delta _2}|}(\tilde{z}_{R_{\delta _2}}^c -\tilde{z}^c_{[n]}) \qquad \text {and}\qquad (-1)^{|I{\setminus } R_{\delta _4}|}(\tilde{z}_{R_{\delta _4}}^c -\tilde{z}^c_{[n]}). \end{aligned}$$If \(\gamma \in \mathsf U _c\), then \(\delta =\delta _2\) and we obtain$$\begin{aligned} (-1)^{|I{\setminus } R_{\delta _2}|}\tilde{z}_{R_{\delta _2}}^c + (-1)^{|I{\setminus } R_{\delta _4}|}\tilde{z}_{R_{\delta _4}}^c. \end{aligned}$$as contribution for \(y_I\).$$\begin{aligned} (-1)^{|I{\setminus } R_{\delta _2}|}(\tilde{z}_{R_{\delta _2}}^c-\tilde{z}^c_{[n]}) \end{aligned}$$
- ii.There is an \(S\subseteq I\) with \(\delta \in {\mathcal {D}}_S\) such that \(\delta \) is of type \(\tilde{z}_{R_{\delta }}^c\), see Fig. 14. Since \(\delta \) must be the ‘rightmost’ diagonal associated to \(S\) if \(\delta \) (associated to \(S\)) is of type \(\tilde{z}_{R_{\delta }}^c\) (in the expression for \(y_I\)), we conclude \(R_\delta =R_\delta \cap I\). In particular, we have \(\Gamma =n\) and \(b=n+1\) and thus \((\tilde{\alpha },b)\cap I\ne \varnothing \) and \((\tilde{\alpha },b)\cap I=(\tilde{\alpha },b)\). If \((a,\tilde{a})\cap I\ne \varnothing \), then collecting terms for \(\tilde{z}_{R_{\delta }}^c\) and \(\tilde{z}^c_{[n]}\) in the expression for \(y_I\) again yields no contribution. We may therefore assume \((a,\tilde{a})\cap I= \varnothing \), that is \(\tilde{a}\in \{a,\gamma \}\cap \mathsf D _c\). First suppose that \(\gamma \in \mathsf D _c\). Then \(\delta \) is either \(\delta _a=\{a,\tilde{\alpha }\}\) or \(\delta _\gamma =\{\gamma ,\tilde{\alpha }\}\). Now \(\delta \) is of type \(\tilde{z}_{R_{\delta }}^c\) in the expression of \(y_I\) if and and only if \(\delta \) is associated to \(R_{\delta _a}\) or \(R_{\delta _\gamma }\). In all other situations, \(\delta \) is of type \((\tilde{z}_{R_{\delta }}^c-\tilde{z}^c_{[n]})\) in the expression of \(y_I\) and is associated to a set \(R_{\delta _a}{\setminus } M\) or \(R_{\delta _\gamma }{\setminus } M\) with non-empty \(M\subseteq (\tilde{\alpha },n+1)\). Collecting terms for \(\tilde{z}_{R_{\delta _a}}^c,\,\tilde{z}_{R_{\delta _\gamma }}^c\), and \(\tilde{z}_{[n]}^c\) yields a vanishing contribution as desired (collecting the terms for \(\tilde{z}_{[n]}^c\) for fixed \(\delta \) does not yield a vanishing contribution, but the terms from \(\delta _a\) and \(\delta _\gamma \) cancel). If \(\gamma \in \mathsf U _c\) then a similar argument givesas contribution for \(y_I\).$$\begin{aligned} (-1)^{|I{\setminus } R_\delta |}\tilde{z}_{[n]}^c \qquad \text {for}\,\delta =\{a,\tilde{\alpha }\}\,\text {with}\,\tilde{\alpha }\in \mathsf U _c\,\text {and}\,R_\delta =R_\delta \cap I \end{aligned}$$

- i.

