Derangements and the 𝑝-adic incomplete gamma function

O’Desky, Andrew; Richman, David

doi:10.1090/tran/8716

Derangements and the $p$-adic incomplete gamma function
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by Andrew O’Desky and David Harry Richman;

Trans. Amer. Math. Soc. 376 (2023), 1065-1087

DOI: https://doi.org/10.1090/tran/8716

Published electronically: December 1, 2022

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Abstract:

We introduce a $p$-adic analogue of the incomplete gamma function. We also introduce quantities ($m$-values) associated to a function on natural numbers and prove a new characterization of $p$-adic continuity for functions with $p$-integral $m$-values. Combinatorial interpretations for the integral values of the incomplete gamma function and functions with $m$-values zero or one are obtained, which show that these functions count derangements in generalized symmetric groups and permutations with restricted cycle lengths.

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Derangements and the $p$-adic incomplete gamma function
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Abstract: