[Paper Review] Major Index Over Descent for Pattern-avoiding Permutations

William J. Keith|arXiv (Cornell University)|Jul 5, 2017

Advanced Combinatorial Mathematics3 references3 citations

TL;DR

This paper provides a formula for the distribution of the major index over 321-avoiding permutations with a fixed number of descents, proving unimodality for the case of two descents. It refines the $q$-analogue of the Frame-Robinson-Thrall hook-length formula for two-rowed partitions and establishes a connection to standard Young tableaux, offering a pathway toward proving the long-standing conjecture on unimodal major index distribution in 321-avoiding permutations.

ABSTRACT

An open conjecture in pattern avoidance theory is that the distribution of the major index among 321-avoiding permutations is distributed unimodally. We construct a formula for this distribution, and in the case of 2 descents prove unimodality, with unimodality for 3 through 5 descents likely being little more complicated. The formula refines the $q$-analogue of the Frame-Robinson-Thrall hooklength formula for two-rowed partitions, and in the latter part of the paper we discuss another theorem of the same type, and further exploration toward this question. We also give observations on the analogous behaviors for other permutation patterns of length 3.

Motivation & Objective

• To resolve the open conjecture that the major index distribution over 321-avoiding permutations with a fixed number of descents is unimodal.
• To derive a closed-form formula for the generating function of the major index over standard Young tableaux of shape $(n-k,k)$ with $i$ descents.
• To establish unimodality for the case of two descents using combinatorial and algebraic techniques.
• To explore connections between pattern-avoiding permutations, standard Young tableaux, and $q$-analogue hook-length formulas.
• To lay groundwork for generalizing unimodality proofs to higher descent numbers.

Proposed method

• Derives a closed-form formula for $f_{(n-k,k),i}(q)$, the generating function of the major index over standard Young tableaux of shape $(n-k,k)$ with $i$ descents.
• Uses the Frame-Robinson-Thrall hook-length formula as a foundation, refining it via $q$-binomial coefficients and $q$-integers.
• Applies recurrence relations and bijections between tableaux of different shapes to prove base cases, particularly for $i=2$.
• Employs symbolic computation to verify complex $q$-binomial identities arising from recurrence substitutions.
• Establishes a bijection between tableaux of shape $(m,k,1)$ with 2 descents and those of shape $(m+1,k+1)$ with 2 descents that increases the major index by 1.
• Leverages the Robinson-Schensted correspondence and known results on $q$-symmetric and unimodal polynomials to support conjectures.

Experimental results

Research questions

• RQ1 Is the distribution of the major index over 321-avoiding permutations with $k$ descents unimodal for all $k$?
• RQ2 Can a closed-form formula be derived for the generating function of the major index over standard Young tableaux of shape $(n-k,k)$ with $i$ descents?
• RQ3 Does the polynomial $f_{(n-k,k),i}(q)$ exhibit symmetry and unimodality for all $n,k,i$?
• RQ4 What is the structural connection between pattern-avoiding permutations and the Stanley hook-length formula for standard Young tableaux?
• RQ5 Can the recurrence method used for two-rowed and three-rowed shapes be generalized to arbitrary partitions?

Key findings

• The paper derives a closed-form formula: $f_{(n-k,k),i}(q)=\frac{q^{k+i^{2}-i}(1-q^{n-2k+1})}{1-q^{i}}\left[{{k-1}\atop{i-1}}\right]_{q}\left[{{n-k}\atop{i-1}}\right]_{q}$, which gives the generating function for the major index over standard Young tableaux of shape $(n-k,k)$ with $i$ descents.
• For $i=2$, the polynomial $f_{(n-k,k),2}(q)$ is symmetric and unimodal with central term at $n$, which implies that $A_{n,2}(q)$ is unimodal.
• The case $i=1$ is unimodal by a direct combinatorial argument: $A_{n,1}(q) = (1+q)^n - \sum_{j=0}^{n} q^j$.
• Empirical verification supports that $f_{(n-k,k),i}(q)$ is symmetric and unimodal for $n \leq 30$, suggesting the full conjecture may hold.
• The formula for three-rowed shapes $(m,k,1)$ is derived as $f_{(m,k,1),i}(q)=q^{k+i^{2}-2i+2}\frac{(1-q^{m-k+1})(1-q^{i-1})}{(1-q^{i})(1-q)}\left[{k\atop{i-1}}\right]_{q}\left[{{m+1}\atop{i-1}}\right]_{q}$, verified via recurrence and bijection.
• The recurrence method is limited to small $i$ and specific shapes, suggesting the need for deeper symmetric function theory to generalize beyond two-rowed and three-rowed cases.