This article is cited in 1 paper

Durfee squares in compositions

M. Archibald^a, A. Blecher^a, Ch. Brennan^a, A. Knopfmacher^a, T. Mansour<sup>b

a</sup> The John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand
^b Department of Mathematics, University of Haifa, Israel

Abstract: We study compositions (ordered partitions) of $n$. More particularly, our focus is on the bargraph representation of compositions which include or avoid squares of size $s \times s$. We also extend the definition of a Durfee square (studied in integer partitions) to be the largest square which lies on the base of the bargraph representation of a composition (i.e., is ‘grounded’). Via generating functions and asymptotic analysis, we consider compositions of $n$ whose Durfee squares are of size less than $s \times s$. This is followed by a section on the total and average number of grounded $s \times s$ squares. We then count the number of Durfee squares in compositions of $n$.

Keywords: composition, generating function, Durfee square.

UDC: 519.115

Received: 22.08.2017

DOI: 10.4213/dm1535

 English version: Discrete Mathematics and Applications, 2018, 28:6, 359–367

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