Journal of Integer Sequences, Vol. 27 (2024), Article 24.5.6

Shade in Compositions of Integers


Aubrey Blecher and Arnold Knopfmacher
The John Knopfmacher Centre for Applicable Analysis and Number Theory
School of Mathematics
University of the Witwatersrand
Private Bag 3, P O WITS 2050
South Africa

Michael Mays
School of Mathematical and Data Sciences
West Virginia University
Morgantown, WV 26506
USA

Abstract:

Integer compositions of n can be viewed as bargraphs, in which the ith part of the composition xi is given by the ith column of the bargraph with xi cells. The sun is at infinity in the northwest of our two-dimensional model, and each composition casts a shadow in accordance with the rules of physics. We find the number of unit squares in this shadow (but not being part of the composition) through a bivariate generating function tracking composition size and shadow.


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(Concerned with sequences A008930 A079500 A219282 A372768.)


Received May 15 2023; revised version received May 14 2024. Published in Journal of Integer Sequences, May 15 2024.


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