A bijective enumeration of $3$-strip tableaux

TL;DR

This paper provides a bijective proof for counting 3-strip tableaux, generalizing classical permutation enumeration results and employing decomposition techniques for a direct combinatorial enumeration.

Contribution

It introduces a novel bijective approach to enumerate 3-strip tableaux, extending previous identities and applying decomposition methods to permutations.

Findings

Established a bijective enumeration method for 3-strip tableaux

Connected 3-strip tableaux enumeration with permutation decompositions

Extended classical combinatorial identities to new tableau classes

Abstract

Baryshnikov and Romik derived the combinatorial identities for the numbers of the $m$-strip tableaux. This generalized the classical Andr\'e's theorem for the number of up-down permutations. They asked for a bijective proof for the enumeration of $3$-strip tableaux. In this paper we will provide such a bijective proof. First we count the $3$-strip tableaux by decomposition. Secondly we will apply this "decomposition" idea on the up-down permutations and down-up permutations to enumerate the $3$-strip tableaux bijectively.