[Paper Review] Interlaced rectangular parking functions

Jean-Christophe Aval, François Bergeron|arXiv (Cornell University)|Mar 13, 2015

Advanced Combinatorial Mathematics15 references3 citations

TL;DR

This paper extends the Grossman-Bizley paradigm to derive an explicit formula for the bi-Frobenius characteristic of interlaced rectangular parking functions in an $m\times n$-rectangle, where vertical and horizontal steps are independently labeled by $1$ to $n$ and $1$ to $m$, respectively. The formula characterizes the decomposition of the $\mathbb{S}_n \times \mathbb{S}_m$-module of such functions into irreducibles, generalizing prior results for coprime $m,n$ and subsuming the Armstrong-Loehr-Warrington formula as a special case.

ABSTRACT

The aim of this work is to extend to a general $S_m imes S_n$-module context the Grossman-Bizley paradigm that allows the enumeration of Dyck paths in a $m imes n$-rectangle. We obtain an explicit formula for the the "bi-Frobenius" characteristic of what we call {\em interlaced} rectangular parking functions in an $m imes n$-rectangle. These are obtained by labelling the $n$ vertical steps of an $m imes n$-Dyck path by the numbers from $1$ to $n$, together with an independent labelling of its horizontal steps by integers from $1$ to $m$. Our formula specializes to give the Frobenius characteristic of the $S_n$-module of $m imes n$-parking functions in the general situation. Hence, it subsumes the result of Armstrong-Loehr-Warrington which furnishes such a formula for the special case when $m$ and $n$ are coprime integers.

Motivation & Objective

• To extend the Grossman-Bizley enumeration paradigm to the context of $\mathbb{S}_n \times \mathbb{S}_m$-modules acting on interlaced rectangular parking functions.
• To provide an explicit formula for the bi-Frobenius characteristic of the $\mathbb{S}_n \times \mathbb{S}_m$-module of $m\times n$-parking functions with independently labeled vertical and horizontal steps.
• To generalize the Armstrong-Loehr-Warrington formula for the Frobenius characteristic of $m\times n$-parking functions to the non-coprime case.
• To establish a connection between the enumeration of $m\times n$-Dyck paths and symmetric functions via a ring homomorphism $\Theta_{a,b}$.
• To explore the Schur-positivity of images of symmetric functions under $\Theta_{a,b}$, particularly for hook Schur functions and general partitions.

Proposed method

• Represents $m\times n$-Dyck paths as decreasing integer sequences $\alpha = a_1a_2\cdots a_n$, where $a_k$ gives the x-coordinate of the south-step at height $k$, constrained by the $(m,n)$-staircase $\delta_{m,n}$.
• Defines interlaced rectangular parking functions as $m\times n$-Dyck paths with vertical steps labeled by $1$ to $n$ and horizontal steps labeled by $1$ to $m$, independently.
• Introduces a ring homomorphism $\theta_{a,b}$ on the ring of symmetric functions $\Lambda$, defined by $\theta_{a,b}(p_k(\mathbf{x})) = \frac{1}{a+b}\binom{ak+bk}{ak}$, which generalizes the Grossman-Bizley formula.
• Constructs a homomorphism $\Theta_{a,b}$ from $\Lambda$ to operators on symmetric functions, isomorphic to the elliptic Hall algebra, to encode the bi-Frobenius characteristic.
• Uses the scalar product $\langle -, h_n(\mathbf{y}) \rangle$ to extract the $\mathbb{S}_n$-module Frobenius characteristic from the bi-Frobenius formula.
• Applies bijections between Dyck paths, compositions, and partitions to relate the Schur function expansion of the characteristic to multinomial coefficients and part multiplicities.

Experimental results

Research questions

• RQ1 How can the Grossman-Bizley enumeration formula for $m\times n$-Dyck paths be extended to the case where both vertical and horizontal steps are independently labeled?
• RQ2 What is the explicit formula for the bi-Frobenius characteristic of the $\mathbb{S}_n \times \mathbb{S}_m$-module of interlaced rectangular parking functions?
• RQ3 How does the formula specialize to recover the Armstrong-Loehr-Warrington result in the coprime case?
• RQ4 Are the images of symmetric functions under the homomorphism $\Theta_{a,b}$ Schur-positive, particularly for hook Schur functions?
• RQ5 Can the elliptic Hall algebra framework be extended to describe three-parameter generating functions involving area and dinv statistics on parking functions?

Key findings

• The paper provides an explicit formula (Theorem 4) for the bi-Frobenius characteristic of the $\mathbb{S}_n \times \mathbb{S}_m$-module of interlaced rectangular parking functions, expressed as a sum over compositions with part multiplicities.
• The formula specializes to the Armstrong-Loehr-Warrington formula when $m$ and $n$ are coprime, recovering the Frobenius characteristic of the $\mathbb{S}_n$-module of $m\times n$-parking functions.
• The homomorphism $\Theta_{a,b}$ maps symmetric functions to operators on $\Lambda$, and its action on $(-1)^j s_{(k|j)}(\mathbf{x})$ yields an $h$-positive expression, implying Schur-positivity.
• Extensive experiments suggest that $\Theta_{a,b}((-1)^{\iota(\mu)} s_\mu(\mathbf{x}))$ is Schur-positive for all partitions $\mu$, where $\iota(\mu)$ counts cells below the diagonal.
• The formula for the bi-Frobenius characteristic is derived by collecting compositions with identical part structures in the generating function, using multinomial coefficients $\binom{j}{\lambda}$.
• The generating function $\sum_{\alpha \in \mathscr{D}_{m,n}} q^{\mathrm{area}(\alpha)} \alpha(\mathbf{x};t) \alpha'(\mathbf{y};r)$, involving area and dinv statistics, specializes to the right-hand side of the main formula (4.1) upon setting parameters appropriately.