Partitions, Durfee symbols, and the Atkin–Garvan moments of ranks
نویسندگان
چکیده
منابع مشابه
On the Positive Moments of Ranks of Partitions
By introducing k-marked Durfee symbols, Andrews found a combinatorial interpretation of 2k-th symmetrized moment η2k(n) of ranks of partitions of n in terms of (k + 1)-marked Durfee symbols of n. In this paper, we consider the k-th symmetrized positive moment η̄k(n) of ranks of partitions of n which is defined as the truncated sum over positive ranks of partitions of n. As combintorial interpret...
متن کاملCounting k-Marked Durfee Symbols
An alternative characterization of k-marked Durfee symbols defined by Andrews is given. Some identities involving generating functions of k-marked Durfee symbols are proven combinatorially by considering the symbols not individually, but in equivalence classes. Also, a related binomial coefficient identity is obtained in the course. A partition λ of a positive integer n is a nonincreasing seque...
متن کاملAutomorphic Properties of Generating Functions for Generalized Rank Moments and Durfee Symbols
Abstract. We define two-parameter generalizations of two combinatorial constructions of Andrews: the kth symmetrized rank moment and the k-marked Durfee symbol. We prove that three specializations of the associated generating functions are so-called quasimock theta functions, while a fourth specialization gives quasimodular forms. We then define a two-parameter generalization of Andrews’ smalle...
متن کاملThe combinatorics of k - marked Durfee symbols Kathy
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 relation between k-marked ...
متن کاملThe odd moments of ranks and cranks
have motivated much research. Here, p(n) denotes the number of partitions of n. In particular, toward a combinatorial explanation of the above congruences many partition statistics have been studied. Among them, the rank suggested by F. Dyson [6] and the crank suggested by the first author and F.G. Garvan [2] have proven successful and their own properties have been extensively studied. Here, t...
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ژورنال
عنوان ژورنال: Inventiones mathematicae
سال: 2007
ISSN: 0020-9910,1432-1297
DOI: 10.1007/s00222-007-0043-4