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

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 $(S_{k} \times S_{n}, diag(S_{k}) \times S_{n-k})$ . We use this combinatorial formula to give new Delsarte linear programming bounds on the size of codes over injections.