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

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.

18 pages