- a.
- 3.\(R_\delta \) has up and down decomposition of type \((1,2)\). If \(R_\delta \) is of type \((1,2)\) then \(\delta =\{\alpha ,\beta \}\) with \(\alpha ,\beta \in \mathsf U _c\) and there is \(u\in \mathsf U _c\) such that \(a<\alpha <u<\beta < b\). This in turn givesas up and down interval decomposition for \(R_\delta \). By arguments as before, we conclude that collecting terms for \(\tilde{z}_{R_\delta }^c\) and \(\tilde{z}^c_{[n]}\) yields a vanishing contribution to \(y_I\) if \((a,\alpha )\cap I\ne \varnothing \). We therefore assume that \((a,\alpha )\cap I= \varnothing \) which is equivalent to \(\gamma =\alpha \in \mathsf U _c\). As a consequence, \(\delta \) is an associated diagonal of \(S\subseteq I\) if and only if \(S=(R_\delta \cap I){\setminus } M\) for some \(M\subseteq (\beta ,b)\cap I\). We now distinguish two cases: either there is an \(S\subseteq I\) with \(\delta \in {\mathcal {D}}_S\) such that \(\delta \) (associated to \(S\)) is of type \(\tilde{z}_{R_{\delta }}^c\) (in the expression for \(y_I\)) or not.$$\begin{aligned} R_{\delta }=(0, n+1)_\mathsf{D _c}\cup [u_1,\alpha ]_\mathsf{U _c}\cup [\beta ,u_m]_\mathsf{U _c} \end{aligned}$$
- a.
There is no \(S\subseteq I\) with \(\delta \in {\mathcal {D}}_S\) such that \(\delta \) (associated to \(J\)) is of type \(\tilde{z}_{R_{\delta }}^c\), see Fig. 15. If \((\beta ,b)\cap I\ne \varnothing \) then collecting the terms \(\tilde{z}_{R_\delta }^c\) and \(\tilde{z}^c_{[n]}\) cancel respectively. If \((\beta ,b)\cap I= \varnothing \) then we have \(\Gamma =\beta \in \mathsf U _c\) and \(\delta =\delta _4\). In this situation, \(\delta \) has a unique contribution to \(y_I\) which equals \((-1)^{|I{\setminus } R_{\delta _4}|}(\tilde{z}_{R_{\delta _4}}^c - \tilde{z}^c_{[n]})\).

- b.
There is a set \(S\subseteq I\) with \(\delta \in {\mathcal {D}}_S\) such that \(\delta \) is of type \(\tilde{z}_{R_{\delta }}^c\), see Fig. 16. Since \(\delta \) is the ‘rightmost’ diagonal associated to \(S\subseteq I\) and since \((a,\alpha ){\cap } I {=}\varnothing \), we conclude that \(b\!=\!n\!+\!1\) and \(\Gamma =n\in \mathsf D _c\) (recall that we also have \(\alpha \!=\!\gamma \in \mathsf U _c\)). Now observe that the set \(S\subseteq I\) with \(\delta \in {\mathcal {D}}_S\) such that \(\delta \) (associated to \(S\)) is of type \(\tilde{z}_{R_{\delta }}^c\) (in the expression for \(y_I\)) is unique: it is \(R_\delta \cap I\). In particular, we have \([\beta ,n]\cap I=[\beta ,n]\). Collecting terms \(\tilde{z}_{R_\delta }^c\) for all subsets \(S\subseteq I\) with \(\delta \in {\mathcal {D}}_S\) cancel, but collecting the terms \(\tilde{z}^c_{[n]}\) does not vanish: we have a contribution of \((-1)^{|I{\setminus } R_\delta |}\tilde{z}_{[n]}^c\) to \(y_I\). We conclude that every diagonal \(\delta =\{\gamma ,\beta \}\) with \(\beta \in \mathsf U _c,\,[\beta ,n]\cap I=[\beta ,n]\) and \(\{ \gamma ,\beta \} \ne \{ u_r,u_{r+1}\}\) contributes \((-1)^{|I{\setminus } R_\delta |}\tilde{z}_{[n]}^c\) to \(y_I\).

- a.

- 1.
\(\gamma , \Gamma \in \mathsf D _c\). Then \(\delta _1,\,\delta _2,\,\delta _3\), and \(\delta _4\) contribute \((-1)^{|I{\setminus } R_\delta |}\tilde{z}_{R_{\delta }}^c\) to \(y_I\) according to Case 1 and no other diagonal contributes according to the previous analysis. The claim follows immediately.

- 2.
\(\gamma \in \mathsf D _c\) and \(\Gamma \in \mathsf U _c\). Then \(\delta _1\) and \(\delta _3\) contribute \((-1)^{|I{\setminus } R_\delta |}\tilde{z}_{R_{\delta }}^c\) to \(y_I\) according to Case 1, while \(\delta _2\) and \(\delta _4\) contribute \((-1)^{|I{\setminus } R_\delta |}\tilde{z}_{R_{\delta }}^c\) to \(y_I\) according to Case 2(b)i. No other diagonal contributes to \(y_I\). The claim follows immediately.

