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 × 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 × s. This is followed by a section on the total and average number of grounded s × s squares. We then count the number of Durfee squares in compositions of n.

The Finite Heine Transformation and Conjugate Durfee Squares

We introduce the idea of a conjugate Durfee square and use it to answer a combinatorial question regarding a finite form of the Heine transformation posed by G. E. Andrews in a recent paper.

Multipartitions, generalized Durfee squares and affine Lie algebra characters

We give some higher dimensional analogues of the Durfee square formula and point out their relation to dissections of multipartitions. We apply the results to write certain affine Lie algebra characters in terms of Universal Chiral Partition Functions.

Durfee Polynomials

Let ${\bf F}(n)$ be a family of partitions of $n$ and let ${\bf F}(n,d)$ denote the set of partitions in ${\bf F}(n)$ with Durfee square of size $d$. We define the Durfee polynomial of ${\bf F}(n)$ to be the polynomial $P_{{\bf F},n}= \sum |{\bf F}(n,d)|y^d$, where $ 0 \leq d \leq \lfloor \sqrt{n} \rfloor.$ The work in this paper is motivated by empirical evidence which suggests that for several families ${\bf F}$, all roots of the Durfee polynomial are real. Such a result would imply that the corresponding sequence of coefficients $\{|{\bf F}(n,d)|\}$ is log-concave and unimodal and that, over all partitions in ${\bf F}(n)$ for fixed $n$, the average size of the Durfee square, $a_{{\bf F}}(n)$, and the most likely size of the Durfee square, $m_{{\bf F}}(n)$, differ by less than 1. In this paper, we prove results in support of the conjecture that for the family of ordinary partitions, ${\bf P}(n)$, the Durfee polynomial has all roots real. Specifically, we find an asymptotic formula for $|{\bf P}(n,d)|$, deriving in the process a simple upper bound on the number of partitions of $n$ with at most $k$ parts which generalizes the upper bound of Erdös for $|{\bf P}(n)|$. We show that as $n$ tends to infinity, the sequence $\{|{\bf P}(n,d)|\},\ 1 \leq d \leq \sqrt{n},$ is asymptotically normal, unimodal, and log concave; in addition, formulas are found for $a_{{\bf P}}(n)$ and $m_{{\bf P}}(n)$ which differ asymptotically by at most 1. Experimental evidence also suggests that for several families ${\bf F}(n)$ which satisfy a recurrence of a certain form, $m_{{\bf F}}(n)$ grows as $c \sqrt{n}$, for an appropriate constant $c=c_{{\bf F}}$. We prove this under an assumption about the asymptotic form of $|{\bf F}(n,d)|$ and show how to produce, from recurrences for the $|{\bf F}(n,d)|$, analytical expressions for the constants $c_{{\bf F}}$ which agree numerically with the observed values.

Two Enumerative Proofs of an Identity of Jacobi

In (7), Wright gives an enumerative proof of an identity algebraically equivalent to that of Jacobi, namelyHere, and in the sequel, products run from 1 to oo and sums from - oo to oo unless otherwise indicated. We give here a simplified version of his argument by working directly with (1), the substitution leading to equation (3) of his paper being omitted. We then supply an alternative proof of (1) by means of a generalisation of the Durfee square concept utilising the rectangle of dimensions v by v + r for fixed r and maximal v contained in the Ferrers graph of a partition.