Compute the full bivariate generating function for height 7 GSAWs (non‑probabilistic model)

Determine the full bivariate generating function f_7(x,y) = \sum_{W \in W_{(0,6)}(G_7)} x^{|W|} y^{\|W\|} that enumerates all growing self‑avoiding walks on the half‑infinite square‑lattice strip of height 7 (grid graph G_7 with vertex set N × {0,1,2,3,4,5,6} and edges between unit‑distance neighbors), starting at (0,6), counted jointly by number of edges (length) and horizontal displacement.

Background

The paper develops a finite‑state machine framework and transfer‑matrix method to compute rational generating functions for growing self‑avoiding walks (GSAWs) on half‑infinite strips of fixed height. The authors successfully obtain full bivariate generating functions for heights up to 6.

For height 7, the directed graph constructed has 954,791 states before minimization (2,311 after), and while the authors could compute the univariate specializations f_7(x,1) and f_7(1,y), they report not being able to obtain the full bivariate generating function f_7(x,y). Computing f_7(x,y) would extend exact enumerations to the next height and enable precise derivation of expectations, recurrences, and asymptotics for length and displacement at height 7.