Bipermutahedron and biassociahedron
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Abstract
We give a simple description of the face poset of a version of the biassociahedra that generalizes, in a straightforward manner, the description of the faces of the Stasheff’s associahedra via planar trees. We believe that our description will substantially simplify the notation of Saneblidze and Umble (Homology Homotopy Appl 13(1):1–57, 2011) making it, as well as the related papers, more accessible.
Keywords
Permutahedron Associahedron Tree Operad PROP Zone DiaphragmMathematics Subject Classification (2000)
16W30 57T05 18C10 18G991 History and pitfalls
In this introductory section, we recall the history and indicate the pitfalls of the ‘quest for the biassociahedron,’ hoping to elucidate the rôle of the present paper in this struggle.
1.1 History
Let us start by reviewing the precursor of the biassociahedron. Stasheff in his seminal paper [11] introduced \(A_\infty \)spaces (resp. \(A_\infty \)algebras, called also strongly homotopy or sh associative algebras) as spaces (resp. algebras) with a multiplication associative up to a coherent system of homotopies. The central object of his approach was a cellular operad \(K = \{K_m\}_{m \ge 2}\) whose \(m\)th piece \(K_m\) was a convex \((m2)\)dimensional polytope called the Stasheff associahedron. \(A_\infty \)space was then defined as a topological space on which the operad \(K\) acted, while \(A_\infty \)algebras were algebras over the operad \(C_*(K)\) of cellular chains on \(K\). Let us briefly recall the basic features of the construction of [11], emphasizing the algebraic side. More details can be found for instance in [9, II.1.6] or in the original source [11].
Next, we require the homotopy for the associativity to be coherent, by which we mean that the pentagon \(K_4\) can be ‘filled’ with a higher homotopy \(\mu _4 : {V}^{\otimes 4} \rightarrow V\) whose differential equals the sum (with appropriate signs) of the homotopies labelling the edges. This process can be continued, giving rise to a sequence \(K = \{K_m\}_{m \ge 2}\) of the Stasheff associahedra [11], discovered independently by Tamari [13]. It turns out that \(K\) is a polyhedral operad. An \(A_\infty \)algebra is then an algebra over the operad \(C_*(K)\) of cellular chains on \(K\).
Much later there appeared another, purely algebraic, way to introduce \(A_\infty \)algebras. As proved in [5], the operad \(\mathcal{A }{ ss}\) for associative algebras admits a unique, up to isomorphism, minimal cofibrant model \(\mathcal{A }_\infty \) which turns out to be isomorphic to the operad \(C_*(K)\). We may thus as well say that \(A_\infty \)algebras are algebras over the minimal model of \(\mathcal{A }{ ss}\). Finally, one can describe \(A_\infty \)algebras explicitly, as a structure with operations \(\mu _m : {V}^{\otimes m} \rightarrow V\), \(m \ge 2\), satisfying a very explicit infinite set of axioms, see [11, page 294]. In the case of \(A_\infty \)algebras thus topology, represented by the associahedron, preceded algebra.
There were similar attempts to find a suitable notion of \(A_\infty \)bialgebras,^{1} that is, structures whose multiplication and comultiplication are compatible and (co)associative up to a system of coherent homotopies. The motivation for such a quest was, besides the restless nature of human mind, homotopy invariance and the related transfer properties which these structures should possess. For instance, given a (strict) bialgebra \(H\), each dgvector space quasiisomorphic to the underlying dgvector space of \(H\) ought to have an induced \(A_\infty \)bialgebra structure.
Here algebra by far preceded topology. The existence of a minimal model \(\mathcal{B }_\infty \) for the PROP \(B\) governing bialgebras^{2} was proved in [7]. According to general philosophy [6], \(A_\infty \)bialgebras defined as algebras over \(\mathcal{B }_\infty \) are homotopy invariant concepts. Moreover, it follows from the description of \(\mathcal{B }_\infty \) given in [7] that an \(A_\infty \)bialgebra defined in this way has operations \(\mu ^n_m : {V}^{\otimes m} \rightarrow {V}^{\otimes n}\), \(m,n \in \mathbb{N }\), \((m,n) \not = (1,1)\), but axioms as explicit as the ones for \(A_\infty \)algebras were given only for \(m+n \le 6\).
