Obtain the specializations of the uniform‑model generating functions for heights 6 and 7

Ascertain the univariate specializations by length and by displacement for the uniform‑probability model of growing self‑avoiding walks on half‑infinite strips of heights 6 and 7, specifically compute f^U_6(x,1), f^U_6(1,y), f^U_7(x,1), and f^U_7(1,y), where f^U_h(x,y) = \sum_{W \in W_{(0,h-1)}(G_h)} p^U_h(W) x^{|W|} y^{\|W\|} and p^U_h(W) is the walk’s probability under uniform neighbor choice.

Background

The uniform model assigns equal probability to each unvisited neighbor at every step, producing a probability‑weighted generating function for GSAWs. The authors’ finite‑state approach yields full or specialized generating functions for several heights, enabling exact expected values and variances.

However, for heights 6 and 7, despite constructing large minimized finite‑state machines (7,294 and 53,808 states, respectively), the authors could not compute either specialization (by length or by displacement). Obtaining these specializations would provide exact probabilistic enumerations and moments for these cases.