- 3.\(\gamma ,\Gamma \in \mathsf U _c\). The only diagonals with a contribution to \(y_I\) are \(\delta _1\) (Case 1), \(\delta _2\) (Case 2(b)i), \(\delta _3\) (Case 2a) and \(\delta _4\) (Case 3a). Taking their contribution into account, we obtainThe claim follows since \(I{\setminus } R_{\delta _2}\) and \(I{\setminus } R_{\delta _4}\) differ by \(\gamma \).$$\begin{aligned} y_I&= (-1)^{|I{\setminus } R_{\delta _1}|}\tilde{z}_{R_{\delta _1}}^c + (-1)^{|I{\setminus } R_{\delta _2}|}(\tilde{z}_{R_{\delta _2}}^c-\tilde{z}_{[n]}^c) + (-1)^{|I{\setminus } R_{\delta _3}|}\tilde{z}_{r_{\delta _3}}^c\\&+ (-1)^{|I{\setminus } R_{\delta _4}|}(\tilde{z}_{R_{\delta _4}}^c-\tilde{z}_{[n]}^c). \end{aligned}$$
- 4.\(\gamma \in \mathsf U _c\) and \(\Gamma \in \mathsf D _c\). We distinguish the two sub-cases \(\Gamma \ne n\) and \(\Gamma =n\).
- (a)
\(\Gamma \ne n\) implies that there is no \(u\in \mathsf U _c\) such that \([u,n]=[u,n]\cap I\). In this situation, \(\delta _1\) and \(\delta _3\) contribute \((-1)^{|I{\setminus } R_\delta |}\tilde{z}_{R_{\delta }}^c\) to \(y_I\) according to Case 1 and \(\delta _2\) and \(\delta _4\) contribute \((-1)^{|I{\setminus } R_\delta |}\tilde{z}_{R_{\delta }}^c\) according to Case 2a. No other diagonal contributes, so the claim follows immediately.

- (b)\(\Gamma =n\). If there is no \(u\in \mathsf U _c\) such that \([u,n]=[u,n]\cap I\) then \(\delta _1\) and \(\delta _2\) contribute according to Case 1 and \(\delta _3\) and \(\delta _4\) contribute according to Case 2a. No other diagonal contributes, so the claim follows immediately. If there exists \(u\in \mathsf U _c\) such that \([u,n]=[u,n]\cap I\) then denote by \(u_\mathrm{{min}}\) the smallest element of \(\mathsf U _c\) such that \([u_\mathrm{{min}},n]=[u_{min},n]\cap I\). Now diagonals \(\delta _1\) and \(\delta _2\) contribute to \(y_I\) according to Case 1 and diagonals \(\delta _3,\,\delta _4\) according to Case 2a. But in this situation, according to Cases 2(b)ii and 3b, we also have contributions of diagonals \(\{a,u\}\) and \(\{\gamma ,u\}\) for \(u\in [u_{min},u_m]_\mathsf{U _c}\). This yields But the second and third sum cancel, so we end up with the claim. \(\square \)

- (a)

In fact, the methods used in the proof of Proposition 6.2 suffice to prove the degenerate cases \({\fancyscript{D}}_I\ne \{\delta _1,\delta _2,\delta _3,\delta _4\}\) as well. But before we try to analyse these cases, we remark that some subsets of \(\{ \delta _1,\delta _2,\delta _3,\delta _4\}\) never form a set \({\fancyscript{D}}_I\) associated to \(I\subseteq [n]\) and \([n]=\mathsf{D _c} \sqcup \mathsf{U _c}\).

**Lemma 6.3**

- (a)There is no partition \([n]=\mathsf D _c\sqcup \mathsf U _c\) induced by a Coxeter element \(c\) and no non-empty \(I\subset [n]\) such that \({\fancyscript{D}}_I\) is one of the following sets:$$\begin{aligned} \varnothing ,\quad \{ \delta _2\}, \quad \{\delta _3\},\quad \{\delta _4\},\quad \{\delta _1,\delta _2\},\quad \{\delta _1,\delta _3\},\quad \{\delta _2,\delta _4\},\quad \text {or}\quad \{\delta _3,\delta _4\}. \end{aligned}$$
- (b)There is a partition \([n]=\mathsf D _c\sqcup \mathsf U _c\) induced by a Coxeter element \(c\) and a non-empty \(I\subset [n]\) such that \({\fancyscript{D}}_I\) is one of the following sets:$$\begin{aligned}&\{\delta _1\},\,\{ \delta _1,\delta _4\},\,\{\delta _2,\delta _3\},\, \{\delta _1,\delta _2,\delta _3\},\,\{\delta _1,\delta _2,\delta _4\},\, \{\delta _1,\delta _3,\delta _4\},\,\{\delta _2,\delta _3,\delta _4\},\\&\quad \text { or } \{\delta _1,\delta _2,\delta _3,\delta _4\}. \end{aligned}$$

