One construction for the Miura-ori flip-graph degree sequence

The flip graph of an origami crease pattern has the flat-foldable mountain-valley assignments as vertices, and an edge joins two of them that differ by a single face flip. A basic invariant of this graph is the degree sequence, which counts the vertices of each degree. On the $m\times n$ Miura-ori, this sequence is known as a bivariate polynomial only for small degrees, each count obtained by a separate argument whose casework grows with the degree. This paper gives one uniform construction that expresses, for every degree $d$, the number of degree-$d$ vertices as a single symmetric polynomial $p_d(m,n)$ for all sufficiently large $m,n$. Subject to a single degree bound, this polynomial has total degree $d-2$, growing for $d\ge5$ as an explicit multiple of $m^{d-2}+n^{d-2}$; the bound is proved here when the count splits into independent row and column factors, and open otherwise. The region is $m,n\ge\max(d-1,2)$; through $d=7$, the polynomials are computed in closed form and the bound is verified in every case. Below this region, the count departs from $p_d$ by a correction whose leading coefficient, through degree eleven, is $-4$ times a Baxter number. Each $p_d$ thus counts the Miura-ori's flat-foldable assignments admitting exactly $d$ single face flips.