A Formula for the Partition Function That “Counts”

Yuriy Choliy; Andrew V. Sills

doi:10.1007/s00026-016-0305-1

A Formula for the Partition Function That “Counts”

Yuriy Choliy, Andrew V. Sills

Mathematical Sciences

Rutgers, The State University of New Jersey

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

We derive a combinatorial multisum expression for the number D(n, k) of partitions of n with Durfee square of order k. An immediate corollary is therefore a combinatorial formula for p(n), the number of partitions of n. We then study D(n, k) as a quasipolynomial. We consider the natural polynomial approximation D~ (n, k) to the quasipolynomial representation of D(n, k). Numerically, the sum ∑1≤k≤nD~(n,k) appears to be extremely close to the initial term of the Hardy-Ramanujan-Rademacher convergent series for p(n).

Original language	English
Pages (from-to)	301-316
Number of pages	16
Journal	Annals of Combinatorics
Volume	20
Issue number	2
DOIs	https://doi.org/10.1007/s00026-016-0305-1
State	Published - Jun 1 2016

Scopus Subject Areas

Discrete Mathematics and Combinatorics

Keywords

Durfee square
integer partitions
partition function

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10.1007/s00026-016-0305-1

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title = "A Formula for the Partition Function That “Counts”",

abstract = "We derive a combinatorial multisum expression for the number D(n, k) of partitions of n with Durfee square of order k. An immediate corollary is therefore a combinatorial formula for p(n), the number of partitions of n. We then study D(n, k) as a quasipolynomial. We consider the natural polynomial approximation D\textasciitilde{} (n, k) to the quasipolynomial representation of D(n, k). Numerically, the sum ∑1≤k≤nD\textasciitilde{}(n,k) appears to be extremely close to the initial term of the Hardy-Ramanujan-Rademacher convergent series for p(n).",

keywords = "Durfee square, integer partitions, partition function",

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T1 - A Formula for the Partition Function That “Counts”

AU - Choliy, Yuriy

AU - Sills, Andrew V.

PY - 2016/6/1

Y1 - 2016/6/1

N2 - We derive a combinatorial multisum expression for the number D(n, k) of partitions of n with Durfee square of order k. An immediate corollary is therefore a combinatorial formula for p(n), the number of partitions of n. We then study D(n, k) as a quasipolynomial. We consider the natural polynomial approximation D~ (n, k) to the quasipolynomial representation of D(n, k). Numerically, the sum ∑1≤k≤nD~(n,k) appears to be extremely close to the initial term of the Hardy-Ramanujan-Rademacher convergent series for p(n).

AB - We derive a combinatorial multisum expression for the number D(n, k) of partitions of n with Durfee square of order k. An immediate corollary is therefore a combinatorial formula for p(n), the number of partitions of n. We then study D(n, k) as a quasipolynomial. We consider the natural polynomial approximation D~ (n, k) to the quasipolynomial representation of D(n, k). Numerically, the sum ∑1≤k≤nD~(n,k) appears to be extremely close to the initial term of the Hardy-Ramanujan-Rademacher convergent series for p(n).

KW - Durfee square

KW - integer partitions

KW - partition function

U2 - 10.1007/s00026-016-0305-1

DO - 10.1007/s00026-016-0305-1

M3 - Article

SN - 0218-0006

VL - 20

SP - 301

EP - 316

JO - Annals of Combinatorics

JF - Annals of Combinatorics

IS - 2

ER -

A Formula for the Partition Function That “Counts”

Abstract

Scopus Subject Areas

Keywords

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