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Stéphane Vinatier
Permuting the partitions of a prime
Journal de théorie des nombres de Bordeaux, 21 no. 2 (2009), p. 455-465, doi: 10.5802/jtnb.682
Article PDF | Reviews MR 2541437 | Zbl pre05620662
Keywords: Partitions of a prime; sums of resolvents; multinomials.

Résumé - 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.

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