Enumerating alternating matrix spaces over finite fields with explicit coordinates

TL;DR

This paper explores the enumeration of linear subspaces of alternating matrices over finite fields, drawing parallels to classical graph enumeration, and introduces q-analogues of known graph enumeration formulas.

Contribution

It develops a linear algebraic framework for counting alternating matrix subspaces, including q-analogues of classical graph enumeration formulas and their generating functions.

Findings

Derived q-analogues of Gilbert's formula for connected graphs.

Established q-analogues of Read's formula for c-colored graphs.

Created an analogue of Riddell's formula relating graph and connected graph generating functions.

Abstract

We initiate the study of enumerating linear subspaces of alternating matrices over finite fields with explicit coordinates. We postulate that this study can be viewed as a linear algebraic analogue of the classical topic of enumerating labelled graphs. To support this viewpoint, we present q-analogues of Gilbert's formula for enumerating connected graphs (Can. J. Math., 1956), and Read's formula for enumerating c-colored graphs (Can. J. Math., 1960). We also develop an analogue of Riddell's formula relating the exponential generating function of graphs with that of connected graphs (Riddell's PhD thesis, 1951), building on Eulerian generating functions developed by Srinivasan (Discrete Math., 2006).