#### Vol. 13, No. 5, 2020

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Counting profile strings from rectangular tilings

### Anthony Petrosino, Alissa Schembor and Kathryn Haymaker

Vol. 13 (2020), No. 5, 861–870
##### Abstract

We study the enumeration of profiles, which are outlines that occur when tiling a rectangular board with squares, dominoes, and trominoes. Profiles of length $m$ correspond to a special subset of the set ${\left\{0,1,2,3\right\}}^{m}$, called profile strings. Profiles and their corresponding strings first appeared in the enumeration of the tilings of rectangular $2×n$ and $3×n$ boards with squares, dominoes, and trominoes. Profiles also play a role in enumerating the tilings of an $m×n$ board for any fixed $m\ge 2$. We describe how profiles arise when enumerating tilings, and we prove that the number of profile strings of length $m$ equals $m\cdot {3}^{m-1}$.

##### Keywords
rectangular tilings, enumeration, induction
##### Mathematical Subject Classification
Primary: 05A19, 05B45