A permutation code preserving a double Eulerian bistatistic

TL;DR

This paper introduces a permutation code that provides a bijective proof of Visontai's conjecture, establishing the equidistribution of certain permutation and subexcedant sequence statistics related to the double Eulerian polynomial.

Contribution

It defines a new permutation code and proves the equidistribution of five-tuples of set-valued statistics, generalizing and providing a bijective proof of Visontai's conjecture.

Findings

Established a bijective proof of Visontai's conjecture.

Proved the equidistribution of specific set-valued statistics.

Connected permutation statistics with subexcedant sequence statistics through a new code.

Abstract

Visontai conjectured in 2013 that the joint distribution of ascent and distinct nonzero value numbers on the set of subexcedant sequences is the same as that of descent and inverse descent numbers on the set of permutations. This conjecture has been proved by Aas in 2014, and the generating function of the corresponding bistatistics is the double Eulerian polynomial. Among the techniques used by Aas are the M\"obius inversion formula and isomorphism of labeled rooted trees. In this paper we define a permutation code (that is, a bijection between permutations and subexcedant sequences) and show the more general result that two $5$-tuples of set-valued statistics on the set of permutations and on the set of subexcedant sequences, respectively, are equidistributed. In particular, these results give a bijective proof of Visontai's conjecture.