# Permuting the partitions of a prime

• [1] XLIM UMR 6172 CNRS / Université de Limoges Faculté des Sciences et Techniques 123 avenue Albert Thomas 87060 Limoges Cedex, France
• Volume: 21, Issue: 2, page 455-465
• ISSN: 1246-7405

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## Abstract

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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.

## How to cite

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Vinatier, Stéphane. "Permuting the partitions of a prime." Journal de Théorie des Nombres de Bordeaux 21.2 (2009): 455-465. <http://eudml.org/doc/10892>.

@article{Vinatier2009,
abstract = {Given an odd prime number $p$, we characterize the partitions $\underline\{\ell \}$ of $p$ with $p$non negative parts $\ell _0\ge \ell _1\ge \ldots \ge \ell _\{p-1\}\ge 0$ for which there exist permutations $\sigma ,\tau$ of the set $\lbrace 0,\ldots ,p-1\rbrace$ such that $p$ divides $\sum _\{i=0\}^\{p-1\}i\ell _\{\sigma (i)\}$ but does not divide $\sum _\{i=0\}^\{p-1\}i\ell _\{\tau (i)\}$. 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.},
affiliation = {XLIM UMR 6172 CNRS / Université de Limoges Faculté des Sciences et Techniques 123 avenue Albert Thomas 87060 Limoges Cedex, France},
author = {Vinatier, Stéphane},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Partitions of a prime; sums of resolvents; multinomials; permutation of partitions of a prime; sums of th powers of resolvents; Galois module structure problem},
language = {eng},
number = {2},
pages = {455-465},
publisher = {Université Bordeaux 1},
title = {Permuting the partitions of a prime},
url = {http://eudml.org/doc/10892},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Vinatier, Stéphane
TI - Permuting the partitions of a prime
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 2
SP - 455
EP - 465
AB - Given an odd prime number $p$, we characterize the partitions $\underline{\ell }$ of $p$ with $p$non negative parts $\ell _0\ge \ell _1\ge \ldots \ge \ell _{p-1}\ge 0$ for which there exist permutations $\sigma ,\tau$ of the set $\lbrace 0,\ldots ,p-1\rbrace$ such that $p$ divides $\sum _{i=0}^{p-1}i\ell _{\sigma (i)}$ but does not divide $\sum _{i=0}^{p-1}i\ell _{\tau (i)}$. 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.
LA - eng
KW - Partitions of a prime; sums of resolvents; multinomials; permutation of partitions of a prime; sums of th powers of resolvents; Galois module structure problem
UR - http://eudml.org/doc/10892
ER -

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