The proof of Part (a) is left to the reader, while the situation of Part (b) is carefully discussed in Sect. 7.

**Lemma 6.4**

- (a)Suppose that \(I\) satisfies one of the following conditionsThen the Minkowski coefficient \(y_I\) of \(\mathsf As ^c_{n-1}\) is
- (i)
\({\fancyscript{D}}_I=\{ \delta _1\}\) (Lemma 7.1),

- (ii)
\({\fancyscript{D}}_I=\{ \delta _1,\delta _3,\delta _4\},\,(a,b)_\mathsf{D}=\{\Gamma \}\), and \(\gamma \in \mathsf U _c\) (Lemma 7.6 (b)

*and*(c)), - (iii)
\({\fancyscript{D}}_I=\{ \delta _1,\delta _2,\delta _4\},\,(a,b)_\mathsf{{D} }=\{\gamma \}\),

*and*\(\Gamma \in \mathsf U _c\) (*Lemma*7.5 (b)*and*(c)), - (iv)
\({\fancyscript{D}}_I=\{ \delta _1,\delta _2,\delta _3\}\)

*and*\((a,b)_\mathsf{{D} }=\{\gamma ,\Gamma \}\) (Lemma 7.4 (a)) - (v)
\({\fancyscript{D}}_I=\{ \delta _2,\delta _3,\delta _4\}\)

*and*\((a,b)_\mathsf{{D} }=\varnothing (\)*Lemma*7.7).

$$\begin{aligned} y_I = \sum _{\delta \in {\fancyscript{D}}_I}(-1)^{|I{\setminus } R_{\delta }|}z_{R_{\delta }}. \end{aligned}$$ - (i)
- (b)Suppose that \(I\) satisfies one of the following conditionsThen the Minkowski coefficient \(y_I\) of \(\mathsf As ^c_{n-1}\) is
- (i)
\({\fancyscript{D}}_I=\{\delta _1,\delta _4\}\) (

*Lemma*7.2), - (ii)
\({\fancyscript{D}}_I=\{\delta _2,\delta _3\}\) (

*Lemma*7.3), - (iii)
\({\fancyscript{D}}_I=\{ \delta _1,\delta _3,\delta _4\}\)

*and*\(\bigcup _{i=1}^k [\alpha _i,\beta _i]_\mathsf{U _c}=\{\Gamma \}\) (*Lemma*7.6 (a)), - (iv)
\({\fancyscript{D}}_I=\{ \delta _1,\delta _2,\delta _4\}\)

*and*\(\bigcup _{i=1}^k [\alpha _i,\beta _i]_\mathsf{U _c}=\{\gamma \}\) (Lemma 7.5 (a)), - (v)
\({\fancyscript{D}}_I=\{ \delta _1,\delta _2,\delta _3\}\)

*and*\(\bigcup _{i=1}^k [\alpha _i,\beta _i]_\mathsf{U _c}=\{\gamma ,\Gamma \}\) (*Lemma*7.4 (b)).

$$\begin{aligned} y_I = (-1)^{|\{\gamma ,\Gamma \}|}z_{[n]} + \sum _{\delta \in {\fancyscript{D}}_I}(-1)^{|I{\setminus } R_{\delta }|}z_{R_{\delta }}. \end{aligned}$$ - (i)

*Proof*

The proof of the claim is a study of the 14 mentioned cases that characterise the non-empty proper subsets \(I \subset [n]\) with \({\fancyscript{D}}_I\ne \{\delta _1,\delta _2,\delta _3,\delta _4\}\). These 14 cases are described in detail in Sect. 7, the proofs are along the lines of the proof of Proposition 6.2.\(\square \)

