Rank and Conjugation for a Second Frobenius Representation of an Overpartition

By Jeremy Lovejoy. 2005

106 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e8fbdfacc8d8a264bed9714904e4a5a655098379

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Citations in the corpus

Partial theta functions and mock modular forms as q-hypergeometric series (2011, background): “ith w6= qℓ (ℓ∈ Z\{0}), O2(w;q) := X n≥0 (−1)nq n( +1) 2 (wq,w−1q)n , 8 KATHRINBRINGMANN,AMANDAFOLSOM,ANDROBERTC.RHOADES denote the overpartition rank generating function (deﬁned in Proposition 1.1 of [20]). We point out that the overpartition function O2(w;q) is equal to McIntosh’s function K2(w;q),”
DYSON'S RANK, OVERPARTITIONS, AND WEAK MAASS FORMS (2007, background): “+2q+4q2 +8q3 +14q4 +··· . Here η(z) := q241 Q∞ n=1 (1−q n) is Dedekind’s eta function and we write q:= e2πiz. Moreover we denote by N(m,n) the number of overpartitions of nwith rank m. It is shown in [22] that O(u;q) := 1 + X∞ n=1 N(m,n)umqn = X∞ n=0 (−1)nq 1 2 n(n+1) (uq,q/u)n = (−q)∞ (q)∞ 1 +2 X ”
The spt-crank for overpartitions (2013, unknown): “k of π to be the largest part minus the number of parts of π. Let N(m,n) denote the number of overpartitions of n with Dyson rank equal to m. As in Proposition 1.1 and the proof of Proposition 3.2 of [23], the generating function for N(m,n) is given by X∞ n=0 X∞ m=−∞ N(m,n)zmqn = X∞ n=0 (−1;q)n qn(n”
RANK DIFFERENCES FOR OVERPARTITIONS (2007, methodology): “ess the base is something other than q. We should also remark that if the number of overpartitions of n with rank m is denoted by N(m,n), then conjugating Ferrers diagrams shows that N(m,n) = N(−m,n) [15]. Hence the values of s and t considered in Theorems 1.1 and 1.2 are suﬃcient to ﬁnd any rank di”
RANK AND CONGRUENCES FOR OVERPARTITION PAIRS (2008, unknown): “Using the above argument or the bijection in [13, 27 ], one can then recover the definition for the rank that we employ in the present paper.”
Rank and crank moments for overpartitions (2008, background): “Then the M2-rank of an overpartition λ is M2-rank(λ) := ⌈ ℓ(λ) 2 ⌉ − n(λ) + n(λo) − χ(λ), where χ(λ) = 1 if the largest part of λ is odd and non-overlined and χ(λ) = 0 otherwise.”