Unrefinable partitions into distinct parts in a normalizer chain

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 2^n 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 excludan