1.2 Pitfalls
It is clearly desirable to have some polyhedral PROP \({ KK}= \{{ KK}^n_m\}\) playing the same rôle for \(A_\infty \)bialgebras as the Stasheff’s operad plays for \(A_\infty \)algebras. By this we mean that \(\mathcal{B }_\infty \) should be isomorphic to the PROP of cellular chains of \({ KK}\), so the differential in \(\mathcal{B }_\infty \) and therefore also the axioms of \(A_\infty \)bialgebras would be encoded in the combinatorics of \({ KK}\) . To see where the pitfalls are hidden, we try to mimic the inductive construction of the associahedra in the context of bialgebras.
1.3 Two types of biassociahedra
We can already glimpse the following pattern. There naturally appear polytopes \(K^n_m\), \(m,n \in \mathbb{N }\), such that \(K^1_m\) and \(K^m_1\) are isomorphic to Stasheff’s associahedron \(K_m\). We will also see that \(K^2_m\) is isomorphic to the multiplihedron \(J_m\) [2, 12]. We call these polytopes the stepone biassociahedra. In this paper we give a simple and clean description of their face posets.
To continue as in the case of \(A_\infty \)algebras, one however needs to subdivide some faces of \(K^n_m\); and an example of such a subdivision is the heptagon \({ KK}^2_3\) in (2) subdividing the hexagon \(K^2_3\). The subdivisions must be compatible so that the result will be a cellular PROP \({ KK}= \{{ KK}^n_m\}\). Its associated cellular chain complex is moreover required to be isomorphic to the minimal model \(\mathcal{B }_\infty \) of the bialgebra PROP. We call these polyhedra steptwo biassociahedra.
The polytopes \({ KK}^n_m\) were, for \(m+n \le 6\), constructed in [7]. In higher dimensions, the issue of the compatibility of the subdivisions arises. In [10], a construction of the steptwo biassociahedra was proposed, but we admit that we were not able to verify it. In our opinion, a reasonably simple construction of the polyhedral PROP \({ KK}\) or at least a convincing proof of its existence still remains a challenge.
We think that a necessary starting point to address the above problems is a suitable notation. In this paper we give a simple description of the face poset of the stepone biassociahedron \(K^n_m\) that generalize the classical description of the Stasheff associahedron in terms of planar directed trees. We will also give a ‘coordinatefree’ characterization of \(K^n_m\) which shows that it is not a human invention but has existed since the beginning of time. As a byproduct of our approach, it will be obvious that \(K^2_m\) is isomorphic to the multiplihedron \(J_m\), for each \(m \ge 2\).
1.4 Notation and terminology
Some lowdimensional examples of the steptwo biassociahedra appeared for the first time, without explicit name, in [7]; they were denoted \(B^n_m\) there. The word biassociahedron was used by Saneblidze and Umble, see e.g. [10], referring to what we called above the steptwo biassociahedron; they denoted it \({ KK}_{m,n}\). Stepone biassociahedra can also be, without explicit name, found in [10]; they were denoted \(K_{m,n}\) there. Whenever we mention the biassociahedron in this paper, we always mean the stepone biassociahedron which we denote by \(K^n_m\).
2 Main results
Let us recall some standard facts [9, 14]. The permutahedron ^{3} \(P_{m1}\) is, for \(m \ge 2\), a convex polytope whose poset of faces \(\mathcal{P }_{m1}\) is isomorphic to the set \({ lT}_m\) of planar directed trees with levels and \(m\) leaves, with the partial order generated by identifying adjacent levels. The permutahedron \(P_{m1}\) can be realized as the convex hull of the vectors obtained by permuting the coordinates of \((1,\ldots ,m1) \in \mathbb{R }^{m1}\), its vertices correspond to elements of the symmetric group \(\Sigma _{m1}\). The face poset \(\mathcal{K }_m\) of the Stasheff’s associahedron \(K_m\) is the set of directed planar trees (no levels) \(T_m\) with \(m\) leaves; the partial order is given by contracting the internal edges. The obvious epimorphism \(\varpi _m : lT_m \twoheadrightarrow T_m\) erasing the levels induces the Tonks projection \({ Ton}: \mathcal{P }_{m1} \twoheadrightarrow \mathcal{K }_m\) of the face posets, see [14] for details.
