The combinatorics of $k$-marked Durfee symbols

Abstract

Abstract. George E. Andrews recently introduced k-marked Durfee symbols which are connected to moments of Dyson's rank. By these connections, Andrews deduced their generating functions and some combinatorial properties and left their purely combinatorial proofs as open problems. The primary goal of this article is to provide combinatorial proofs in answer to Andrews' request. We obtain a partition identity, which gives a relation between k-marked Durfee symbols and Durfee symbols by constructing bijections, and all identities on k-marked Durfee symbols given by Andrews could follow from this identity. In a similar manner, we also prove the identities due to Andrews on k-marked odd Durfee symbols combinatorially, which resemble ordinary k-marked Durfee symbols with a modified subscript and with odd numbers as entries.

Counting $k$-Marked Durfee Symbols

“…Also, Ji [8] proved the results in §3 along with many more open problems posed by Andrews in [3]. Ji's approach is essentially different from §1 and [4].…”

“…As stated in the introduction, Ji [8] solved many more open problems listed in [3] using generating function techniques. The aim of this paper was to use more elementary methods on conveniently chosen sets of objects.…”

On the Positive Moments of Ranks of Partitions

“…The proofs of the above two interpretations are based on the following partition identity given by Ji [8]. We shall adopt the notation D k (m 1 , m 2 , .…”

“…Andrews [1] proved the above theorem by using the k-fold generalization of Watson's q-analog of Whipple's theorem. Ji [8] gave a combinatorial proof of Theorem 1.2 by establishing a map from k-marked Durfee symbols to ordinary partitions. Kursungoz [9] provided another proof of Theorem 1.2 by using an alternative representation of k-marked Durfee symbols.…”

Distribution of the full rank in residue classes for odd moduli

“…Bringmann and Ben Kane, in addition, are preparing a paper studying 2marked Durfee symbols in general arithmetic progressions; that paper overlaps with this one for the modulus 5 case, for which we have both proved the same result independently. Kathy Ji has a paper in the arXiv [13] on several bijections for Durfee symbols and odd Durfee symbols, with their combinatorial implications. Other doctoral students of George Andrews are also studying Durfee symbols, particularly Kagan Kursungoz ([14]).…”