Durfee squares in compositions

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 $n$.