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 ̲ of p with p non negative parts 0 1 ... p-1 0 for which there exist permutations σ,τ of the set {0,...,p-1} such that p divides i=0 p-1 i σ(i) but does not divide i=0 p-1 i τ(i) . This happens if and only if the maximal number of equal parts of ̲ 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 ̲ de p à p parts positives ou nulles 0 1 ... p-1 0 pour lesquelles il existe des permutations σ,τ de l’ensemble {0,...,p-1} telles que p divise i=0 p-1 i σ(i) mais ne divise pas i=0 p-1 i τ(i) . Cela se produit si et seulement si le nombre maximal de parts égales de ̲ 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/

[A] Andrews G.E., The theory of partitions. Encyclopedia of Mathematics and its applications 2, Addison-Wesley, 1976. | MR 557013 | Zbl 0655.10001

[DM] Dixon J.D., Mortimer B., Permutation groups. Graduate Texts in Mathematics 163, Springer-Verlag, New York, 1996. | MR 1409812 | Zbl 0951.20001

[V] Vinatier S., Galois module structure in wealky ramified 3-extensions. Acta Arithm. 119 (2005), no. 2, 171–186. | MR 2167720 | Zbl 1075.11071