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Congruences for the cycle indicator of the symmetric group
Let $n$ be a positive integer and let $C_n$ be the cycle indicator of the symmetric group $S_n$. Carlitz proved that if $p$ is a prime, and if $r$ is a non negative integer, then we have the congruence $C_{r+np}\equiv (X_1^p-X_p)^nC_r \mod{pZ_p[X_1,\cdots,X_{r+np}]},$ where $Z_p$ is the ring of $p$-...
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Published in: | Communications in Mathematics 2023, Vol.31 (2023), Issue 1 |
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Main Authors: | , |
Format: | Article |
Language: | English |
Online Access: | Get full text |
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Summary: | Let $n$ be a positive integer and let $C_n$ be the cycle indicator of the
symmetric group $S_n$. Carlitz proved that if $p$ is a prime, and if $r$ is a
non negative integer, then we have the congruence $C_{r+np}\equiv
(X_1^p-X_p)^nC_r
\mod{pZ_p[X_1,\cdots,X_{r+np}]},$ where $Z_p$ is the ring of $p$-adic
integers. We prove that for $p\neq 2$, the preceding congruence holds modulo
$npZ_p[X_1,\cdots,X_{r+np}]$. This allows us to prove a Junod's conjecture for
Meixner polynomials. |
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ISSN: | 2336-1298 2336-1298 |
DOI: | 10.46298/cm.10391 |