Partitions with a restriction on the multiplicity of the summands

Hagis, Peter

doi:10.1090/s0002-9947-1971-0272735-1

Cited by 21 publications

(19 citation statements)

References 9 publications

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“…There is a better way to compute an isolated value of Q(n). P. Hagis [15][16][17][18][19][20][21][22][23] used the circle method to study various restricted partition functions. In [16], he provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the partition function Q(n), i.e.,…”

Section: (13)mentioning

confidence: 99%

Distinct partitions and overpartitions

Merca¹

2021

CJM

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In 1963, Peter Hagis, Jr. provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the partition function $Q(n)$ which counts partitions of $n$ into distinct parts. Computing $Q(n)$ by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we investigate new connections between partitions into distinct parts and overpartitions and obtain a surprising recurrence relation for the number of partitions of $n$ into distinct parts. By particularization of this relation, we derive two different linear recurrence relations for the partition function $Q(n)$. One of them involves the thrice square numbers and the other involves the generalized octagonal numbers. The recurrence relation involving the thrice square numbers provide a simple and fast computation of the value of $Q(n)$. This method uses only (large) integer arithmetic and it is simpler to program. Infinite families of linear inequalities involving partitions into distinct parts and overpartitions are introduced in this context.

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Section: (13)mentioning

confidence: 99%

Distinct partitions and overpartitions

Merca¹

2021

CJM

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“…we find that p reg (A; n) equals the number of partitions of n where no part occurs more than A − 1 times. Hagis [11] obtained asymptotics for the number of partitions where no part is repeated more than t times, and letting t = A − 1 in Corollary 4.2 of [11] gives the following theorem.…”

Section: Generalization Of a Theorem Of Erdős And Lehnermentioning

confidence: 99%

Limiting Betti distributions of Hilbert schemes on $n$ points

Griffin¹,

Ono²,

Rolen³

et al. 2021

Preprint

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recently observed that work of Göttsche, combined with a classical result of Erdős and Lehner on integer partitions, implies that the limiting Betti distribution for the Hilbert schemes (C 2 ) [n] on n points, as n → +∞, is a Gumbel distribution. In view of this example, they ask for further such Betti distributions. We answer this question for the quasihomogeneous Hilbert schemes ((C 2 ) [n] ) T α,β that are cut out by torus actions. We prove that their limiting distributions are also of Gumbel type. To obtain this result, we combine work of Buryak, Feigin, and Nakajima on these Hilbert schemes with our generalization of the result of Erdős and Lehner, which gives the distribution of the number of parts in partitions that are multiples of a fixed integer A ≥ 2. Furthermore, if p k (A; n) denotes the number of partitions of n with exactly k parts that are multiples of A, then we obtain the asymptotica result which is of independent interest.

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“…Rademacher's method was used extensively by many practitioners, including Grosswald [15,16], Haberzetle [18], Hagis [19,20,21,22,23,24,25,26,27], Hua [31], Iseki [32,33,34], Lehner [36], Livingood [37], Niven [45], and Subramanyasastri [52] to study various restricted partitions functions.…”

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confidence: 99%