Ask zeta functions associated with small graphs

\(W_{\Gamma}^{+}\) and \(W_{\Gamma}^{-}\)

In the paper

for each (finite simple) graph \(\Gamma\), bivariate rational generating functions \(W_{\Gamma}^{\pm}(q,t) \in \mathbf Q(q,t) \cap \mathbf Q(q)[\![t]\!]\) are defined. These generating functions encode average sizes of kernels in modules of (anti-)symmetric matrices with support constraints defined by \(\Gamma\). The package Zeta for SageMath includes an implementation of an algorithm for explicitly computing \(W_\Gamma(q,t)\). The following lists of the rational functions \(W_{\Gamma}^{\pm}(q,t)\) for all graphs with at most \(7\) vertices were computed using Zeta. In the lists, an “ask[±] zeta function” refers to an instance of \(W_{\Gamma}^{\pm}(q,t)\). These functions are provided as SageMath input and, for graphs with at most six vertices, also rendered using MathJax.

In the files above, graphs are specified using the graph6 format. Plots were created using SageMath.

\(W^\sharp_\Gamma\)

In the paper

we introduced the rational functions \(W^{\sharp}_{\Gamma}(q,t)\). Letting \(\hat\Gamma\) denote the reflexive closure of \(\Gamma\), we have \(W^\sharp_\Gamma(q,t) = W^\pm_{\hat\Gamma}(q,t)\). The following list contains the rational functions \(W_\Gamma^\sharp(q,t)\) for all graphs with at most \(7\) vertices.