# A new “dinv” arising from the two part case of the shuffle conjecture

## Abstract

For a symmetric function *F*, the eigen-operator Δ_{F} acts on the modified Macdonald basis of the ring of symmetric functions by \(\Delta_{F} \tilde{H}_{\mu}= F[B_{\mu}] \tilde{H}_{\mu}\). In a recent paper (Int. Math. Res. Not. 11:525–560, 2004), J. Haglund showed that the expression \(\langle\Delta_{h_{J}} E_{n,k}, e_{n}\rangle\)*q*,*t*-enumerates the parking functions whose diagonal word is in the shuffle 12⋯*J*∪∪*J*+1⋯*J*+*n* with *k* of the cars *J*+1,…,*J*+*n* in the main diagonal including car *J*+*n* in the cell (1,1) by *t*^{area}*q*^{dinv}.

In view of some recent conjectures of Haglund–Morse–Zabrocki (Can. J. Math., doi:10.4153/CJM-2011-078-4, 2011), it is natural to conjecture that replacing *E*_{n,k} by the modified Hall–Littlewood functions \(\mathbf{C}_{p_{1}}\mathbf{C}_{p_{2}}\cdots\mathbf{C}_{p_{k}} 1\) would yield a polynomial that enumerates the same collection of parking functions but now restricted by the requirement that the Dyck path supporting the parking function touches the diagonal according to the composition *p*=(*p*_{1},*p*_{2},…,*p*_{k}). We prove this conjecture by deriving a recursion for the polynomial \(\langle\Delta_{h_{J}} \mathbf{C}_{p_{1}}\mathbf{C}_{p_{2}}\cdots \mathbf{C}_{p_{k}} 1 , e_{n}\rangle \), using this recursion to construct a new \(\operatorname{dinv}\) statistic (which we denote \(\operatorname{ndinv}\)), then showing that this polynomial enumerates the latter parking functions by \(t^{\operatorname{area}} q^{\operatorname{ndinv}}\).

### Keywords

Symmetric functions Macdonald polynomials Parking functions## 1 Introduction

*V*=(

*v*

_{1},

*v*

_{2},…,

*v*

_{n}) on top and their diagonal numbers

*U*=(

*u*

_{1},

*u*

_{2},…,

*u*

_{n}) on the bottom. In the corresponding

*n*×

*n*tableau of lattice cells we have shaded the

*main diagonal*(or 0-diagonal) and drawn the

*supporting*Dyck path. The component

*u*

_{i}gives the number of lattice cells EAST of the

*i*th NORTH step and WEST of the main diagonal. The cells adjacent to the NORTH steps of the path are filled with the corresponding cars from bottom to top.

*U*to give a Dyck path is that

*V*=(

*v*

_{1},

*v*

_{2},…,

*v*

_{n}) is a permutation in

*S*

_{n}satisfying

*u*

_{i}may also be viewed as the order of the diagonal supporting car

*V*

_{i}. In the example above, car 3 is in the third diagonal, 1 and 8 are in the second diagonal, 5,7 and 6 are in the first diagonal and 2 and 4 are in the main diagonal. We have purposely listed the cars by diagonals from right to left starting with the highest diagonal. This gives the

*diagonal word*of

*PF*which we will denote

*σ*(

*PF*). It is easily seen that

*σ*(

*PF*) can also be obtained directly from the 2-line array by successive right to left readings of the components of the vector

*V*=(

*v*

_{1},

*v*

_{2},…,

*v*

_{n}) according to decreasing values of

*u*

_{1},

*u*

_{2},…,

*u*

_{n}. In previous work, each parking function is assigned a

*weight*

*U*and

*V*in the two line representation will be also referred to as

*U*(

*PF*) and

*V*(

*PF*). It will also be convenient to denote by \(\mathcal{PF}_{n}\) the collection of parking functions in the

*n*×

*n*lattice square.

*shuffle conjecture*[10] states that for any partition

*μ*=(

*μ*

_{1},

*μ*

_{2},…,

*μ*

_{ℓ})⊢

*n*we have the identity

*e*

_{n}is the familiar elementary symmetric function, \(h_{\mu_{1}}h_{\mu_{2}}\cdots h_{\mu_{\ell}}\) is the

*homogeneous*symmetric function basis indexed by

*μ*, \(\mathcal{E}_{1},\mathcal{E}_{2},\ldots,\mathcal{E}_{\ell}\) are successive segments of the word 1234⋯

