Index of Seaweed Algebras and Integer Partitions

Abstract

The index of a Lie algebra is an important algebraic invariant.  In 2000, Vladimir Dergachev and Alexandre Kirillov  defined seaweed subalgebras of $\mathfrak{gl}_n$ (or $\mathfrak{sl}_n$) and provided a formula for the index of a seaweed algebra using a certain graph, a so called meander. In a recent paper, Vincent Coll, Andrew Mayers, and Nick Mayers defined a new statistic for partitions, namely  the index of a partition, which arises from seaweed Lie algebras of type A. At the end of their paper, they presented an interesting conjecture, which involves integer partitions into odd parts. Motivated by their work, in this paper, we exploit various index statistics and the index weight generating functions for partitions.  In particular, we examine their conjecture by considering the generating function for partitions into odd parts.  We will also reprove another result  from their paper using generating functions.