Permuting the partitions of a prime
Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 2, p. 455-465

Given an odd prime number $p$, we characterize the partitions $\underline{\ell }$ of $p$ with $p$ non negative parts ${\ell }_{0}\ge {\ell }_{1}\ge ...\ge {\ell }_{p-1}\ge 0$ for which there exist permutations $\sigma ,\tau$ of the set $\left\{0,...,p-1\right\}$ such that $p$ divides ${\sum }_{i=0}^{p-1}i{\ell }_{\sigma \left(i\right)}$ but does not divide ${\sum }_{i=0}^{p-1}i{\ell }_{\tau \left(i\right)}$. This happens if and only if the maximal number of equal parts of $\underline{\ell }$ is less than $p-2$. The question appeared when dealing with sums of $p$-th powers of resolvents, in order to solve a Galois module structure problem.

Étant donné un nombre premier $p$ impair, on caractérise les partitions $\underline{\ell }$ de $p$ à $p$ parts positives ou nulles ${\ell }_{0}\ge {\ell }_{1}\ge ...\ge {\ell }_{p-1}\ge 0$ pour lesquelles il existe des permutations $\sigma ,\tau$ de l’ensemble $\left\{0,...,p-1\right\}$ telles que $p$ divise ${\sum }_{i=0}^{p-1}i{\ell }_{\sigma \left(i\right)}$ mais ne divise pas ${\sum }_{i=0}^{p-1}i{\ell }_{\tau \left(i\right)}$. Cela se produit si et seulement si le nombre maximal de parts égales de $\underline{\ell }$ est strictement inférieur à $p-2$. Cette question est apparue en manipulant des sommes de puissances $p$-ièmes de résolvantes, en lien avec un problème de structure galoisienne.

DOI : https://doi.org/10.5802/jtnb.682
Keywords: Partitions of a prime; sums of resolvents; multinomials.
@article{JTNB_2009__21_2_455_0,
author = {Vinatier, St\'ephane},
title = {Permuting the partitions of a prime},
journal = {Journal de th\'eorie des nombres de Bordeaux},
publisher = {Universit\'e Bordeaux 1},
volume = {21},
number = {2},
year = {2009},
pages = {455-465},
doi = {10.5802/jtnb.682},
mrnumber = {2541437},
zbl = {pre05620662},
language = {en},
url = {http://www.numdam.org/item/JTNB_2009__21_2_455_0}
}

Vinatier, Stéphane. Permuting the partitions of a prime. Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 2, pp. 455-465. doi : 10.5802/jtnb.682. http://www.numdam.org/item/JTNB_2009__21_2_455_0/

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