منابع مشابه
BG-Ranks and 2-Cores
We find the number of partitions of n whose BG-rank is j, in terms of pp(n), the number of pairs of partitions whose total number of cells is n, giving both bijective and generating function proofs. Next we find congruences mod 5 for pp(n), and then we use these to give a new proof of a refined system of congruences for p(n) that was found by Berkovich and Garvan.
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 کاملCombining Hook Length Formulas and Bg-ranks for Partitions via the Littlewood Decomposition
Recently, the first author has studied hook length formulas for partitions in a systematic manner. In the present paper we show that most of those hook length formulas can be generalized and include more variables via the Littlewood decomposition, which maps each partition to its t-core and t-quotient. In the case t = 2 we obtain new formulas by combining the hook lengths and BG-ranks introduce...
متن کاملSwitched symplectic graphs and their 2-ranks
We apply Godsil–McKay switching to the symplectic graphs over F2 with at least 63 vertices and prove that the 2-rank of (the adjacency matrix of) the graph increases after switching. This shows that the switched graph is a new strongly regular graphwith parameters (22ν−1, 22ν−1, 22ν−2, 22ν−2) and 2-rank 2ν+2 when ν ≥ 3. For the symplectic graph on 63 vertices we investigate repeated switching b...
متن کاملTalk: Level Raising Mod 2 and Arbitrary 2-selmer Ranks
rankE(Q) ? = ords=1 L(E/Q, s). This rank conjecture is known when ords=1 L(E/Q, s) ≤ 1 (Gross-Zagier, Kolyvagin, ...). The recent breakthrough of Bhargava-Skinner-W. Zhang shows that the rank conjecture holds for at least 66% of all elliptic curves over Q. Even more interestingly, BSD further predicts a refined BSD formula for the leading term of L(E/Q, s) at s = 1 in terms of various important...
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ژورنال
عنوان ژورنال: The Electronic Journal of Combinatorics
سال: 2006
ISSN: 1077-8926
DOI: 10.37236/1156