Chains of binary paths and shifted tableaux

TL;DR

This paper establishes a bijection between multichains of binary paths and shifted tableaux, enabling enumeration of certain chains and proving related formulas through new tableau bijections.

Contribution

It introduces a natural bijection connecting binary path chains and shifted tableaux, facilitating enumeration and proof of formulas for chains with specific properties.

Findings

Bijection between binary path chains and shifted tableaux

Enumeration formulas for chains with maximum length and small intervals

New bijections on shifted tableaux for proof techniques

Abstract

In this paper, a natural bijection between multichains of binary paths and shifted tableaux is presented, and it is used for the enumeration of the chains with maximum length from a given path $P$ to the maximum path $\mathbf{1}_{|P|}$. By mapping chains to shifted tableaux, the main formulas given in a recent paper by the authors for the enumeration of the $P - \mathbf{1}_{|P|}$ chains having only small intervals and minimum length are proved, using some new bijections on shifted tableaux.