*n*of respective lengths

*μ*

_{1},

*μ*

_{2},…,

*μ*

_{ℓ}and the symbol \(\chi(\sigma(\mathit{PF})\in \mathcal{E}_{1}{\cup\!\cup}\mathcal{E}_{2}{\cup\!\cup}\cdots {\cup\!\cup}\mathcal{E}_{\ell})\) is to indicate that the sum is to be carried out over parking functions in \(\mathcal{PF}_{n}\) whose diagonal word is a shuffle of the words \(\mathcal{E}_{1},\mathcal{E}_{2},\ldots, \mathcal{E}_{\ell}\). In [8] Haglund proved the

*l*=2 case of (1.7). By a remarkable sequence of identities it is shown in [8] that this case is a consequence of the more refined identity

*n*+

*J*)×(

*n*+

*J*) lattice square that have

*k*of the cars

*J*+1,…,

*J*+

*n*in the main diagonal including car

*J*+

*n*in the cell (1,1). Here the

*E*

_{n,k}are certain ubiquitous symmetric functions introduced in [3] with sum

*diagonal composition*of a Parking function, which we denote by

*p*(

*PF*) and is simply the composition which gives the position of the zeros in the vector

*U*=(

*u*

_{1},

*u*

_{2},…,

*u*

_{n}), or equivalently the lengths of the segments of the main diagonal between successive hits of its supporting Dyck path. One of their conjectures is the identity

*p*⊨

*n*and

*μ*⊢

*n*. Here for each integer

*a*,

**C**

_{a}is the operator whose action on a symmetric function

*F*[

*X*], in plethystic notation, can be simply expressed in the form

*E*

_{n,k}is replaced by one of the symmetric polynomials \(\mathbf {C}_{p_{1}}\mathbf{C}_{p_{2}}\cdots\mathbf{C}_{p_{k}} 1\). Note, however, that since the

*k*in (1.8), under the action of \(\Delta_{h_{J}}\) controls the number of

*big cars*on the main diagonal, it natural to suspect that the combination of \(\Delta_{h_{J}}\) and \(\mathbf{C}_{p_{1}}\mathbf {C}_{p_{2}}\cdots \mathbf{C}_{p_{k}} 1\) would result in forcing

*k*of the big cars to hit the diagonal according to the composition

*p*=(

*p*

_{1},

*p*

_{2},…,

*p*

_{k}). Miraculous as this might appear to be, computer data beautifully confirm this mechanism … but up to a point. In fact, following this line of reasoning, one might conjecture the identity

*q*-reduced) version of (1.8) is actually true. Namely This circumstance led to the conjecture that (1.14) could be made true by replacing the classical parking function “

*dinv*” by a new dinv more focused on the positions of the big cars.

The main result of this paper is a proof of this conjecture. Banking on the intuition gained from previous work [7] and using some of the identities developed there with the **C**_{a} and **B**_{b} operators we are able to derive the following basic recursion.

### Theorem 1.1

*For all compositions*

*p*=(

*p*

_{1},

*p*

_{2},…,

*p*

_{k})

*we have*

*with*\(\mathbf{B}_{a}=\omega\widetilde{\mathbf{B}}_{a}\omega\)

*and for any symmetric function*

*F*[

*X*]

**C**operators by the

**B**operators in (1.11) has the effect of allowing the controlled Dyck paths to hit the diagonal everywhere, including the points forced by the composition

*p*. This led us to interpret the first polynomial on the right hand side of (1.8) as a weighted enumeration of the collection of parking functions with diagonal word of a shuffle of 12⋯(

*J*−1) by

*J*(

*J*+1)⋯(

*n*+

*J*−1) whose big cars hit the main diagonal according to the collection of compositions obtained by concatenating (

*p*

_{2},…,

*p*

_{k}) with an arbitrary composition of

*p*

_{1}. Guided by this interpretation, by means of (1.16) we obtained a recursive construction of the appropriate new dinv and proved the identity

*ndinv*” is also given an equivalent somewhat less recursive construction with the hope that it may be conducive to the discovery of a direct formula for the new dinv which, as in the case of the classical dinv, is closely related to the geometry of the corresponding parking function diagram.

## 2 Auxiliary identities from the Theory of Macdonald polynomials

*Λ*. The subspace of homogeneous symmetric polynomials of degree

*m*will be denoted by

*Λ*

^{=m}. We will seldom work with symmetric polynomials expressed in terms of variables but rather express them in terms of one of the six classical symmetric function bases

- (1)
“

*power*” {*p*_{μ}}_{μ}, - (2)
“

*monomial*” {*m*_{μ}}_{μ}, - (3)
“

*homogeneous*” {*h*_{μ}}_{μ}, - (4)
“

*elementary*” {*e*_{μ}}_{μ}, - (5)
“

*forgotten*” {*f*_{μ}}_{μ}and - (6)
“

*Schur*” {*s*_{μ}}_{μ}.