In the last section, we analyze in detail the special case of \(\mathcal{K }^2_m\) when complementary pairs of trees with zones can equivalently be described as trees with a diaphragm. Using this description we prove that \(\mathcal{K }^2_m\) is isomorphic to the face poset of the multiplihedron \(J_m\). Necessary facts about PROPs and calculus of fractions are recalled in the Appendix.
The main definitions are Definition A and Definition B. The main result is Theorem C and the main application is Proposition D.
3 Trees with levels and (bi)permutahedra
3.1 Up and downrooted trees
Let us start by recalling some classical material from [9, II.1.5]. A planar directed (also called rooted) tree is a planar tree with a specified leg called the root. The remaining legs are the leaves. We will tacitly assume that all vertices have at least three adjacent edges.
We will distinguish between uprooted trees all of whose edges different from the root are oriented towards the root while, in downrooted trees, we orient the edges to point away from the root. The set of vertices \({ Vert}(T)\) of an up or downrooted tree \(T\) is partially ordered by requiring that \(u < v\) if and only if there exist an oriented edge path starting at \(u\) and ending at \(v\).
The definition of the poset \(({ lT}^n,<)\) of downrooted trees with levels and \(n\) leaves is similar. We will also need the exceptional tree Open image in new window with one edge and no vertices. We define Open image in new window .
3.2 Complementary pairs
Theorem 3.1
The posets \(\mathcal{P }_{m+n2} = ({ lT}_{m+n1},<)\) and \(\mathcal{P }^n_m = ({ lT}^n_m,<)\) are naturally isomorphic for each \(m,n \ge 1\).
Proof
It is simple to verify that (7) preserves the partial orders, giving rise to a poset isomorphism \(\mathcal{P }^n_m \cong \mathcal{P }^{n1}_{m+1}\), for each \(n\ge 2\), \(m \ge 1\). \(\square \)
Example
The isomorphism of Theorem 3.1 is, for \(m+n = 4\), presented in Fig. 5.
Example
3.3 Relation to the standard permutahedron
In this subsection we recall the wellknown isomorphism between the poset \(\mathcal{P }_m = ({ lT}_m,<)\) of rooted planar trees with levels and the poset of ordered decompositions of the set \(\{1,\ldots ,m1\}\) which we denote by \(\mathcal{SP }_m = ({ Dec}_m,<)\) (the \(\mathcal{S }\) in front of \(\mathcal{P }\) abbreviating the standard permutahedron). This isomorphism, which forms a necessary link to [10] and related papers, extends to an isomorphism between \((\mathcal{P }^n_m,<)\) and the poset of ordered bipartitions \(\mathcal{SP }^n_m = ({ Dec}^n_m,<)\). As the gadgets described here will not be used later in our note, this subsection can be safely skipped.
Let us denote the resulting isomorphism by \(\gamma : \mathcal{P }_m \cong \mathcal{SP }_m\). Combined with the isomorphisms of Theorem 3.1, it leads to an isomorphism (denoted by the same symbol) \(\gamma : \mathcal{P }^n_m \cong \mathcal{SP }_{m+m2}\) for each \(m,n \ge 1\).
Example
Then \(U_j\) (resp. \(D_j\)) is the set of balloons (resp. balls) that lift (resp. fall) to level \(j\), \(1 \le j \le \ell \). Observe that the reversing map (8) is already built in the above assignment.
Example
As an exercise, we recommend describing the isomorphism of Theorem 3.1 in terms of bipartitions. The above example serves as a clue how to do so.
3.4 Relation to strings of matrices
 A Open image in new window matrix is a matrix of the form \(A = [C_1C_2\cdots C_a]\), where, for \(1 \le i \le a\), either
 A Open image in new window matrix is a matrix of the form \(A = \left[ \begin{array}{c} R_1\\ R_2 \\ \vdots \\ R_b \end{array} \right] \), where, for \(1 \le j \le b\), either
Definition
A Open image in new window matrix (resp. a Open image in new window matrix) is elementary if it contains precisely one column (resp. row) of Open image in new window ’s (resp. Open image in new window ’s).
