The gerrymander sequence, or A348456

Abstract

Recently Kauers, Koutschan and Spahn announced a significant increase in the length of the so-called {\em gerrymander sequence}, given as A348456 in the OEIS, extending the sequence from 3 terms to 7 terms. We give a further extension to 11 terms, but more significantly prove that the coefficients grow as $\lambda^{4L^2},$ where $\lambda \approx 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 A348456 grow as $\lambda^{4L^2+dL+e} \cdot L^g,$ where $d=-8.08708 \pm 0.0002,$ $e \approx 7.69$ and $g = 0.75 \pm 0.01,$ with $g$ almost certainly $3/4$ exactly. We also have generated 26 terms in the related gerrymander polynomial (defined below), and have been able to predict the asymptotic behaviour with a satisfying degree of precision. Indeed, it behaves exactly as $L$ times the corresponding coefficient of the generalised gerrymander sequence. 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.