**Lemma 6.5**

- (a)In all cases of Lemma 6.4 (a) we have \(R_\delta =\varnothing \) if \(\delta \in \{ \delta _1,\delta _2,\delta _3,\delta _4\}{\setminus } {\fancyscript{D}}_I\). Thus$$\begin{aligned} y_I = \sum _{i=1}^4(-1)^{|I{\setminus } R_{\delta _i}|}z_{R_{\delta _i}}. \end{aligned}$$
- (b)In all cases of Lemma 6.4 (b) there is precisely one \(\delta \in \{ \delta _1,\delta _2,\delta _3,\delta _4\}{\setminus } {\fancyscript{D}}_I\) with \(R_{\delta }\ne \varnothing \):Moreover, we have \(\gamma \ne \Gamma \) except for Lemma 7.3 (a) where \(\gamma =\Gamma \in \mathsf U _c\).
- (i)\(R_{\delta _2}=[n]\) (Lemma 7.2 (a) and Lemma 7.6 (a)) and we have$$\begin{aligned} y_I = (-1)^{|I{\setminus } R_{\delta _2}|}z_{R_{\delta _2}} + \sum _{\delta \in {\fancyscript{D}}_I}(-1)^{|I{\setminus } R_{\delta }|}z_{R_{\delta }} = \sum _{i=1}^4(-1)^{|I{\setminus } R_{\delta _i}|}z_{R_{\delta _i}}. \end{aligned}$$
- (ii)\(R_{\delta _3}=[n]\) (Lemma 7.2 (b) and Lemma 7.5 (a)) and we have$$\begin{aligned} y_I = (-1)^{|I{\setminus } R_{\delta _3}|}z_{R_{\delta _3}} + \sum _{\delta \in {\fancyscript{D}}_I}(-1)^{|I{\setminus } R_{\delta }|}z_{R_{\delta }} = \sum _{i=1}^4(-1)^{|I{\setminus } R_{\delta _i}|}z_{R_{\delta _i}}. \end{aligned}$$
- (iii)\(R_{\delta _4}=[n]\) (Lemma 7.3 and Lemma 7.4 (b)) and we have$$\begin{aligned} y_I = (-1)^{|I{\setminus } R_{\delta _4}|+|\{\gamma ,\Gamma \}|}z_{R_{\delta _4}} + \sum _{\delta \in {\fancyscript{D}}_I}(-1)^{|I{\setminus } R_{\delta }|}z_{R_{\delta }}. \end{aligned}$$

- (i)

*Proof*

**Lemma 6.6**

*Proof*

We combine Proposition 6.2, Lemmas 6.5 and 6.6 to obtain Theorem 4.2 if \(I\subset [n]\) has an up and down interval decomposition of type \((1,w)\):

**Corollary 6.7**

The techniques to prove Proposition 6.2 also enable us to compute the Minkowski coefficient \(y_I\) of \(\mathsf As ^c_{n-1}\) if the up and down interval decomposition of \(I\) is of type \((v,w),\,v>1\), and \(I\ne [n]\).

**Lemma 6.8**

Let \(I\) be a non-empty proper subset of \([n]\) with up and down interval decomposition of type \((v,w)\) with \(v>1\). Then \(y_I=0\) for the Minkowski coefficient of \(\mathsf As ^c_{n-1}\).

*Proof*

For every proper diagonal \(\delta =\{d_1,d_2\}\) with \(d_1<d_2\) that appears in the expression for \(y_I\), there is a nested component \(N=(a_i,b_i)_\mathsf{{D} }\sqcup \bigsqcup _{j=1}^{w_i}[\alpha _{i,j},\beta _{i,j}]_\mathsf{{U} }\) of \(I\) such that \(a_i\le d_1<d_2\le b_i\). Now \(\delta \) appears in the expression for \(y_I\) for every set \(S\) where \(R_\delta \cap N \subseteq S \subseteq I\). Since \(v>1\), the diagonal \(\delta \) never contributes to \(y_I\).\(\square \)

We now analyse the remaining case \(I=[n]\) and consider \((0,n+1)_{\mathsf{{D} }}\sqcup [u_1,u_m]_{\mathsf{{U} }}\) as up and down interval decomposition of \(I\).

