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Combinatorial Interpretations of Convolutions of the Catalan Numbers

Steven J. Tedford
Published Online: 2011-02-24 | DOI: https://doi.org/10.1515/integ.2011.003


We reintroduce an interpretation of the kth-fold self convolution of the Catalan numbers by showing that they count the number of words in symbols X and Y, where the total number of Y's is k more than the total number of X's, and at no time are there more Y's than k plus the number of X's. Using this, we exhibit some of the wide variety of combinatorial interpretations of the kth-fold self convolution of the Catalan numbers. Finally, we show how these numbers appear as the last column in a truncated Pascal's triangle.

Keywords.: Catalan Numbers; convolutions; Pascal's Triangle

About the article

Received: 2010-05-23

Revised: 2010-08-18

Accepted: 2010-09-27

Published Online: 2011-02-24

Published in Print: 2011-02-01

Citation Information: Integers, Volume 11, Issue 1, Pages 35–45, ISSN (Print) 1867-0652, DOI: https://doi.org/10.1515/integ.2011.003.

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