Skew domino Schensted algorithm and sign-imbalance

TL;DR

This paper introduces a skew domino Schensted algorithm using growth diagrams, extending existing correspondences and properties, and provides new generating functions and formulas for sign-imbalance of skew shapes.

Contribution

It develops a skew domino Schensted correspondence and extends key properties and formulas related to sign-imbalance, generalizing previous results in the field.

Findings

Derived a new generating function for skew domino tableaux

Extended the color-to-spin property to skew shapes

Generalized sign-imbalance formulas for skew shapes

Abstract

Using growth diagrams, we define a skew domino Schensted correspondence which is a domino analogue of the skew Robinson-Schensted correspondence due to Sagan and Stanley. The color-to-spin property of Shimozono and White is extended. As an application, we give a simple generating function for the weighted sum of skew domino tableaux, which is a generalization of Stanley's sign-imbalance formula. The generating function gives a method to calculate the generalized sign-imbalance formula. We also extend Sj{\"o}strand's theorems on sign-imbalance of skew shapes.