Unrefinable partitions into distinct parts in a normalizer chain | [Discrete Mathematics Letters • 2022]

Author(s):

1. Riccardo Aragona: Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Universit`a degli Studi dell’Aquila, Italy

2. Roberto Civino: Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Universit`a degli Studi dell’Aquila, Italy

3. Norberto Gavioli: Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Universit`a degli Studi dell’Aquila, Italy

4. Carlo Maria Scoppola: Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Universit`a degli Studi dell’Aquila, Italy

Abstract:

Unrefinable partitions into distinct parts are those in which no part x can be replaced with integers whose sum is x obtaining a new partition into distinct parts. Such a relationship between the parts does not seem to be much investigated and consequently very little is known on general properties of unrefinable partitions. However, they play a role in the combinatorial nature of a certain chain of subgroups. More precisely, in a recent paper on a study of the Sylow 2-subgroups of the symmetric group with 2n elements it has been shown that the growth of the first (n 2) consecutive indices of a certain normalizer chain is linked to the sequence of partitions of integers into distinct parts. We prove here that the (n 1)-th index of the previously mentioned chain is related to the number of unrefinable partitions into distinct parts satisfying a condition on the minimal excludant.

Page(s): 72-77

DOI: 10.47443/dml.2021.0109

Published: Journal: Discrete Mathematics Letters, Volume: 8, Issue: 0, Year: 2022

Keywords:

unrefinable partitions , Sylow 2subgroups , symmetric group on 2n elements , partitions into distinct parts , minimal excludant

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