**Lemma 6.9**

*Proof*

- (1)\(\mathsf U _c\ne \varnothing \) and \(\mathsf D _c\ne \{ 1,n\}\). Then \({\fancyscript{D}}_{[n]}=\{\delta _1, \delta _2, \delta _3, \delta _4\}\) and each diagonal of \({\fancyscript{D}}_{[n]}\) contributes to \(y_{[n]}\) as well as all proper diagonals \(\{0,u\}\) and \(\{1,u\}\) with \(u\in \mathsf U _c\) since \(a=0\) and \(\gamma =1\). Hence we have for \(\sum _{J\subset [n]}(-1)^{|[n]{\setminus } J|}z_J\). Since \(\{0,u_1\}\) is not a proper diagonal, the second and third sum do not cancel and the term \((-1)^{|[n]{\setminus } R_{\{1,u_1\}}|}z_{[n]}^c\) remains. Now \(|[n]{\setminus } R_{\{1,u_1\}}|=1\) andimply the claim.$$\begin{aligned} \sum _{\delta \in {\fancyscript{D}}_{[n]}}(-1)^{|[n]{\setminus } R_{\delta }|} z_{R_\delta }^c = (-1)^{|[n]{\setminus } R_{\delta _1}|} \big ( z_{R_{\delta _1}}^c-z_{R_{\delta _2}}^c-z_{R_{\delta _3}}^c+z_{R_{\delta _4}}^c \big ) \end{aligned}$$
- (2)\(\mathsf U _c=\varnothing \) and \(\mathsf D _c\ne \{ 1,n\}\). Then \({\fancyscript{D}}_{[n]}=\{\delta _2, \delta _3, \delta _4\}\) and we haveThe claim follows now from \(R_{\delta _1} = [n]\) and Lemma 6.6.$$\begin{aligned} \sum _{J\subset [n]}(-1)^{|[n]{\setminus } J|}z_J=\sum _{\delta \in {\fancyscript{D}}_{[n]}}(-1)^{|[n]{\setminus } R_{\delta }|} z_{R_\delta }^c. \end{aligned}$$
- (3)\(\mathsf U _c\ne \varnothing \) and \(\mathsf D _c= \{ 1,n\}\). We have \({\fancyscript{D}}_{[n]}=\{\delta _1, \delta _2, \delta _3\}\). Now each diagonal of \({\fancyscript{D}}_{[n]}\) and all proper diagonals \(\delta _{0,u}=\) \(\{0,u\}\) and \(\delta _{1,u}=\) \(\{1,u\}\) with \(u\in \mathsf U _c\) contribute to \(y_I\) since \(a=0\) and \(\gamma =1\). Similar to the first case, a term \((-1)^{|[n]{\setminus } R_{\{1,u_1\}}|}z_{[n]}^c\) is not canceled and we obtainSince \(z_{R_{\delta _4}}=z_\varnothing =0\), the claim follows.$$\begin{aligned} y_{[n]} = (-1)^{|[n]{\setminus } R_{\delta _1}|} \big (z_{R_{\delta _1}}^c-z_{R_{\delta _2}}^c-z_{R_{\delta _3}}^c\big ). \end{aligned}$$
- (4)\(\mathsf U _c=\varnothing \) and \(\mathsf D _c= \{ 1,n\}\). We have \({\fancyscript{D}}_{[n]}=\{\delta _2, \delta _3\},\,R_{\delta _1}=[n]\) and \(R_{\delta _4}=\varnothing \). Hence\(\square \)$$\begin{aligned} y_{[n]} = z_{[n]} + \sum _{\delta \in {\fancyscript{D}}_{[n]}}(-1)^{|[n]{\setminus } R_{\delta }|} z_{R_\delta }^c = (-1)^{|[n]{\setminus } R_{\delta _1}} \big ( z_{R_{\delta _1}}^c-z_{R_{\delta _2}}^c-z_{R_{\delta _3}}^c+z_{R_{\delta _4}}^c \big ). \end{aligned}$$

## 7 Characterisation of \({\fancyscript{D}}_I \ne \)\(\{\delta _1,\delta _2,\delta _3,\delta _4\}\) for \(I\subset [n]\)

As stated in Lemma 6.3, not all 15 proper subsets of \(\{\delta _1,\delta _2,\delta _3,\delta _4\}\) appear as set of proper diagonals \({\fancyscript{D}}_I\) for \(I\subset [n]\) with up and down decomposition of type \((1,w)\) and some Coxeter element \(c\). The proof that a subset does not appear is not difficult, for example, we can show that if \({\fancyscript{D}}_I\) contains certain diagonal(s) then \({\fancyscript{D}}_I\) is forced to contain certain others. In this section we discuss Lemma 6.3(b) in detail and study the sets \({\fancyscript{D}}_I\) with \(|{\fancyscript{D}}_I|<4\). The seven proper subset of \(\{\delta _1,\delta _2,\delta _3,\delta _4\}\) that are possible are characterised in Lemmas 7.1–7.7. We identified 14 conditions that characterise these seven subsets.

