Unrefinable partitions into distinct parts in a normalizer chain

TL;DR

This paper explores the relationship between the indices of a normalizer chain in symmetric groups and unrefinable partitions into distinct parts, revealing new combinatorial connections and properties.

Contribution

It establishes a novel link between the (n-1)-th index of a normalizer chain and the count of unrefinable partitions with specific minimal excludant conditions.

Findings

The (n-1)-th index relates to unrefinable partitions with minimal excludant constraints.

Growth of indices is connected to partitions into distinct parts.

Provides new combinatorial insights into normalizer chains and partition theory.

Abstract

In a recent paper on a study of the Sylow 2-subgroups of the symmetric group with 2^n elements it has been show 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. 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. 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.