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.
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