**Lemma 7.1**

If \({\fancyscript{D}}_I=\{ \delta _1\}\), then \(I=\{d_r\}\) with \(1\le r\le \ell \).

*Proof*

\(\delta _1\in {\fancyscript{D}}_I\) implies \((a,b)_\mathsf{D _c}\ne \varnothing \) and \(\delta _2,\delta _3,\delta _4\not \in {\fancyscript{D}}_I\) imply \(\gamma =\Gamma \in \mathsf D _c\). \(\square \)

**Lemma 7.2**

- (a)
\(I=\{d_1,u_1\}\) and \(u_1<d_2\), or

- (b)
\(I=\{u_m,d_{\ell }\}\) and \(d_{\ell -1}<u_m\).

*Proof*

\(\delta _2,\delta _3\not \in {\fancyscript{D}}_I\) imply that \(\{a,\Gamma \}\) and \(\{\gamma ,b\}\) are (non-degenerate) edges of \(Q_{c}\). In particular, neither \(\gamma ,\Gamma \in \mathsf D _c\) nor \(\gamma ,\Gamma \in \mathsf U _c\) is possible.

Firstly, suppose \(\gamma \in \mathsf D _c\) and \(\Gamma \in \mathsf U _c\). Then \(\delta _2\not \in {\fancyscript{D}}_I\) implies \(a=0,\,\Gamma =u_1\), and \(\gamma =d_1=1\). Now \(\delta _3\not \in {\fancyscript{D}}_I\) yields \(b=d_2\) and \(\Gamma =u_1\) requires \(u_1<d_2\) and we have shown (a).

Secondly, suppose \(\gamma \in \mathsf U _c\) and \(\Gamma \in \mathsf D _c\). Then \(\delta _3\not \in {\fancyscript{D}}_I\) implies \(b=n+1,\,\gamma =u_m\), and \(\Gamma =d_\ell =n\). Now \(\delta _2\not \in {\fancyscript{D}}_I\) yields \(a=d_{\ell -1}\) and \(\gamma =u_m\) requires \(d_{\ell -1}<u_m\). This gives (b). \(\square \)

**Lemma 7.3**

- (a)
\(I=\{u_s\}\) with \(1\le s\le m\), or

- (b)
\(I=\{u_s,u_{s+1}\}\) with \(1\le s<m\).

*Proof*

**Lemma 7.4**

- (a)
\(I=\{d_r,d_{r+1}\}\sqcup M\) with \(1\le r< \ell \) and \(M\subseteq [d_r,d_{r+1}]\cap \mathsf U _c\) or

- (b)
\(I= M \sqcup \{u_s,u_{s+1}\}\) with \(1\le s < m\) and \(M=[u_s,u_{s+1}]\cap \mathsf D _c \ne \varnothing \).

*Proof*

\(\delta _1\in {\fancyscript{D}}_I\) implies \((a,b)_{\mathsf{{D} }_c}\ne \varnothing \), while \(\delta _4\not \in {\fancyscript{D}}_I\) implies that \(\{\gamma ,\Gamma \}\) is either an edge of \(Q_{c}\) or \(\gamma =\Gamma \). Suppose first \(\gamma =\Gamma \). Then \(\gamma =\Gamma \in \mathsf D _c\) implies the contradiction \({\fancyscript{D}}_I=\{\delta _1\}\), while \(\gamma =\Gamma \in \mathsf U _c\) implies \((a,b)_{\mathsf{{D} }_c}=\varnothing \), contradicting \(\delta _1\in {\fancyscript{D}}_I\). We therefore assume \(\gamma \ne \Gamma \) and only have to distinguish the cases \(\gamma ,\Gamma \in \mathsf D _c\) and \(\gamma ,\Gamma \in \mathsf U _c\), the other cases \(\gamma \in \mathsf D _c,\,\Gamma \in \mathsf U _c\) and \(\gamma \in \mathsf U _c\) and \(\Gamma \in \mathsf D _c\) are not possible since \(\delta _4\not \in {\fancyscript{D}}_I\).

Lemma 7.5 is symmetric to Lemma 7.6, the proofs are along the same lines.

