[Paper Review] Equidistributions of MAJ and STAT over pattern avoiding permutations

Joanna N. Chen|arXiv (Cornell University)|Jul 22, 2017

Advanced Combinatorial Mathematics8 references3 citations

TL;DR

This paper constructs a novel bijection from $S_n(213)$ to $S_n(231)$ that maps the pair of statistics $(maj, stat)$ to $(stat, maj)$, providing bijective proofs for several conjectures by Amini on equidistributions of Mahonian statistics over pattern-avoiding permutations. The key result establishes a direct symmetry between the major index and the $stat$ statistic across these permutation classes.

ABSTRACT

Babson and Steingr\'ımsson introduced generalized permutation patterns and showed that most of the Mahonian statistics in the literature can be expressed by the combination of generalized pattern functions. Particularly, they defined a new Mahonian statistic in terms of generalized pattern functions, which is denoted $stat$. Recently, Amini investigated the equidistributions of these Mahonian statistics over sets of pattern avoiding permutations. Moreover, he posed several conjectures. In this paper, we construct a bijection from $S_n(213)$ to $S_n(231)$, which maps the statistic $(maj,stat)$ to the statistic $(stat,maj)$. This allows us to give solutions to some of Amini's conjectures.

Motivation & Objective

• To resolve open conjectures by Amini concerning equidistributions of Mahonian statistics over pattern-avoiding permutations.
• To establish a bijective connection between $S_n(213)$ and $S_n(231)$ that interchanges the statistics $maj$ and $stat$.
• To provide a constructive, combinatorial proof of equidistribution results previously suggested via generating functions or enumeration.
• To extend the understanding of generalized permutation patterns and their role in Mahonian statistic equidistributions.
• To offer a new bijection that does not preserve the $adj$ statistic, distinguishing it from prior constructions by Burstein and Chen-Li.

Proposed method

• Constructs a map $\alpha: S_n(213) \to S_n(231)$ based on block decomposition of permutations by their first element $k$.
• Uses a recursive decomposition $\pi = k\pi'\pi''$ where $\pi'$ contains elements less than $k$ and $\pi''$ contains elements greater than $k$.
• Applies a transformation $\phi$ to reorder the blocks $\pi'$ and $\pi''$ using a reversal and shift mechanism to preserve descent structure.
• Employs the identity $maj(\pi) + stat(\pi) = (n+1)des(\pi) - (k-1)$ to relate $maj$ and $stat$ across the bijection.
• Leverages the fact that $des(\alpha(\pi)) = des(\pi)$ and $ides(\alpha(\pi)) = ides(\pi)$ to maintain descent symmetry.
• Derives a recurrence for the generating function $M_n(q,t) = \sum_{\pi \in S_n(213)} q^{maj(\pi)} t^{des(\pi)}$ via block decomposition and the bijection.

Experimental results

Research questions

• RQ1 Is the statistic $maj$ over $S_n(213)$ equidistributed with $stat$ over $S_n(231)$, as conjectured by Amini?
• RQ2 Can a bijective proof be constructed to show that $maj$ and $stat$ are interchanged under a map between $S_n(213)$ and $S_n(231)$?
• RQ3 Does the recurrence for $M_n(q,t)$ derived from the bijection match known generating function identities for $S_n(213)$?
• RQ4 How does the new bijection compare to prior constructions by Burstein and Chen-Li in terms of preserved statistics?
• RQ5 Can the $mak$ statistic over $S_n(132)$ be shown to equidistribute with $stat$ over $S_n(213)$ via this framework?

Key findings

• The map $\alpha: S_n(213) \to S_n(231)$ is a well-defined bijection that satisfies $maj(\pi) = stat(\alpha(\pi))$ and $stat(\pi) = maj(\alpha(\pi))$.
• The bijection preserves the descent number: $des(\alpha(\pi)) = des(\pi)$, and satisfies $ides(\alpha(\pi)) = ides(\pi)$.
• The recurrence $M_n(q,t) = M_{n-1}(q,t) + \sum_{k=1}^{n-1} q^k t M_{k-1}(q, qt) M_{n-k}(q, q^k t)$ holds for the generating function $M_n(q,t)$ over $S_n(213)$.
• The construction resolves Conjecture 1.2: $\sum_{\pi \in S_n(213)} q^{maj(\pi)} = \sum_{\pi \in S_n(231)} q^{stat(\pi)}$.
• The result implies that the joint distribution $(maj, stat)$ on $S_n(213)$ is equidistributed with $(stat, maj)$ on $S_n(231)$.
• The bijection is not a restriction of prior maps by Burstein or Chen-Li, indicating a distinct structural approach to equidistribution.