Signed enumeration of ribbon tableaux

TL;DR

This paper extends the classical Schensted correspondence to ribbon tableaux with varying sizes, incorporating signs into the enumeration, and provides a combinatorial proof related to symmetric group character tables.

Contribution

It introduces a signed enumeration extension of the Schensted correspondence for ribbon tableaux of different sizes, expanding combinatorial and algebraic understanding.

Findings

Extended Fomin's growth diagram approach to signed ribbon tableaux

Provided a combinatorial proof for symmetric group character table column sums

Generalized classical correspondence to include signs and variable ribbon sizes

Abstract

We give an extension of the classical Schensted correspondence to the case of ribbon tableaux, where ribbons are allowed to be of different sizes. This is done by extending Fomin's growth diagram approach of the classical correspondence between permutations and pairs of standard tableaux of the same shape, in particular by allowing signs in the enumeration. As an application we give a combinatorial proof for the column sums of the character table of the symmetric group.