Journal of Integer Sequences, Vol. 26 (2023), Article 23.1.4

Generating Functions for Straight Polyomino Tilings of Narrow Rectangles


Mudit Aggarwal and Samrith Ram
Indraprastha Institute of Information Technology Delhi
New Delhi 110020
India

Abstract:

Let m, k be fixed positive integers. Determining the generating function for the number of tilings of an m × n rectangle by k × 1 rectangles is a longstanding open problem to which the answer is only known in certain special cases. We give an explicit formula for this generating function in the case where m < 2k. This result is used to obtain the generating function for the number of tilings of an m × n × k box with k × k × 1 bricks.


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(Concerned with sequences A000045 A000930 A001835 A003269 A003520 A005708 A005709 A005710 A049086 A083066 A099390 A236576 A236579 A236580 A236581.)


Received November 15 2022; revised versions received December 9 2022; December 13 2022; December 20 2022. Published in Journal of Integer Sequences, January 1 2023.


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