The gerrymander sequence, or A348456

Abstract

The gerrymander sequence, gL, given as A348456 in the OEIS, counts the number of ways to bisect a 2L×2L chessboard into two polyominoes of equal area. Recently Kauers, Koutschan and Spahn announced a significant increase in the length of this sequence from 3 to 7 terms. We give a further extension to 11 terms, but more significantly prove that the coefficients grow as λ4L2, where λ≈1.7445498, and is equal to the corresponding quantity for self-avoiding walks crossing a square (WCAS), or self-avoiding polygons crossing a square (PCAS). These are, respectively, OEIS sequences A007764 and A333323. Thus we have established a close connection between these previously separate problems.We have also related the sub-dominant behaviour to that of WCAS and PCAS, allowing us to conjecture that the coefficients of the gerrymander sequence grow as gL∼λ4L2+2dL+e⋅(2L)h, where d=−4.04354±0.0001, e≈8 and h=0.75±0.01, with h almost certainly 3/4 exactly.We have also generated 26 terms of the related OEIS sequence A068416, which counts the number of ways to partition a L×L square into two connected components (not necessarily of equal area). We have thus been able to predict the asymptotic behaviour of this sequence with a satisfying degree of precision. Indeed, it behaves exactly as L times the corresponding coefficient of the generalised gerrymander sequence (defined below).The improved algorithm we give for counting these sequences is a variation of that which we recently developed for extending a number of sequences for SAWs and SAPs crossing a domain of the square or hexagonal lattices. It makes use of a minimal perfect hash function and in-place memory updating of the arrays for the counts of the number of paths.