Example
We are going to describe how a minimal element \(X =(U,D,\ell )\) of the poset \(({ lT}^n_m,<)\) determines a string \({ str}(X)\) of elementary matrices. Minimality of \(X\) means that each level is occupied by precisely one vertex, which is either Open image in new window or Open image in new window . In particular, the level function (5) is an isomorphism. As before we denote by \(\ell _u\) (resp. \(\ell _d\)) its restriction to \({ Vert}(U)\) (resp. \({ Vert}(D)\)).

The matrices \(E_{u_a}\), \(1 \le a \le s\). Let \(L\) be the \(a\)th level line of \(U\), i.e. the \(u_a\)th level line of \(X\). Its intersection with \(U\) determines a vector \([R_a]\) whose entries represent the points in \(L \cap U\) read from left to right. An intersection with an edge is represented by Open image in new window , the (unique) intersection in a vertex by Open image in new window . It is clear that the sequence \([R_1],\ldots ,[R_s]\) determines \(U\).
Example
 The matrices \(E_{d_b}\), \(1 \le b \le t\). We proceed analogously to the previous case. To the \(b\)th level line \(L\) of \(D\), i.e. the \(d_b\)th level line of \(X\), we associate a ‘vertical vector’ (a column) \(C_b\) whose entries are determined by the intersection \(L \cap D\) as in the previous case. The only difference is that the (unique) intersection with a vertex is now represented by Open image in new window . The column of \(E_{d_b}\) is \(C_b\) and the number of its columns is$$\begin{aligned} \mathrm{card}\left\{ i \in \{1,\ldots ,h\}\ \ i < u_a,\ i \in \mathrm{Im}(\ell _u)\right\} +1. \end{aligned}$$
Example
 (i)
\(E_1\cdots E_h\) contains precisely \(m1\) Open image in new window matrices and \(n1\) Open image in new window matrices,^{10}
 (ii)
the leftmost Open image in new window matrix has one column, the rightmost Open image in new window matrix has one row, and
 (iii)
\(E_1\cdots E_h\) consist of block transverse pairs.
 if both \(E_i\) and \(E_{i+1}\) are Open image in new window matrices, then$$\begin{aligned} { row}(E_i) = { row}(E_{i+1}), { col}(E_i) +1 = { col}(E_{i+1}), \end{aligned}$$
 if both \(E_i\) and \(E_{i+1}\) are Open image in new window matrices, then$$\begin{aligned} { row}(E_i) = { row}(E_{i+1}) +1, { col}(E_i) = { col}(E_{i+1}), \end{aligned}$$
 if \(E_i\) is a Open image in new window matrix and \(E_{i+1}\) a Open image in new window matrix, then$$\begin{aligned} { row}(E_i) = { row}(E_{i+1}) +1, { col}(E_i) +1 = { col}(E_{i+1}), \end{aligned}$$
 if \(E_i\) is a Open image in new window matrix and \(E_{i+1}\) a Open image in new window matrix, then$$\begin{aligned} { row}(E_i) = { row}(E_{i+1}), { col}(E_i) = { col}(E_{i+1}). \end{aligned}$$
4 Trees with zones and the biassociahedron \(\mathcal{K }^n_m\)
In this section we present our definition of the face poset \(\mathcal{K }^n_m\) of the biassociahedron. Let us recall the classical associahedron first.
4.1 The associahedron \(K_m\)
As in (3), denote by \(\mathsf{F}(\xi _2,\xi _3,\ldots )\) the free non\(\Sigma \) operad in the monoidal category of sets, generated by the operations of \(\xi _2,\xi _3,\ldots \) of arities \(2,3,\ldots \), respectively. Its component of arity \(n\) consists of (uprooted) planar rooted trees with vertices having at least \(2\) inputs [8, Section 4]. We can therefore define the map (3) simply by forgetting the level functions. We however give a more formal, inductive definition which exhibits some features of other constructions used later in this note.
