Let $c(n)$ be the number of Self avoiding walks (SAW) of length $n$ on an infinite lattice $L$. Are there any known non-geometric interpretations of $c(n)$?. For example, is there a number theoretic version of SAW's? We know for example that Catalan numbers count a myriad of things so perhaps SAW appear elsewhere? For reference, $c(n)$ for the usual 2-D integer lattice looks like 1,4,12,36,100,284,780,,... for $n=0,1,2,...

The online integer sequence library gives no results other than the number of SAWs.

Note: I'm not exactly interested in characterizations, such as the ones in this question. Something like the Hammersley and Welsh characterization in terms of excursion width is closer to what I'm looking for. In fact one sees partition functions crop up naturally, with theorems of Hardy and Ramanujan used for growth estimates.