Compute the full energetic‑model generating function for height 4

Determine the full trivariate generating function f^E_4(x,y,C) = \sum_{W \in W_{(0,3)}(G_4)} p^E_4(W) x^{|W|} y^{\|W\|}, where p^E_4(W) is the walk’s probability under the energetic model with attraction parameter C, for growing self‑avoiding walks on the half‑infinite square‑lattice strip of height 4.

Background

The energetic model biases steps toward neighbors adjacent to previously visited sites via an energy parameter C, requiring width‑4 frames and substantially larger state spaces. The authors computed specialized generating functions f^E_4(x,1,C) and f^E_4(1,y,C), but could not obtain the full trivariate generating function.

An explicit expression for f^E_4(x,y,C) would enable exact evaluation of expected length, displacement, and higher moments across attraction strengths, offering a rigorous counterpart to Monte Carlo findings on neighbor attraction in GSAWs.