The Number of Convex Polyominoes with Given Height and Width

TL;DR

This paper provides a new combinatorial proof for counting convex polyominoes with fixed dimensions, introduces a subclass with specific corner inclusion, and discusses sampling methods and path intersection statistics.

Contribution

It presents a novel combinatorial proof for convex polyomino enumeration and extends the analysis to directed polyominoes and path intersection moments.

Findings

New combinatorial proof for convex polyomino count

Enumeration of directed polyominoes with corner constraints

Calculation of moments for path intersection points

Abstract

We give a new combinatorial proof for the number of convex polyominoes whose minimum enclosing rectangle has given dimensions. We also count the subclass of these polyominoes that contain the lower left corner of the enclosing rectangle (directed polyominoes). We indicate how to sample random polyominoes in these classes. As a side result, we calculate the first and second moments of the number of common points of two monotone lattice paths between two given points.