A natural generalization of Balanced Tableaux

TL;DR

This paper introduces a new concept called 'type' of a tableau, unifying various tableau families and providing new combinatorial insights into permutation decompositions and tableau enumeration.

Contribution

It generalizes balanced and standard tableaux through the notion of 'type', offering new proofs and expanding the combinatorial framework of tableaux.

Findings

Balanced and standard tableaux are equinumerous.

New families of tableaux with similar properties are identified.

A unified framework for various tableau types is established.

Abstract

We introduce the notion of "type" of a tableau, that allows us to define new families of tableaux including both balanced and standard Young tableaux. We use these new objects to describe the set of reduced decompositions of any permutation. We then generalize the work of Fomin \emph{et al.} by giving, among other things, a new proof of the fact that balanced and standard tableaux are equinumerous, and by exhibiting many new families of tableaux having similar combinatorial properties to those of balanced tableaux.