On the Algebraic Combinatorics of Injections and its Applications to Injection Codes

TL;DR

This paper explores the algebraic combinatorics of injections between finite sets, providing new formulas for spherical functions and applying these to improve bounds on injection codes.

Contribution

It introduces a novel combinatorial formula for spherical functions of a specific Gelfand pair and applies it to enhance linear programming bounds for injection codes.

Findings

New combinatorial formula for spherical functions

Improved Delsarte bounds on injection code sizes

Enhanced understanding of algebraic structures in combinatorics

Abstract

We consider the algebraic combinatorics of the set of injections from a $k$-element set to an $n$-element set. In particular, we give a new combinatorial formula for the spherical functions of the Gelfand pair $(S_k \times S_n, \text{diag}(S_k) \times S_{n-k})$. We use this combinatorial formula to give new Delsarte linear programming bounds on the size of codes over injections.