On some combinatorial sequences associated to invariant theory

TL;DR

This paper explores sequences derived from invariant theory of Lie groups, revealing their combinatorial, recursive, and hypergeometric properties, and connecting them through group inclusion and boundary conditions.

Contribution

It introduces new sequences from $G_2$ and $SL(3)$ invariant theory, analyzes their properties, and links them via binomial transforms and group branching rules.

Findings

Sequences are P-recursive with explicit recurrence relations.

Associated differential operators are third order and solvable in hypergeometric functions.

Sequences relate through group inclusion and boundary conditions.

Abstract

We study the enumerative and analytic properties of some sequences constructed using tensor invariant theory. The octant sequences are constructed from the exceptional Lie group $G_2$ and the quadrant sequences from the special linear group $SL(3)$. In each case we show that the corresponding sequences are related by binomial transforms. The first three octant sequences and the first four quadrant sequences are listed in the On-Line Encyclopedia of Integer Sequences (OEIS). These sequences all have interpretations as enumerating two-dimensional lattice walks but for the octant sequences the boundary conditions are unconventional. These sequences are all P-recursive and we give the corresponding recurrence relations. In all cases the associated differential operators are of third order and have the remarkable property that they can be solved to give closed formulae for the ordinary generating functions in terms of classical Gaussian hypergeometric functions. Moreover, we show that the octant sequences and the quadrant sequences are related by the branching rules for the inclusion of $SL(3)$ in $G_2$.