Codes and shifted codes of partitions

TL;DR

This paper explores the combinatorial identities of partition codes, generalizing from Schur functions to Schur Q-functions, and applies these to classical rules like Littlewood-Richardson and Pieri.

Contribution

It demonstrates that certain identities are direct consequences of classical results and extends the approach to include Schur Q-functions using partition codes.

Findings

Identifies that the combinatorial identity is a consequence of classical results.

Generalizes the approach to Schur Q-functions.

Applies the framework to Littlewood-Richardson and Pieri Rules.

Abstract

In a recent paper, Carrell and Goulden found a combinatorial identity of the Bernstein operators that they then used to prove Bernstein's Theorem. We show that this identity is a straightforward consequence of the classical result. We also show how a similar approach using the codes of partitions can be generalized from Schur functions to also include Schur $Q$-functions and derive the combinatorial formulation for both cases. We then apply them by examining the Littlewood-Richardson and Pieri Rules.