*ω*may be defined by setting for the power basis indexed by

*μ*=(

*μ*

_{1},

*μ*

_{2},…,

*μ*

_{k})⊢

*n*

*v*=(

*v*

_{1},

*v*

_{2},…,

*v*

_{k}) we set \(|v|=\sum_{i=1}^{k} v_{i} \) and

*l*(

*v*)=

*k*.

*MAPLE*or

*MATHEMATICA*for computer experimentation. We simply set for any expression

*E*=

*E*(

*t*

_{1},

*t*

_{2},…) and any power symmetric function

*p*

_{k}

*F*we set

*Q*

_{F}is the polynomial yielding the expansion of

*F*in terms of the power basis. Note that in writing

*E*(

*t*

_{1},

*t*

_{2},…) we are tacitly assuming that

*t*

_{1},

*t*

_{2},

*t*

_{3},… are all the variables appearing in

*E*and in writing \(E(t_{1}^{k},t_{2}^{k},\ldots)\) we intend that all the variables appearing in

*E*have been raised to their

*k*th power.

*plethystic*minus sign, we will carry out the

*ordinary*sign change by prepending our expressions with a superscripted minus sign, or, as the case may be, by means of a new variable

*ϵ*which outside of the plethystic bracket is simply replaced by −1. For instance, these conventions give for

*X*

_{k}=

*x*

_{1}+

*x*

_{2}+⋯+

*x*

_{n}

*X*=

*x*

_{1}+

*x*

_{2}+

*x*

_{3}+⋯

*F*∈

*Λ*and any expression

*E*we have

*F*∈

*Λ*

^{=k}we may also rewrite this as

*μ*and a cell

*c*∈

*μ*, Macdonald introduces four parameters

*l*=

*l*

_{μ}(

*c*), \(l'=l'_{\mu}(c)\),

*a*=

*a*

_{μ}(

*c*) and \(a'=a'_{\mu}(c)\) called

*leg*,

*coleg*,

*arm*and

*coarm*which give the number of lattice cells of

*μ*strictly NORTH, SOUTH, EAST and WEST of

*c*(see attached figure). Following Macdonald we will set

*μ*′ the conjugate of

*μ*, the basic ingredients playing a role in the theory of Macdonald polynomials are

*star*scalar product, defined by setting for the power basis

*z*

_{μ}gives the order of the stabilizer of a permutation with cycle structure

*μ*.

*μ*

*F*[

*X*]

*e*

_{n}[

*B*

_{μ}]=

*T*

_{μ}for

*μ*⊢

*n*, the operator ∇ itself reduces to \(\Delta_{e_{n}}\) when acting on symmetric polynomials that are homogeneous of degree

*n*.

*u*) out of the denominators on both sides of (2.11) then setting

*u*=1 gives

*u*by 1/

*u*and letting

*u*=0 in (2.11) gives

*β*the empty partition we can take \(\tilde{H}_{\beta}=1\) and

*D*

_{β}=−1, (2.11) in this case for

*α*=

*μ*reduces to

*X*=1−

*u*, combined with (2.14) gives for

*μ*⊢

*n*

*β*=(1) we have \(\tilde{H}_{\beta}=1\) and

*Π*

_{β}=1, formula (2.12) reduces to the surprisingly simple identity

*ν*→

*μ*simply means that the sum is over

*ν*’s obtained from

*μ*by removing a corner cell and

*μ*←

*ν*means that the sum is over

*μ*’s obtained from

*ν*by adding a corner cell.

*f*,

*g*

*ϕ*be the operator defined by setting for any symmetric function

*f*

*ϕ*is usually written in the form

*f*,

*g*

*n*in

*X*and

*Y*reduces to

Note that the orthogonality relations in (2.10) yield us the following Macdonald polynomial expansions:

### Proposition 2.1

*μ*

### Remark 2.2

*S*

_{n}is given by the symmetric function

*e*

_{n}. To see this we simply combine the relation in (2.18) with the degree

*n*restricted Macdonald–Cauchy formula (2.29) obtaining

## 3 Proof of the basic recursion

To establish Theorem 1.1 we need some preliminary observations. To begin we have the following reduction.

### Theorem 3.1

*For all*

*p*=(

*p*

_{1},

*p*

_{2},…,

*p*

_{k})⊨

*n*

*and*

*j*≥0

*we have*

*if and only if*,

*with*\(\mathbf{C}_{a}^{*}\)

*and*\(\mathbf{B}_{a}^{*}\)

*the*∗-

*scalar product duals of*

**C**

_{a}

*and*

**B**

_{a},

*we have*

*for all*

*j*≥0

*and*1≤

*a*≤

*n*.

We will give a proof of (3.1) first, and then in the following pages we will establish (3.2) after developing a few necessary identities.