**Lemma 7.5**

- (a)
\(I=\{d_{r+1}, \ldots , d_\ell \}\cup \{u_m\}\) with \(d_r<u_m<d_{r+1}<d_\ell \), or

- (b)
\(I=\{d_r\}\cup M\) with \(1<r<\ell \) and \(\varnothing \ne M\subseteq [d_r,d_{r+1}]\cap \mathsf U _c\) or

- (c)
\(I=\{d_1\}\cup M\) with \(M\subseteq [d_1,d_2]\cap \mathsf U _c\) and \(M{\setminus }\{u_1\} \ne \varnothing \).

*Proof*

Since \(\delta _1\in {\fancyscript{D}}_I\), we have \((a,b)_{\mathsf{{D} }_c}\ne \varnothing \), that is, \(a,b\) are not consecutive numbers in \(\mathsf D _c\). From \(\delta _3\not \in {\fancyscript{D}}_I\), we deduce that \(\{\gamma ,b\}\) is an edge of \(Q_{c}\) and \(\gamma ,\Gamma \in \mathsf U _c\) is therefore impossible unless \(\gamma =\Gamma \). Moreover, \(\delta _4\in {\fancyscript{D}}_I\) implies that \(\gamma =\Gamma \) is impossible. We now have two cases to distinguish.

Firstly, suppose \(\gamma =u_m\) and \(b=n+1\). Then \(\Gamma =d_\ell =n\) and \(\delta _2\in {\fancyscript{D}}_I\) implies \((a,\Gamma )_{\mathsf{{D} }_c}\ne \varnothing \). Together with \(a=\max \{d\in \mathsf D _c\vert d<u_m\}\) we have \(a=d_r\) for some \(1\le r\le \ell -2\) with \(u_m<d_{r+1}\) and \(I=(d_r,n+1)_{\mathsf{{D} }_c}\cup [u_m,u_m]_\mathsf{U _c}\), this shows (a).

Secondly, we suppose \(\gamma =d_r\) and \(b=d_{r+1}\) for some \(1\le r\le \ell -1\) and \(\Gamma \in (\gamma ,b)\cap \mathsf{{U} }_c\). If \(\gamma =1\) then \(\delta _2\in {\fancyscript{D}}_I\) implies \(\Gamma \ne u_1\), so we distinguish the cases \(\gamma \,{=}\,1\) and \(\gamma \,{\ne }\, 1\). Suppose first that \(\gamma {=}d_r\) with \(r\,{>}\,1\). If \([d_r,d_{r{+}1}]\,{\cap }\, \mathsf U _c{\ne } \varnothing \) then we immediately have the claim for every non-empty \(M\subseteq [d_r,d_{r+1}]\cap \mathsf U _c\). If \([d_r,d_{r+1}]\cap \mathsf U _c= \varnothing \) then \(\gamma =\Gamma \in \mathsf D _c\) which is impossible. Thus we have shown (b). Suppose now that \(\gamma =d_1=1\). Then \(a=0,\,b=d_2\), and \(\delta _2\in {\fancyscript{D}}_I\) implies \(\Gamma \in \mathsf U _c{\setminus } \{u_1\}\). This proves (c). \(\square \)

**Lemma 7.6**

- (a)
\(I=\{d_1,\ldots ,d_{r-1}\}\cup \{u_1\}\) with \(d_1<d_{r-1}<u_1<d_r\), or

- (b)
\(I=\{d_r\}\cup M\) with \(1 < r < \ell \) and \(\varnothing \ne M\subseteq [d_{r-1},d_r]\cap \mathsf U _c\) or

- (c)
\(I=\{d_\ell \}\cup M\) with \(M\subseteq [d_{\ell -1},d_\ell ]\cap \mathsf U _c\) and \(M{\setminus }\{u_m\}\ne \varnothing \).

**Lemma 7.7**

*Proof*

## Footnotes

- 1.
A proper diagonal is a line segment connecting a pair of vertices of \(Q\) whose relative interior is contained in the interior of \(Q\). A non-proper diagonal is a diagonal that connects vertices adjacent in \(\partial Q\) and a degenerate diagonal is a diagonal where the end-points are equal.

## Notes

### Acknowledgments

The author was partially supported by DFG Forschergruppe 565 *Polyhedral Surfaces*. Extended abstracts of preliminary versions were presented at FPSAC 2011 and CCCG 2011. I thank various anonymous referees for their helpful comments, in particular, one referee for *Discrete and Computational Geometry.*

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