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

Metric intersection problems in Cayley graphs and the Stirling recursion