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 {0,1,2,3}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 2. We describe how profiles arise when enumerating tilings, and we prove that the number of profile strings of length m equals m 3m1 .

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Keywords
rectangular tilings, enumeration, induction
Mathematical Subject Classification
Primary: 05A19, 05B45
Milestones
Received: 4 June 2020
Revised: 16 August 2020
Accepted: 17 August 2020
Published: 5 December 2020

Communicated by Arthur T. Benjamin
Authors
Anthony Petrosino
Villanova University
Villanova, PA
United States
Alissa Schembor
Villanova University
Villanova, PA
United States
Kathryn Haymaker
Villanova University
Villanova, PA
United States