Equidistribution of (X,Y)-descents, (X,Y)-adjacent pairs, and (X,Y)-place-value pairs on permutations

TL;DR

This paper studies the distribution of specific permutation statistics related to element positions and values, revealing equidistribution phenomena, deriving distribution formulas, and providing combinatorial proofs of identities, thus generalizing previous results.

Contribution

It introduces a unified framework for analyzing (X,Y)-descent, adjacency, and place-value pairs, establishing their equidistribution and deriving new distribution formulas.

Findings

Certain permutation statistics are equidistributed for specific X and Y.

Distribution formulas for (X,Y)-place-value pairs are derived.

Provides combinatorial proofs of notable identities.

Abstract

An $(X,Y)$-descent in a permutation is a pair of adjacent elements such that the first element is from $X$, the second element is from $Y$, and the first element is greater than the second one. An $(X,Y)$-adjacency in a permutation is a pair of adjacent elements such that the first one is from $X$ and the second one is from $Y$. An $(X,Y)$-place-value pair in a permutation is an element $y$ in position $x$, such that $y$ is in $Y$ and $x$ is in $X$. It turns out, that for certain choices of $X$ and $Y$ some of the three statistics above become equidistributed. Moreover, it is easy to derive the distribution formula for $(X,Y)$-place-value pairs thus providing distribution for other statistics under consideration too. This generalizes some results in the literature. As a result of our considerations, we get combinatorial proofs of several remarkable identities. We also conjecture existence of a bijection between two objects in question preserving a certain statistic.