On compositions associated to seasweed subalgebras of sl(n)

TL;DR

This paper provides formulas and generating functions for counting seaweed subalgebras of sl(n) with a given index, based on parametrization by pairs of compositions, and discusses extending this framework to other classical Lie algebras.

Contribution

It introduces explicit formulas and generating functions for counting seaweed subalgebras of sl(n) by index, advancing combinatorial understanding of these structures.

Findings

Closed-form formulas for C(n,k) are derived.

Generating functions for C(n,k) are established.

Framework for similar analysis in other classical families is proposed.

Abstract

A standard seaweed subalgebra of $A_{n-1}=\mathfrak{sl}(n)$ may be parametrized by a pair of compositions of the positive integer $n$. For all $n$ and certain $k(n)$, we provide closed-form formulas and the generating functions for $C(n,k)$ -- the number of parametrizing pairs which yield a seaweed subalgebra of $\mathfrak{sl}(n)$ of index $k$. Our analysis sets the framework for addressing similar questions in the other classical families.