Generating functions for inscribed polyominoes

The goal of this paper is to propose a method to construct exact expressions and generating functions for the enumeration of general polyominoes up to translation with respect to area. We illustrate the proposed method with the construction of the generating functions for the polyominoes inscribed in a given b×kb×k rectangle with area min+1min+1 and min+2min+2. These polyominoes are not convex and we use geometric arguments to construct their generating functions. We use a statistic on polyominoes that we call the index and the multiplicative property of a diagonal product of polyominoes. From the main generating functions, we extract the generating functions and exact formulas for convex polyominoes of index one and two. The formulas obtained suggest an asymptotic evaluation for the number p(n)p(n) of polyominoes of area nn different from the usual evaluation based on numerical results.