I am thinking of the polynomials $f_{s_1,s_2,..,s_k}$ as in the definition 4.3 in this paper http://annals.math.princeton.edu/wp-content/uploads/annals-v182-n1-p07-p.pdf
In the use of these polynomials in the main proof of the paper $s_i \in \{ 1,-1\}$ correspond to the sign assigned to the edge $i$.
- I want to know as to if something special is known about these polynomials $f_{s_1,s_2,..,s_k}$ if say the corresponding sequence of $k$ edges consist of having gone through sets of (and may be one partial) mutually disjoint perfect matchings of the $(n.n)$-bipartite graph being constructed.
Imagine arranging the edges of a $d$-regular $(n,n)$-bipartite graph as $e_{ij}$ where $1 \leq i \leq d$ denote an indexing over an arbitrary sequence of $d$ disjoint perfect matchings into which the edge set can be decomposed and $1 \leq j \leq n$ are arbitrary labels of the $n$ edges in the $i^{th}$ perfect matching. Now imagine going through the edges in the order $e_{11}, e_{12},..,e_{1n},e_{21},e_{22},...$ and so on. And after having gone through $k$ edges in such a sequence and having assigned $\pm 1$ signs to them say one looks at the corresponding $f_{s_1,s_2,..,s_k}$.