### Proof of (3.1)

**C**

_{a}and

**B**

_{b}satisfy the commutativity relations

*F*[

*X*] that is homogeneous of degree

*n*−

*p*

_{1}is equivalent to (3.1) since when

*p*

_{2},…,

*p*

_{k}are the parts of a partition the polynomials \(\mathbf{C}_{p_{2}}\cdots\mathbf{C}_{p_{k}} 1 \) are essentially elements of the Hall–Littlewood basis. Now, since all the operators Δ

_{F}are self adjoint with respect to the ∗-scalar product, (3.3) in turn can be rewritten in the form

*p*

_{1}≥1) is equivalent to (3.2) due to the arbitrariness of

*F*[

*X*]. This completes our proof. □

Our next goal is to prove (3.2). To begin we have the following auxiliary identity.

### Proposition 3.2

### Proof

Next we have

### Proposition 3.3

### Proof

*P*[

*X*]∈

*Λ*we have

*v*≥

*r*and in (3.10) we have

*r*≥

*a*≥1 we obtain

## 4 The construction of the new dinv

*J*+

*n*)×(

*J*+

*n*) lattice square whose diagonal word is a shuffle of the two words \(\mathcal{E}_{J}=12\cdots J\) and \(\mathcal{E}_{J,n}=J+1\cdots J+n\) with car

*J*+

*n*in the (1,1) lattice square. In symbols

Before we can proceed with our construction of the new dinv, we need some preliminary observations about this family of parking functions. To begin we should note that the condition that the diagonal word be a shuffle of 12⋯*J* with *J*+1⋯*J*+*n*, together with the column increasing property of parking functions, forces the columns of the Dyck path where by the *length* of column *i* of a Dyck path *D* we refer to the number of NORTH steps of *D* of abscissa *i* supporting a \(\mathit{PF}\in\mathcal{PF}(J,n)\) to be of length at most 2. The reason for this is simple: as we read the cars of *PF* to obtain *σ*(*PF*) from right to left by diagonals starting from the highest and ending with the lowest the big cars (*J*+1,…,*J*+*n*) as well as the small cars (1,2,…,*J*) will be increasing. Thus we will never see a big car on top of a big car nor a small car on top of a small car. So the only possibility is a big car on top of a small car, i.e. columns of length most 2 as we asserted.

This yields an algorithm for constructing all the elements of the family \(\mathcal{PF}(J,n)\). Let us denote by “\(\operatorname{red}(\mathit{PF})\)”, and call it the “*reduced tableau*” of *PF*, the configuration obtained by replacing in a \(\mathit{PF}\in\mathcal{PF}(J,n)\) all big cars by a 2 and all small cars by a 1. We can simply obtain all the reduced tableaux of elements of \(\mathcal{PF}(J,n)\) by constructing first the family \(\mathcal{D}_{J,n}\) of Dyck paths of length *n*+*J* with no more than *J* columns of length 2 and all remaining columns of length 1. Then for each Dyck path \(D\in\mathcal{D}_{J,n}\) fill the cells adjacent to the NORTH steps of each column of length 2 by a 1 under a 2, then fill the columns of length 1 by a 1 or a 2 for a total of *J* ones and *n* twos.

Clearly each \(\mathit{PF}\in\mathcal{PF}(J,n)\) can be uniquely reconstructed from its reduced tableau by replacing all the ones by 1,2,…,*J* and all the twos by *J*+1,…,*J*+*n* by diagonals from right to left starting from the highest and ending with the lowest. It will also be clear that we need only work with reduced tableaux to construct our new dinv. However, being able to refer to the original cars will turn out to be more convenient in some of our proofs. For this reason we will work with a *PF* or its \(\operatorname{red}(\mathit{PF})\) interchangeably depending on the context.

This given, we have the following basic fact.

### Proposition 4.1

*For any*

*if we set*{

*i*

_{1}<

*i*

_{2}<⋯<

*i*

_{k}}={

*i*∈[1,

*J*+

*n*]:

*v*

_{i}>

*J*}

*then the vector*

*gives the area sequence of a Dyck path*,

*which here and after will be referred to as the Dyck path “supporting” the big cars of PF*.

