Polynomials from tilings of rectangles
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.
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