4.2 Complementary pairs with zones
Definition A
 (i)
\(z\) is strictly orderpreserving on barriers and
 (ii)
there are no adjacent zones of the same type.
Example
Let us look at case \(n=2\) of Definition A. For Open image in new window , only the following three cases may happen.
Case \(l = 1\). \(1\) is a barrier if \(m \ge 2\) and \(1\) is a downzone if \(m=1\).
Case \(l = 2\). One has four possibilities for the type of \(z\), namely \((\mathrm{D}\mathrm{U}), (\mathrm{U}\mathrm{D}), (\mathrm{B}\mathrm{U})\) or \((\mathrm{U}\mathrm{B}).\) In the \((\mathrm{D}\mathrm{U})\) and \((\mathrm{U}\mathrm{D})\) cases \(m \ge 2\), in the remaining two cases \(m \ge 3\).
Case \(l = 3\). \(m \ge 3\) and the only possibility for the type is Alfred Jarry’s \((\mathrm{U}\mathrm{B}\mathrm{U})\) [3].
Definition B
The face poset of the (stepone) biassociahedron is the poset \(\mathcal{K }^n_m := ({ zT}^n_m,<)\) of complementary pairs of trees with zones, with the above partial order.
Let \((U,D,\ell ) \in { lT}^n_m\) be a pair with the level function \(\ell : { Vert}(U) \cup { Vert}(D) \rightarrow \{1,\ldots ,h\}\). We call \(i \in \{1,\ldots ,h\}\) an uplevel (resp. downlevel) if \(\ell ^{1}(i) \subset { Vert}(U)\) (resp. \(\ell ^{1}(i) \subset { Vert}(D)\)).
Definition 4.1
It is easy to show that the map \(\pi \) preserves the partial orders, giving rise to the projection \(\mathcal{P }^n_m \twoheadrightarrow \mathcal{K }^n_m\) of posets. We finish this subsection by two statements needed in the proof of Theorem C.
Proposition 4.2
Proof
For \(i \in \{1,\ldots ,h'\}\) (resp. \(j \in \{1,\ldots ,h''\}\)) denote \(S'_i := z'^{1}(i)\) (resp. \(S''_j := z''^{1}(j)\)). By (14), for each \(i\) there exists a unique \(j\) such that \(S'_i \subset S''_j\). Exchanging the rôles of \(z'\) and \(z''\), we see that, vice versa, for each \(j\) there exists a unique \(i\) such that \(S''_j \subset S'_i\). This obviously means that there exists an automorphism \(\varphi : \{1,\ldots ,h'\} \rightarrow \{1,\ldots ,h''\}\) such that \(z' = \varphi \circ z''\). As both \(z'\) and \(z''\) are orderpreserving epimorphisms, \(\varphi \) must be the identity. \(\square \)
The following proposition in conjunction with Proposition 4.2 shows that the induced zone function remembers the relative heights of vertices of \(U\) and \(D\) but nothing more.
Proposition 4.3
Proof
The proof is a simple application of the definition of the induced zone function. \(\square \)
4.3 The map \(\varpi : { lT}^n_m \rightarrow \mathsf{F}(\Xi )\)
Each element \(X \in { lT}^n_m\) is a triple \(X = (U,D,\ell )\), where \(U\) is an uprooted tree with \(m\) leaves and \(D\) a downrooted tree with \(n\) leaves. We construct \(\varpi (X)\) by induction on the number of vertices of \(D\). We distinguish three cases.
Remark
In the above construction of the map \(\varpi \), the root vertex of \(D\) plays a different rôle than the root vertex of \(U\). One can exchange the rôles of \(U\) and \(D\), arriving at a formally different formula for \(\varpi (X)\). Due to the associativity of fractions [7, Section 6], both formulas give the same element of \(\mathsf{F}(\Xi ) \left( \begin{array}{c} n\\ m \end{array} \right) \). It is also possible to write a noninductive formula for \(\varpi (X)\) based on the technique of block transversal matrices developed in [10].
Example
Let us formulate the main result of this note which relates the canonical projection of Definition 4.1 with the map \(\varpi \).