### Proof

*J*+

*n*is in the (1,1) lattice square it follows that \(u_{i_{1}}=0\). Thus we need only show that

*PF*that means that for

*i*

_{s−1}<

*j*<

*i*

_{s}the car

*v*

_{j}is small and thus, except perhaps for

*j*=

*i*

_{s}−1, the car

*v*

_{j}must be in a column of length 1. In particular we see that we must have \(u_{j}\le u_{i_{s-1}}\) for, the first violation of this inequality would put a small car above a big car (for

*j*=

*i*

_{s−1}+1) or a small car above a small car for

*j*>

*i*

_{s−1}+1. This gives \(u_{i_{s}} \le u_{i_{s}-1}+1\) as desired with equality only if car \(v_{i_{s}}\) is at the top of a column of length 2 and all the small cars in between \(v_{i_{s-1}}\) and \(v_{i_{s}}\) are in the same diagonal as \(v_{i_{s-1}}\). □

*p*=(

*p*

_{1},

*p*

_{2},…,

*p*

_{k})⊨

*n*. Our goal is to construct a statistic “

*ndinv*” which yields the equality

*p*=(3,2). In this case there are only two possible Dyck paths as given below on the left. On the right we added the 2’s and their corresponding diagonal numbers. Now the least number of 1’s we need to add to get a legal reduced diagram is 3 for the first and 2 for the second as shown below

*t*

^{area}

*q*

^{dinv}it is better to have a look at the non-reduced versions of the two tableau above. Namely

*PF*, the pairs (3,7), (1,5) and (6,2), are the only ones contributing to the dinv and the sum of the area numbers is 5, so its classical weight is

*t*

^{5}

*q*

^{3}. Similarly, the pairs contributing to the dinv on the

*PF*on the right are (2,6), (5,1) and (4,1) and the area numbers add to 3, so its classical weight is

*t*

^{3}

*q*

^{3}. The latter is not the same as what comes out of (4.3). The area is OK but the dinv in not. The calculation in (4.4) thus asserts that the “

*new dinv*” should be 4. Similarly, as we will show in a moment, the calculation in (4.5) yields the result that the new dinv of the

*PF*on the left of (4.6) should be 4 again. In fact, it turns out that none of the 8 parking functions we obtain by inserting an extra 1 in the reduced tableau on the right of (4.3) have area 5 thus the last term in (4.5) can only be produced by the

*PF*on the left of (4.6). We give below the 8 above mentioned reduced tableaux with the extra 1 shaded

For a while in our investigation this appeared to be a challenging puzzle. The discovery of the recursion of Theorem 1.1 completely solved this puzzle but, as we shall see, it created another puzzle.

*p*=(

*p*

_{1},

*p*

_{2},…,

*p*

_{k})⊨

*n*

*p*

_{2},…,

*p*

_{k},

*r*] represents the concatenation of the compositions (

*p*

_{2},…,

*p*

_{k}) and

*r*. This strongly suggests what should recursively happen to the new weight of our parking functions by the removal of a single (appropriate) car. That is, if the chosen car is

*small*there should be a loss of area of

*p*

_{1}−1 and a loss of

*ndinv*of

*k*−1 and if the chosen car is

*big*no loss of any kind.

Starting from this observation and further closer analysis of (4.8) led us to the following recursive algorithm for constructing “*ndinv*”.

*vertical dominos*. For instance the \(\operatorname{red}(\mathit{PF})\) below, which is none other than the

*minimal ones*obtained from the Dyck path on the right will be viewed as the sequence of dominos

*PF*belongs to the family \(\mathcal{PF}_{5}([3,3,2])\) and as such will be divided into 3 sections, one for each part of [3,3,2]. To do this we simply cut the sequence in (4.10) before each domino \(\bigl[{2\atop0}\bigr]\) obtaining the three sections

*p*

_{1}=3>1, (4.8) suggests that we should remove a 1 from the first section, then process it somewhat to cause a loss of dinv of 2 (=

*k*−1), and loss of area 2 (=

*p*

_{1}−1). Taking a clue from the classical dinv, we can see that the first small car in the corresponding

*PF*would contribute a unit to the classical dinv with the big cars to its right in the main diagonal. The latter of course correspond to the dominoes \(\bigl[{2\atop0}\bigr]\) that begin each of the following sections. Thus the desired loss of dinv can be simply obtained by bodily moving the first section to the end, and removing the \(\bigl[{1\atop0}\bigr]\) obtaining

*p*

_{1}−1 then it must be equal to the number of big cars in the moved section, minus one. This means that we can fix both problems by making the domino replacements \(\bigl[{2\atop1}\bigr] {\rightarrow}\bigl[{2\atop0}\bigr]\) and \(\bigl[{2\atop2}\bigr] {\rightarrow} \bigl[{2\atop1}\bigr]\), obtaining

In the case that *p*_{1}=1 there will be only one big car in the first section and if there are small cars they all must be on the main diagonal. In this case we can process the first section as we did for *p*_{1}>1. If there are no small cars then the first section consists of the single domino \(\bigl[{2\atop0}\bigr]\) and (4.8) suggests that we should simply remove it with no further ado.

