Rank and crank moments for overpartitions
نویسندگان
چکیده
منابع مشابه
Rank and Crank Moments for Overpartitions
We study two types of crank moments and two types of rank moments for overpartitions. We show that the crank moments and their derivatives, along with certain linear combinations of the rank moments and their derivatives, can be written in terms of quasimodular forms. We then use this fact to prove exact relations involving the moments as well as congruence properties modulo 3, 5, and 7 for som...
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In 2003, Atkin and Garvan initiated the study of rank and crank moments for ordinary partitions. These moments satisfy a strict inequality. We prove that a strict inequality also holds for the first rank and crank moments of overpartitions and consider a new combinatorial interpretation in this setting.
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Moments of the partition rank and crank statistics have been studied for their connections to combinatorial objects such as Durfee symbols, as well as for their connections to harmonic Maass forms. This paper proves a conjecture due to two of the authors that re ned a conjecture of Garvan. Garvan's original conjecture states that the moments of the crank function are always larger than the mome...
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Andrews, Chan, and Kim recently introduced a modified definition of crank and rank moments for integer partitions that allows the study of both even and odd moments. In this paper, we prove the asymptotic behavior of these moments in all cases. Our main result states that the two families of moment functions are asymptotically equal, but the crank moments are also asymptotically larger than the...
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The rank of a partition is the largest part minus the number of parts. This statistic was introduced by Dyson [14], who observed empirically that it provided a combinatorial explanation for Ramanujan’s congruences p(5n + 4) ≡ 0 (mod 5) and p(7n + 5) ≡ 0 (mod 7). Here p(n) denotes the usual partition function. Specifically, Dyson conjectured that if N(s,m, n) denotes the number of partitions of ...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2009
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2008.10.017