[Paper Review] Mahonian STAT on rearrangement class of words

Shishuo Fu, Ting Hua+1 more|arXiv (Cornell University)|Aug 26, 2017

semigroups and automata theory8 references3 citations

TL;DR

This paper constructs an explicit involution on the rearrangement class of a word that establishes a joint equidistribution of the sextuple statistics (des, Id, F, MAJ, STAT) and (des, Id, F, STAT, MAJ), proving that STAT and MAJ are equidistributed when swapped under this involution. The method combines Burstein's involution, Foata-Sch"utzenberger's involution, and the RSK algorithm to preserve key statistics while swapping MAJ and STAT, extending prior equidistribution results from permutations to rearrangement classes of words.

ABSTRACT

In 2000, Babson and Steingrímsson generalized the notion of permutation patterns to the so-called vincular patterns, and they showed that many Mahonian statistics can be expressed as sums of vincular pattern occurrence statistics. STAT is one of such Mahonian statistics discoverd by them. In 2016, Kitaev and the third author introduced a words analogue of STAT and proved a joint equidistribution result involving two sextuple statistics on the whole set of words with fixed length and alphabet. Moreover, their computer experiments hinted at a finer involution on $R(w)$, the rearrangement class of a given word $w$. We construct such an involution in this paper, which yields a comparable joint equidistribution between two sextuple statistics over $R(w)$. Our involution builds on Burstein's involution and Foata-Schützenberger's involution that utilizes the celebrated RSK algorithm.

Motivation & Objective

• To extend the joint equidistribution of Mahonian statistics (MAJ, STAT) from permutations to rearrangement classes of words.
• To resolve a conjecture from Kitaev and Vajnovszki (2016) by constructing an explicit involution on R(w), the rearrangement class of a word w.
• To preserve the inverse descent set Id and other key statistics while swapping MAJ and STAT.
• To provide a refined equidistribution result over R(w) that replaces the statistic Adj with Id, as Adj is not well-defined in rearrangement classes.
• To generalize prior equidistribution theorems on permutations and words to the setting of rearrangement classes, using RSK-based involutions.

Proposed method

• Constructs an involution φ on permutations that preserves des, Id, and F, and swaps MAJ and STAT.
• Uses the RSK algorithm to define a map j on sequences, which is applied to the top and bottom parts of a permutation to define the image under φ.
• Applies the coding map c to translate between words and permutations, enabling the involution to be lifted to rearrangement classes R(w).
• Employs the inverse RSK algorithm to reconstruct permutations from standard Young tableaux pairs, ensuring bijectivity and invertibility.
• Relies on the fact that φ is an involution (φ² = 1), which ensures the equidistribution of the swapped statistics.
• Verifies that the involution preserves the number of fixed points F and the inverse descent set Id, crucial for the equidistribution result.

Experimental results

Research questions

• RQ1 Can a joint equidistribution of (des, Id, F, MAJ, STAT) and (des, Id, F, STAT, MAJ) be established over the rearrangement class R(w) of a word w, analogous to known results on permutations and full word sets?
• RQ2 Is there an explicit involution on R(w) that swaps MAJ and STAT while preserving des, Id, and F?
• RQ3 How can the RSK algorithm be used to construct such an involution in the context of rearrangement classes of words?
• RQ4 Can the equidistribution result be extended to k-extensions of the alphabet, where letters are partitioned into small and large, and statistics are redefined accordingly?
• RQ5 What are the implications of this equidistribution for the classification of Euler-Mahonian statistics on rearrangement classes?

Key findings

• The paper constructs an explicit involution φ on the symmetric group S_n that preserves des, Id, and F, and satisfies MAJ(φ(π)) = STAT(π) and STAT(φ(π)) = MAJ(π), proving joint equidistribution of (des, Id, F, MAJ, STAT) and (des, Id, F, STAT, MAJ).
• The involution φ is lifted to the rearrangement class R(w) via the coding map c, yielding an involution φ_{R(w)} on R(w) that preserves des, Id, and F while swapping MAJ and STAT.
• The key result is Corollary 1.5, which establishes that (des, Id, F, MAJ, STAT) and (des, Id, F, STAT, MAJ) have the same joint distribution on R(w) for any word w.
• The involution preserves the inverse descent set Id, which uniquely determines both des and ides, leading to Corollary 1.6: the joint equidistribution holds with IMAJ replacing Adj.
• An example is provided where v = 4 3 4 4 2 1 6 5 1 in R(w) with (des, Id, F, MAJ, STAT) = (5, {2,3,4,8}, 4, 25, 21), and φ_{R(w)}(v) = 4 1 6 4 3 2 4 5 1, which yields (des, Id, F, STAT, MAJ) = (5, {2,3,4,8}, 4, 25, 21), confirming the swap.
• The result confirms that (des, STAT) forms a new Euler-Mahonian pair over rearrangement classes, extending the known families of such statistics.