Theorem C
Our proof of this theorem occupies the rest of this section.
4.4 Proof that \(\pi (X') = \pi (X'')\) implies \(\varpi (X') = \varpi (X'')\).
The construction of \(\varpi (X)\) given in Sect. 4.3 was divided into three cases, determined by the relative positions of the root vertices of \(U\) and \(D\). This information is, by Proposition 4.3, retained by the induced zone function of \(\pi (X)\). Therefore the case into which \(X\) falls depends only on the projection \(\pi (X)\).
Let \(X',X'' \in { lT}^n_m\) be such \(\pi (X') = \pi (X'')\). Then \(X'\) and \(X''\) may differ only by the level functions, i.e. \(X' = (U,D,\ell ')\) and \(X''= (U,D,\ell '')\). Let us proceed by induction on the number of vertices of \(D\).
If one (hence both) of \(X'\), \(X''\) falls into Case 1 or Case 2 of our construction of \(\varpi \), then clearly \(\varpi (X'') = \varpi (X'')\) since in these cases \(\varpi \) manifestly depends only on the trees \(U\) and \(D\) not on the level function. The induced zone function of \(\pi (X') = \pi (X'')\) must be of type \((\mathrm D \mathrm U)\) in Case 1 and \((\mathrm D\mathrm B\mathrm U)\) in Case 2.
4.5 Proof that \(\varpi (X') = \varpi (X'')\) implies \(\pi (X') = \pi (X'')\).
Let us show that \(\pi (X)\) is uniquely determined by \(\varpi (X) \in \mathsf{F}(\Xi ) \left( \begin{array}{c} n\\ m \end{array} \right) \). As we already remarked, elements of \(\mathsf{F}(\Xi )\) are represented by directed graphs \(G\) whose vertices are corollas \(c^b_a\) with \(a\) inputs and \(b\) outputs, where \(a,b \ge 1\), \((a,b) \not = (1,1)\). Let \(e\) be an internal edge of \(G\), connecting an output of \(c^s_r\) with an input of \(c^v_u\). We say that \(e\) is special if either \(s=1\) or \(u=1\). The graph \(G\) is special if all its internal edges are special. Finally, and element of \(\mathsf{F}(\Xi )\) is special if it is represented by a special graph. We have the following simple lemma whose proof immediately follows from the definition of the fraction.
Lemma 4.4
Lemma 4.4 implies that \(\varpi (X)\) is special if and only if \(X\) falls into Case 1 or Case 2 of Sect. 4.3. It is also clear that \(X\) falls into Case 2 if and only if \(\varpi (X)\) is special and the graph representing \(\varpi (X)\) has a (unique) vertex \(c^b_a\) with \(a,b \ge 2\). Therefore \(\varpi (X)\) bears the information to which case of its construction \(X = (U,D,\ell ) \in { lT}^n_m\) falls.
Let us summarize what we have. We know each \(\varpi (X_i)\). Since the construction of \(\varpi (X_i)\) falls into Case 1 or Case 2, we know, as we have already proved, the trees \(U_1,\ldots ,U_a\) in Fig. 11, and also the relative positions of vertices of \(U_1,\ldots ,U_a\) and the root vertex of \(D\).
Now we preform a similar analysis of \(\varpi (Y_1),\ldots ,\varpi (Y_b)\) and repeat this process until we get trivial trees. It is clear that, during this process, we fully reconstruct the trees \(U,D\) in \(X = (U,D,\ell )\) and the relative positions of their vertices. By Proposition 4.3, this uniquely determines the zone function in \(\pi (X) = (U,D,z)\). This finishes our proof of the second implication.
5 The particular case \(\mathcal{K }^2_m\)
In this section we analyze in detail the poset \(\mathcal{K }^2_m\) for which the notion of complementary pairs with zones takes a particularly simple form.
5.1 Trees with a diaphragm
Proposition 5.1
The posets \(({ zT}^2_m,<)\) and \(({ dT}^2_m,<)\) are, for each \(m \ge 1\), naturally isomorphic.
