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Recursions for Excedance number in some permutations groups

Eli Bagno·David Garber·Toufik Mansour·Robert Shwartz
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Abstract

The excedance number for S_n is known to have an Eulerian distribution. Nevertheless, the classical proof uses descents rather than excedances. We present a direct recursive proof which seems to be folklore and extend it to the colored permutation groups G_r,n. The generalized recursion yields some interesting connection to Stirling numbers of the second kind. We also show some logconcavity result concerning a variant of the excedance number. Finally, we show that the generating function of the excedance number defined on G_r,n is symmetric.

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Eli Bagno, David Garber, Toufik Mansour, Robert Shwartz. 2008-06-03. Recursions for Excedance number in some permutations groups. https://arxiv.org/abs/math/0702452

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