*ndinv*” rigorously and in full generality, we will break our argument into three separate steps. In the first step we use the ideas stemming from the above example to construct a bijection

*Π*

_{J}([

*p*

_{1},

*p*

_{2},…,

*p*

_{k}]) satisfy the same recursion as the polynomials \(\langle\Delta_{h_{J}}\mathbf{C}_{p_{1}}\mathbf{C}_{p_{2}}\cdots \mathbf{C}_{p_{k}}1, e_{n}\rangle\).

In the third step we establish the equality in (4.2) by verifying the equality in the base cases.

*Φ*(

*PF*) by the following procedure.

*Cut the domino sequence of*\(\operatorname{red}(\mathit{PF})\)*into sections starting at the dominos*\(\bigl[{2\atop0}\bigr]\)

- (1)If the first section does not contain a domino \(\bigl[{1\atop0}\bigr]\)
*remove its only domino*\(\bigl[{2\atop0}\bigr]\)*from the sequence of dominos*

- (2)If the first section contains a domino \(\bigl[{1\atop0}\bigr]\), work on the first section as follows:
*remove its first domino*\(\bigl[{1\atop0}\bigr]\)*for each (but the first) domino*\(\bigl[{2\atop a}\bigr]\)*make the replacement*\(\bigl[{2\atop a}\bigr] {\rightarrow}\bigl[{2\atop a-1}\bigr]\)*if adjacent pairs*\(\bigl[{2\atop a-1}\bigr] \bigl[{1\atop a}\bigr]\)*are created make the replacements*\(\bigl[{2\atop a-1}\bigr] \bigl[{1\atop a}\bigr] {\rightarrow}\allowbreak \bigl[{1\atop a-1}\bigr] \bigl[{2\atop a}\bigr]\)*cycle the modified first section to the end of the sequence of dominos*

*PF*′ be the parking function corresponding to the resulting domino sequence and set

It is clear that *Φ* maps the left hand side of (4.15) into the right hand side. To show that *Φ* is a bijection we need only show that the procedure above can be reversed to reconstruct *PF* from *PF*′ for any *PF*′ in the right hand side of (4.15). We will outline the salient steps of the reversed procedure.

*PF*=

*Φ*

^{−1}(

*PF*′) is to be in \(\mathcal{PF}_{J}([p_{1},p_{2},\ldots, p_{k}])\) we already know the diagonal composition of the Dyck path of the big cars of

*PF*. Thus we can proceed as follows:

- (1)Say \(\mathit{PF}'\in\mathcal{PF}_{J}([p_{2},\ldots, p_{k}])\) (which will only occur when
*p*_{1}=1)*Then**PF**is the parking function obtained by prepending*\(\bigl[{2\atop0}\bigr]\)*to the domino sequence of**PF*′*.*

- (2)Say \(\mathit{PF}'\in\mathcal{PF}_{J-1}([p_{2},\ldots, p_{k},1])\) (which will only occur when
*p*_{1}=1)*Then**PF**is the parking function obtained by inserting*\(\bigl[{1\atop0}\bigr]\)*immediately after the first*\(\bigl[{2\atop0}\bigr]\)*in the last section of**PF*′*, then cycle back the last section to be the first in the domino sequence.*

- (3)Say \(\mathit{PF}'\in\mathcal{PF}_{J-1}([p_{2},\ldots, p_{k},q])\) for a
*q*⊨*p*_{1}−1>0*Let*\(\operatorname{last}(\mathit{PF}')\)*be the domino sequence obtained by removing the first**k*−1*sections from the domino sequence of**PF*′*.**Modify*\(\operatorname{last}(\mathit{PF}')\)*by inserting a*\(\bigl[{1\atop0}\bigr] \)*immediately after its first*\(\bigl[{2\atop0}\bigr]\)*.**For**a*≥1*replace, in*\(\operatorname{last}(\mathit{PF}')\)*, each pair*\(\bigl[{1\atop a-1}\bigr] \bigl[{2\atop a}\bigr]\)*by the pair*\(\bigl[{2\atop a}\bigr] \bigl[{1\atop a}\bigr]\)*.*(note that for this to put a big car on top of a big car we must have a \(\bigl[{2\atop a-1}\bigr]\) preceding the \(\bigl[{1\atop a-1}\bigr]\), but that \(\bigl[{2\atop a-1}\bigr]\) will also be replaced either by this step or by the next steps)

*For**a*≥1*replace each*\(\bigl[{2\atop a}\bigr]\)*preceded by a*\(\bigl[{1\atop a}\bigr]\)*in*\(\operatorname{last}(\mathit{PF}')\)*by*\(\bigl[{2\atop a+1}\bigr]\)*Replace each*\(\bigl [{2\atop0}\bigr]\)*, except the first by a*\(\bigl[{2\atop1}\bigr]\)(note if a replaced \(\bigl[{2\atop0}\bigr]\) is preceded by a \(\bigl[{2\atop0}\bigr]\) then that \(\bigl[{2\atop0}\bigr]\) itself will also be replaced by \(\bigl[{2\atop1}\bigr]\))

*The modified*\(\operatorname{last}(\mathit{PF}')\)*followed by the first**k*−1*sections of**PF*′*gives then the domino sequence of our target**PF.*

*Φ*is bijective.

