On the explicit construction of higher deformations of partition statistics

Bringmann, Kathrin

doi:10.1215/00127094-2008-035

Cited by 87 publications

(134 citation statements)

References 26 publications

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“…Therefore we have that ord ℓ ½F ℓ ðzÞ ¼ þ∞, which in turn implies that M ℓ ðzÞ ≡ 0 ðmod ℓÞ. The theorem now follows from [3], [5], and [6].…”

Section: Proofmentioning

confidence: 81%

Congruences for the Andrews spt function

Ono

2010

Proc. Natl. Acad. Sci. U.S.A.

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Ramanujan-type congruences for the Andrews sptðnÞ partition function have been found for prime moduli 5 ≤ ℓ ≤ 37 in the work of Andrews [Andrews GE, (2008 24≡ 0 ðmod ℓÞ. This congruence gives ðℓ − 1Þ∕2 arithmetic progressions modulo ℓ 3 which support a mod ℓ congruence. This result follows from the surprising fact that the reduction of a certain mock theta function modulo ℓ, for every ℓ ≥ 5, is an eigenform of the Hecke operator T ðℓ 2 Þ.harmonic Maass form | partitions

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“…Therefore we have that ord ℓ ½F ℓ ðzÞ ¼ þ∞, which in turn implies that M ℓ ðzÞ ≡ 0 ðmod ℓÞ. The theorem now follows from [3], [5], and [6].…”

Section: Proofmentioning

confidence: 81%

Congruences for the Andrews spt function

Ono

2010

Proc. Natl. Acad. Sci. U.S.A.

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“…/ is a holomorphic modular form of weight 2 on 0 .4/. The proof of the claims about v 1 .q/ follows from (3)(4) and the following theorem of Bringmann [2008]. …”

Section: Proof Of Theorem 12mentioning

confidence: 93%

Untitled

2013

Algebra Number Theory

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We prove that limits of multiplicities associated to graded families of ideals exist under very general conditions. Most of our results hold for analytically unramified equicharacteristic local rings with perfect residue fields. We give a number of applications, including a "volume = multiplicity" formula, generalizing the formula of Lazarsfeld and Mustat ,ȃ , and a proof that the epsilon multiplicity of Ulrich and Validashti exists as a limit for ideals in rather general rings, including analytic local domains. We prove a generalization of this to generalized symbolic powers of ideals proposed by Herzog, Puthenpurakal and Verma. We also prove an asymptotic "additivity formula" for limits of multiplicities and a formula on limiting growth of valuations, which answers a question posed by the author, Kia Dalili and Olga Kashcheyeva. Our proofs are inspired by a philosophy of Okounkov for computing limits of multiplicities as the volume of a slice of an appropriate cone generated by a semigroup determined by an appropriate filtration on a family of algebraic objects.

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“…This can be seen in the asymptotic formulas of [9]. Again, since our asymptotic aim is n 3/2 p(n), (2.11) implies that we may ignore all terms with weight k < 3.…”

Section: Corollary 23 Garvan's Conjecture Is True If K = 4 and N Ismentioning

confidence: 98%

Inequalities between ranks and cranks

Bringmann

Mahlburg

2009

Proc. Amer. Math. Soc.

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Abstract. Higher 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 the first several cases of (and strengthens) a conjecture due to Garvan, which states that the moments of the crank function are always larger than the moments of the rank function. Furthermore, asymptotic estimates for these differences are also proven.

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On the explicit construction of higher deformations of partition statistics

Cited by 87 publications

References 26 publications

Congruences for the Andrews spt function

Congruences for the Andrews spt function

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Inequalities between ranks and cranks

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