Abstract
Combinatorics This work is concerned with the perimeter enumeration of column-convex polyominoes. We consider both the rectangular lattice and the hexagonal lattice. For the rectangular lattice, two formulas for the generating function (gf) already exist and, to all appearances, neither of them admits of a further simplification. We first rederive those two formulas (so as to make the paper self-contained), and then we enrich the rectangular lattice gf with some additional variables. That done, we make a change of variables, which (practically) produces the hexagonal lattice gf. This latter gf was first found by Lin and Wu in 1990. However, our present formula, in addition to having a simpler form, also allows a substantially easier Taylor series expansion. As to the methods, our one is descended from algebraic languages, whereas Lin and Wu used the Temperley methodology.
Highlights
The enumeration of polyominoes is a topic of great interest to physicists, chemists and combinatorialists alike
Column-convex polyominoes were first presented in Temperley’s 1956 paper [15], and directed animals were first presented in Dhar, Phani and Barma’s 1982 paper [5]. (It is plausible that directed animals appeared comparatively late because they do not submit to column-by-column approaches.)
As to column-convex polyominoes with hexagonal cells, their area generating function was first found by Klarner [9], and their perimeter generating function was first found by Lin and Wu [11]
Summary
The enumeration of polyominoes (by perimeter and/or area) is a topic of great interest to physicists, chemists and combinatorialists alike. Temperley [15] dealt with column-convex polyominoes with square cells He computed the area generating function and started computing the perimeter generating function (which we denote Gsq), but he had to stop because the formulae became too bulky. As to column-convex polyominoes with hexagonal cells, their area generating function was first found by Klarner [9], and their perimeter generating function (which we denote Ghex) was first found by Lin and Wu [11]. (The repeated derivation appears in the present one as an essential part.) In particular, in Section 4 we find K, the perimeter generating function for wall polyominoes.
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