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

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{DM_2018_30_3_a0,
     author = {M. Archibald and A. Blecher and Ch. Brennan and A. Knopfmacher and T. Mansour},
     title = {Durfee squares in compositions},
     journal = {Diskretnaya Matematika},
     pages = {3--13},
     publisher = {mathdoc},
     volume = {30},
     number = {3},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2018_30_3_a0/}
}
TY  - JOUR
AU  - M. Archibald
AU  - A. Blecher
AU  - Ch. Brennan
AU  - A. Knopfmacher
AU  - T. Mansour
TI  - Durfee squares in compositions
JO  - Diskretnaya Matematika
PY  - 2018
SP  - 3
EP  - 13
VL  - 30
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DM_2018_30_3_a0/
LA  - ru
ID  - DM_2018_30_3_a0
ER  - 
%0 Journal Article
%A M. Archibald
%A A. Blecher
%A Ch. Brennan
%A A. Knopfmacher
%A T. Mansour
%T Durfee squares in compositions
%J Diskretnaya Matematika
%D 2018
%P 3-13
%V 30
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DM_2018_30_3_a0/
%G ru
%F DM_2018_30_3_a0
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/

[1] Andrews G. E., Eriksson K., Integer Partitions, Cambridge University Press, 2004 | MR | Zbl

[2] Flajolet P., Sedgewick R., Analytic Combinatorics, Cambridge University Press, 2009 | MR | Zbl

[3] Heubach S., Mansour T., Combinatorics of Compositions and Words, Chapman Hall/CRC, Taylor Francis Group, Boca Raton, London, New York, 2009 | MR

[4] Sloane N.J.A. https://oeis.org/