The Number of Convex Polyominoes and the Generating Function of Jacobi Polynomials

Abstract

Lin and Chang gave a generating function of convex polyominoes with an $m+1$ by $n+1$ minimal bounding rectangle. Gessel showed that their result implies that the number of such polyominoes is $$\frac{m+n+mn}{m+n}\left(\genfrac{}{}{0ex}{}{2m+2n}{2m}\right)-\frac{2mn}{m+n}{\left(\genfrac{}{}{0ex}{}{m+n}{m}\right)}^2.$$ We show that this result can be derived from some binomial coefficients identities related to the generating function of Jacobi polynomials.

8 pages