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Metric intersection problems in Cayley graphs and the Stirling recursion

In the symmetric group Sym(n) with n at least 5 let H be a conjugacy class of elements of order 2 and let Γbe the Cayley graph whose vertex set is the group G generated by H (so G is Sym(n) or Alt(n)) and whose edge set is determined by H. We are interested in the metric structure of this graph. In particular, for g\in G let B_{r}(g) be the metric ball in Γof radius r and centre g. We show that the intersection numbers Φ(Γ; r, g):=|\,B_{r}(e)\,\cap\,B_{r}(g)\,| are generalized Stirling functions in n and r. The results are motivated by the study of error graphs and related reconstruction problems.