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- Title
PERIODICITY AND THE INDEX OF INTEGER PARTITIONS.
- Authors
Mayers, Nicholas W.
- Abstract
Seaweed subalgebras of sl(n), referred to here simply as seaweed algebras, are defined by an ordered pair of compositions of n. The index theory of these algebras has recently been used to define statistics on integer partitions. Given a partition λ, one natural choice of such statistic, denoted here by ind(λ), arises from taking the index of the seaweed algebra defined by the pair consisting of λ and its weight w(λ). Here, we examine ind restricted to partitions whose parts come from the set f1; 2; dg for d > 1 odd. In particular, we find (partial) formulae for (1) the number of such partitions with fixed ind value, and (2) when d = 3 (mod 4), the difference between the number of such partitions with ind even and the number with ind odd. Interestingly, we find that for (1) the corresponding sequences of values are eventually periodic and for (2) the sequences of values are periodic. Periodic phenomena are not new to integer partition statistics arising from the index theory of seaweed algebras. Consequently, the addition of the results established in this paper further suggest the existence of a more general theorem involving index, integer partitions, and periodicity.
- Subjects
ODD numbers; PARTITIONS (Mathematics); INTEGERS; ALGEBRA; MARINE algae
- Publication
Integers: Electronic Journal of Combinatorial Number Theory, 2023, Vol 23, p1
- ISSN
1553-1732
- Publication type
Article
- DOI
10.5281/zenodo.8214826