New Partition Statistic Bg-rank and Its Applications
نویسندگان
چکیده
Let π denote partition into parts λ1 ≥ λ2 ≥ λ3 . . .. In [2] we defined BG-rank(π) as
منابع مشابه
On the Andrews-stanley Refinement of Ramanujan’s Partition Congruence modulo 5 and Generalizations
In a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic srank(π) = O(π)−O(π′), where O(π) denotes the number of odd parts of the partition π and π′ is the conjugate of π. In a forthcoming paper, Andrews proved the following refinement of Ramanujan’s partition congruence mod 5: p0(5n+ 4) ≡ p2(5n+ 4) ≡ 0 (mod 5), p(n) = p0(n) + p2(n), where pi(n)...
متن کاملThe Bg-rank of a Partition and Its Applications
Let π denote a partition into parts λ1 ≥ λ2 ≥ λ3 . . .. In a 2006 paper we defined BG-rank(π) as
متن کاملNew identities for 7-cores with prescribed BG-rank
Let π be a partition. BG-rank(π) is defined as an alternating sum of parities of parts of π [1]. In [2], Berkovich and Garvan found theta series representations for the t-core generating functions P n≥0 at,j(n)q , where at,j(n) denotes the number of t-cores of n with BG-rank = j. In addition, they found positive eta-quotient representations for odd t-core generating functions with extreme value...
متن کاملDyson’s Rank, Overpartitions, and Weak Maass Forms
In a series of papers the first author and Ono connected the rank, a partition statistic introduced by Dyson, to weak Maass forms, a new class of functions related to modular forms. Naturally it is of wide interest to find other explicit examples of Maass forms. Here we construct a new infinite family of such forms, arising from overpartitions. As applications we obtain combinatorial decomposit...
متن کاملRank Differences for Overpartitions
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 ...
متن کامل