On the enumeration of k-omino towers

TL;DR

This paper introduces a new class of fixed polyominoes called k-omino towers, providing a recurrence relation and enumeration via hypergeometric functions, and establishing a connection to classical hypergeometric identities.

Contribution

It defines k-omino towers, derives a recurrence relation for their enumeration, and links the counting problem to hypergeometric functions and identities.

Findings

Enumeration of k-omino towers using hypergeometric functions

Recurrence relations for counting k-omino towers

Connection to classical hypergeometric identities

Abstract

We describe a class of fixed polyominoes called $k$-omino towers that are created by stacking rectangular blocks of size $k\times 1$ on a convex base composed of these same $k$-omino blocks. By applying a partition to the set of $k$-omino towers of fixed area $kn$, we give a recurrence on the $k$-omino towers therefore showing the set of $k$-omino towers is enumerated by a Gauss hypergeometric function. The proof in this case implies a more general hypergeometric identity with parameters similar to those given in a classical result of Kummer.