Geodesic
Durfee squares in compositions
Diskretnaya Matematika, Tome 30 (2018) no. 3, pp. 3-13

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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. 
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     title = {Durfee squares in compositions},
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     url = {http://geodesic.mathdoc.fr/item/DM_2018_30_3_a0/}
}
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M. Archibald; A. Blecher; Ch. Brennan; A. Knopfmacher; T. Mansour. Durfee squares in compositions. Diskretnaya Matematika, Tome 30 (2018) no. 3, pp. 3-13. http://geodesic.mathdoc.fr/item/DM_2018_30_3_a0/