Polynomials from tilings of rectangles

TL;DR

This paper investigates tilings of rectangles with squares and Ferrers-shaped tiles, deriving generating functions, analyzing polynomial properties, and connecting to known sequences, with implications for polyomino tilings.

Contribution

It introduces new results on the real-rootedness and interlacing properties of tiling polynomials for Ferrers shapes, expanding understanding of tiling enumeration and polynomial behavior.

Findings

Tiling polynomials for two-column Ferrers shapes are real-rooted.

These polynomials form interlacing sequences.

Connections established with several OEIS sequences.

Abstract

We study tilings of rectangular boards using unit squares together with a single type of big tile shaped as a Ferrers diagram. We derive generating functions for these tilings, prove real-rootedness and interlacing properties of associated independence polynomials, and establish connections with several sequences in the OEIS. Our results touch on tilings involving L-shaped polyominoes, fault-free tilings, and cylindric variants. We prove that tiling polynomials for two-column Ferrers shapes are real-rooted and form interlacing sequences.