Inverse descent statistic for Andr\'e and simsun permutations

TL;DR

This paper proves that the inverse descent statistic, along with descent and major index, is equidistributed across simsun, Andre I, and Andre II permutations, revealing new combinatorial symmetries related to Euler numbers.

Contribution

It establishes the equidistribution of the inverse descent statistic over three permutation classes sharing the same tree shape, extending Stanley's shuffle theorem.

Findings

Inverse descent is equidistributed over the three permutation sets.

New refinements of Stanley's shuffle theorem are developed.

The results connect permutation statistics with combinatorial tree structures.

Abstract

Simsun permutations, Andr\'e I permutations and Andr\'e II permutations are three combinatorial models for Euler numbers. It's known that the descent statistic is equidistributed over the set of Andr\'e I permutations and the set of simsun permutations. In this paper, we prove that the trivariate statistic (ides, des, maj), comprising the inverse descent, descent, and major index, are equidistributed over these three sets. This result is equivalent to showing that the inverse descent is equidistributed over these three sets that share the same tree shape. The proof of the equidistribution of the inverse descent over the set of Andr\'e I permutations and the set of Andr\'e II permutations with the same tree shape reduces to establishing new refinements of Stanley's shuffle theorem.