Since *Φ* moves EAST, by one cell, *p*_{1}−1 big cars it causes a loss of area equal to *p*_{1}−1. Thus the definition in (4.16) combined by the bijectivity of *Φ* proves the recursion in (4.9).

It remains to show equality in the base cases which, in view of the definition in (4.16) should be characterized by the absence of small cars.

*p*are equal to 1. To see this note that it is only the presence of small cars that allows the supporting Dyck path of one of our

*PF*′

*s*to have columns of lengths 2. But if all the columns are of length 1, the “area” statistic is 0 and the Dyck path supporting the big cars can only have area sequence a string of 0′

*s*. But in this case the family reduces to a single parking function which consists of cars 1,2,…,

*n*placed on the main diagonal from top to bottom. Thus it follows from our definition of

*Π*

_{J}(

*p*) and (4.16) that

### Theorem 4.2

*For*

*p*=(

*p*

_{1},

*p*

_{2},…,

*p*

_{k})⊨

*n*

*we have*

### Proof

*F*[

*X*] we have

*s*

_{λ}we have

*a*+|

*λ*|=

*n*

*a*+|

*μ*|=1, Thus for

*a*≥1 (4.19) reduces to

*p*

_{i}are equal to 1 in successive applications of

**C**

_{1}only the term corresponding to \(s_{1^{m}}\) in the Schur function expansion of \(\mathbf{C}_{1}^{m} 1\) will survive in the scalar product

This completes of proof of (4.17). This was the last fact we need to establish the equality in (1.17). □

### Remark 4.3

As we already mentioned, our definition of *ndinv* creates another puzzle. Indeed, the classical *dinv* can be immediately computed from the geometry of the parking function or directly from (1.6) which expresses it explicitly in terms of the two line array representation. For this reason we made a particular effort to obtain a non-recursive construction of *ndinv* and in the best scenario derive form it an explicit formula similar to (1.6). However, our efforts yielded only a partially non-recursive construction. In our original plan of writing we decided to include this further result even though in the end it yields a more complex algorithm for computing *ndinv* than from the original recursion. This was in the hope that our final construction may be conducive to the discovery of an explicit formula. It develops that during the preparation of this manuscript a new and better reason emerged for the inclusion of our final construction. It turns out that Angela Hicks and Yeonkyung Kim have very recently succeeded in discovering the desired explicit formula by a careful analysis of the combinatorial identities we are about to present. The results of Hicks–Kim will appear in a separate publication [13].

*ndinv*it will be convenient to make a few changes in the domino sequences. To begin, we shall use the actual car numbers at the top of the dominos rather than 1 or 2. We do this, so that we may refer to individual dominos by their car as the corresponding area number on the bottom is being changed. But now, to distinguish big cars from small cars we must in each case specify the number

*J*of small cars. Secondly, we will have sections end with a big car, rather than begin with a big car. This only requires, moving the initial big car to the end of the domino sequence. For example, the parking function below whose domino sequence was given in (4.10) has

*J*=5 thus cars 1,2,3,4,5 are

*small*and 5,6,…,13 are

*big*.

*ndinv*we will begin by modifying our first construction to adapt to these new domino sequences.

- (1)
*The recursive construction will now consist of as many steps as there are dominos in the domino sequence* - (2)
*At each step the first domino of the first section is removed*- (a)
*when we remove an*\(\bigl[{s\atop0}\bigr]\)*, the section is cycled to the end after it is processed as before* - (b)
*when we remove a*\(\bigl[{b\atop0}\bigr]\)*, it is because the section consisted of a single big car domino.*

- (a)
- (3)
*The removal of an*\(\bigl[{s\atop0}\bigr]\)*contributes to**ndinv**the number of*\(\bigl[{b\atop0}\bigr]\)*’s to its right minus one.*

There are a few observations to be made about the effect of the cycling process. To begin note that when the domino sequence consists of a single section, no visible cycling occurs. However, even in this case, for accounting purposes, it is convenient to consider all of its dominos to have been cycled. With this provision, each domino in the original domino sequence will have an associated cycling number *c* that counts the number of times it has been cycled before it is removed.