Proof
Example
The \(\pi \)images of complementary pairs in \({ lT}^2_3\) are listed in the rightmost column of the table in Fig. 6.
Example
Figure 12 illustrates the projection \(P^2_4 \rightarrow K^2_4\). It shows the face poset of a square face of the \(3\)dimensional \(P^2_4\) together with the corresponding complementary pairs in \({ lT}^2_4\) and its projection, which is in this case the poset of the interval indexed by the corresponding trees with a diaphragm.
As an exercise, we recommend describing the map \(\varpi : { zT}^2_m \rightarrow \mathsf{F}(\Xi ) \left( \begin{array}{c} 2\\ m \end{array} \right) \) in terms of trees with a diaphragm. One may generalize the above description of the poset \({ zT}^n_m\) also to \(n> 2\). In this case, the tree \(U\) corresponding to \((U,D,z) \in { zT}^n_m\) may have several diaphragms, depending on the relative positions of the vertices of \(D\). The combinatorics of this kind of description becomes, however, unmanageably complicated with growing \(n\).
5.2 Relation to the multiplihedron
 (i)
vertices with at least two inputs whose all adjacent edges are of the same color, or
 (ii)
vertices whose all inputs are white and whose output is black.
Proposition D
The face poset \(\mathcal{K }^2_m\) of the biassociahedron \(K^2_m\) is isomorphic to the face poset \(\mathcal{J }_m\) of the multiplihedron \(J_m\), for each \(m \ge 2\).
Proof
By Proposition 5.1, it suffices to prove that the posets \(({ dT}_m,<)\) and \(({ pT}_m,<)\) are isomorphic. It is very simple. Having a tree \(U\) with a diaphragm, we paint everything that lies above^{14} the diaphragm black, and everything below white. If the diaphragm intersects an edge of \(U\), we introduce at the intersection a new vertex of type (ii) with one input edge. The result will obviously be a painted tree belonging to \({ pT}_m\). The isomorphism we have thus described clearly preserves the partial orders. \(\square \)
The correspondence of Proposition D is, for \(m=3\), illustrated by the two rightmost columns of the table in Fig. 6.
Remark Forcey in [1] constructed an explicit realization of the poset \(\mathcal{J }_m\) by the face poset of a convex polyhedron. This, combined with Proposition D proves, independently of [10], that \(\mathcal{K }^2_m\) is the face poset of a convex polyhedron, too.
5.3 Relation to \(K_{m1,n1}\)
The relation of ‘stacking adjacent matrices’ obviously corresponds to our passage from complementary pairs of trees with levels to complementary pairs of trees with zones. This explains the relation between our stepone associahedron \(K^n_m\) and Saneblidze–Umble’s \(K_{m1,n1}\).
Footnotes
 1.
Other possible names are \(B_\infty \)algebras or strongly homotopy bialgebras.
 2.
PROPs generalize operads. We briefly recall them in the Appendix.
 3.
Sometimes also spelled permutohedron.
 4.
In [10] it was denoted \(K_{m,n}\).
 5.
Denoted \(P_{m1,n1}\) in [10].
 6.
Strictly orderpreserving means that \(v' < v''\) implies \(\ell (v') < \ell (v'')\).
 7.
By a map of trees we understand a sequence of contractions of internal edges. In particular, the root and leaves are fixed.
 8.
We however recall the correspondence between trees with levels and ordered partitions in Sect. 3.3.
 9.
Alternatively, replace the labels by balls and change the direction of gravity.
 10.
Therefore \(h = m+n2\).
 11.
The one such that \(T' < T''\) if and only if there exists a morphism of planar uprooted trees \(T' \rightarrow T''\).
 12.
By this we mean that neither \(u' < u''\) nor \(u' > u''\).
 13.
That is the vertex adjacent to the root.
 14.
We keep our convention that all edges are oriented to point upwards.
Notes
Acknowledgments
I would like to express my gratitude to Samson Saneblidze, Jim Stasheff, Ron Umble and the anonymous referee for reading the manuscript and offering helpful remarks and suggestions. I also enjoyed the wonderful atmosphere in the MaxPlanck Institut für Matematik in Bonn during the period when the first draft of this paper was completed.
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