Based on these observations, a step by step study of our recursive construction of *ndinv* led us to the following somewhat less recursive algorithm. It consists of two stages. In the first stage, the domino sequence is doctored and wrapped around a circle to be used in the second stage. The second stage uses circular motion to mimic the cycling of sections that takes place in the recursive procedure. To facilitate the understanding of the resulting algorithm we will illustrate each stage by applying it to the parking function in (4.21). More precisely we work as follows:

**Stage I**

*Move each*\(\bigl[{s\atop a}\bigr]\)*in the domino sequence**a**places to its left and increase the area number by*1*of each domino*\(\bigl[{b\atop a}\bigr]\)*that is being by-passed*. For instance the domino sections in (4.23) become$$ \everymath{\displaystyle} \begin{array}{@{}l} \left[ \left[ \begin{array}{c} 5\\0 \end{array} \right], \left[ \begin{array}{c} 2\\1 \end{array} \right], \left[ \begin{array}{c} 10\\2 \end{array} \right], \left[ \begin{array}{c} 7\\2 \end{array} \right], \left[ \begin{array}{c} 12\\0 \end{array} \right] \right], \qquad\left[ \left[ \begin{array}{c} 4\\0 \end{array} \right], \left[ \begin{array}{c} 1\\1 \end{array} \right], \left[ \begin{array}{c} 9\\2 \end{array} \right], \left[ \begin{array}{c} 6\\2 \end{array} \right], \left[ \begin{array}{c} 11\\0 \end{array} \right] \right], \\[4mm] \left[ \left[ \begin{array}{c} 3\\0 \end{array} \right], \left[ \begin{array}{c} 8\\1 \end{array} \right], \left[ \begin{array}{c} 13\\0 \end{array} \right] \right]. \end{array} $$(4.24)*Next wrap the resulting sequence clockwise around a circle with positions marked by a “*∘*”*

*bar*“|” to separating beginning and ending dominos; the ∘’s will be successively changed to •’s during the second stage)

**Stage II**

*Set**ndinv*=0*and set the auxiliary parameter**c**to*1*.**Mark the first domino by changing its “*∘*” to a “*•*”.**Cycling clockwise from the first domino to the**bar**find the first*\(\bigl[{b\atop0}\bigr]\)*, call it “endsec”.**Cycling clockwise from**endsec**to the**bar**add*1*to**ndinv**each time we meet a*\(\bigl[{b\atop0}\bigr]\)*.*

*endsec*and lightly boxed the two

*ndinv*contributing big car dominos.)

*While there is a domino that has not been marked repeat the following steps:*

*cycling clockwise from the last**endsec**mark the first unmarked domino**if in so doing the bar is crossed add*1*to**c.*

*If the domino is a*\(\bigl[{s\atop a}\bigr]\)

*then clockwise from it find the first*\(\bigl[{b\atop a}\bigr]\)

*with*

*a*<

*c, call it “endsec” then cycle clockwise from*

*endsec*

*back to this*\(\bigl[{s\atop a}\bigr]\)

*for each encountered unmarked*\(\bigl[{b\atop a}\bigr]\)*add*1*to**ndinv**provided**a*<*c**if the bar is not crossed or**a*<*c*+1*after the bar is crossed*

*ndinv*is reached after all the small car dominos are marked).

*c*value increases to 2 and we obtain)

*ndinv*=14, which is the total number of lightly boxed dominos in the previous five configurations).

### Remark 4.4

We will not include a proof of the validity of this second algorithm, since A.S. Hicks and Y. Kim, using their discoveries, are able to provide in [13] a much simpler and more revealing validity argument than we can offer with our present tools. Here it should be sufficient to acknowledge that the auxiliary domino sequence resulting from Phase I together with the *c* statistic constructed in Phase II have ultimately been put to such beautiful use in subsequent work.

*ndinv*may have an extension that can be used in a more general settings than the present one. To see this, let us recall that the 2 part case of the Shuffle Conjecture, proved by J. Haglund in [8], may be stated as follows:

*n*by

*n*+1−

*J*in (1.17), for (

*p*

_{1},

*p*

_{2},…,

*p*

_{k})⊨

*n*+1−

*J*we get

*n*+1, that we also have

*n*+1)×(

*n*+1) lattice square which have the biggest car

*n*+1 in cell (1,1). But it was also shown in [8] that we have

*ndinv*, or a suitable extension of it, may give a new parking function interpretation to any of the polynomials occurring on the left hand side of (1.7). If that were the case then that would provide an alternate form of the Shuffle Conjecture. It is interesting to note that computer exploration has led us to conjecture that for

*p*=(

*p*

_{1},

*p*

_{2},…,

*p*

_{k})⊨

*n*the polynomials

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