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Relevant bibliographies by topics / Integer partition theory

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Journal articles

Academic literature on the topic 'Integer partition theory'

Author: Grafiati

Published: 4 June 2021

Last updated: 10 February 2022

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Journal articles on the topic "Integer partition theory"

1

GARVAN, FRANK G., and HAMZA YESILYURT. "SHIFTED AND SHIFTLESS PARTITION IDENTITIES II." International Journal of Number Theory 03, no. 01 (2007): 43–84. http://dx.doi.org/10.1142/s1793042107000808.

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Let S and T be sets of positive integers and let a be a fixed positive integer. An a-shifted partition identity has the form [Formula: see text] Here p(S,n) is the number partitions of n whose parts are elements of S. For all known nontrivial shifted partition identities, the sets S and T are unions of arithmetic progressions modulo M for some M. In 1987, Andrews found two 1-shifted examples (M = 32, 40) and asked whether there were any more. In 1989, Kalvade responded with a further six. In 2000, the first author found 59 new 1-shifted identities using a computer search and showed how these c

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2

Yan, Xiao-Hui. "Partitions of the set of nonnegative integers with identical representation functions." International Journal of Number Theory 15, no. 10 (2019): 1969–75. http://dx.doi.org/10.1142/s1793042119501070.

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Let [Formula: see text] be the set of nonnegative integers. For any set [Formula: see text], let [Formula: see text] denote the number of representations of [Formula: see text] as [Formula: see text] with [Formula: see text]. Chen and Wang proved that the set of positive integers can be partitioned into two subsets [Formula: see text] and [Formula: see text] such that [Formula: see text] for all [Formula: see text]. In this paper, we prove that, for a given integer [Formula: see text] and a partition [Formula: see text], there is an integer [Formula: see text] such that [Formula: see text] doe

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3

RØDSETH, ØYSTEIN J., and JAMES A. SELLERS. "PARTITIONS WITH PARTS IN A FINITE SET." International Journal of Number Theory 02, no. 03 (2006): 455–68. http://dx.doi.org/10.1142/s1793042106000644.

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For a finite set A of positive integers, we study the partition function pA(n). This function enumerates the partitions of the positive integer n into parts in A. We give simple proofs of some known and unknown identities and congruences for pA(n). For n in a special residue class, pA(n) is a polynomial in n. We examine these polynomials for linear factors, and the results are applied to a restricted m-ary partition function. We extend the domain of pA and prove a reciprocity formula with supplement. In closing we consider an asymptotic formula for pA(n) and its refinement.

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4

Ballantine, Cristina, and Mircea Merca. "Combinatorial proof of the minimal excludant theorem." International Journal of Number Theory 17, no. 08 (2021): 1765–79. http://dx.doi.org/10.1142/s1793042121500615.

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The minimal excludant of a partition [Formula: see text], [Formula: see text], is the smallest positive integer that is not a part of [Formula: see text]. For a positive integer [Formula: see text], [Formula: see text] denotes the sum of the minimal excludants of all partitions of [Formula: see text]. Recently, Andrews and Newman obtained a new combinatorial interpretation for [Formula: see text]. They showed, using generating functions, that [Formula: see text] equals the number of partitions of [Formula: see text] into distinct parts using two colors. In this paper, we provide a purely combi

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5

BRENNAN, CHARLOTTE, ARNOLD KNOPFMACHER, and STEPHAN WAGNER. "The Distribution of Ascents of Size d or More in Partitions of n." Combinatorics, Probability and Computing 17, no. 4 (2008): 495–509. http://dx.doi.org/10.1017/s0963548308009073.

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A partition of a positive integer n is a finite sequence of positive integers a1, a2, . . ., ak such that a1+a2+ċ ċ ċ+ak=n and ai+1 ≥ ai for all i. Let d be a fixed positive integer. We say that we have an ascent of size d or more if ai+1 ≥ ai+d.We determine the mean, the variance and the limiting distribution of the number of ascents of size d or more (equivalently, the number of distinct part sizes of multiplicity d or more) in the partitions of n.

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6

Andrews, George E. "The Bhargava-Adiga Summation and Partitions." Journal of the Indian Mathematical Society 84, no. 3-4 (2017): 151. http://dx.doi.org/10.18311/jims/2017/15836.

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The Bhargava-Adiga summation rivals the 1ψ1􀀀summation of Ramanujan in elegance. This paper is devoted to two applications in the theory of integer partitions leading to partition questions related to Gauss's celebrated three triangle theorem.

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7

Calkin, Neil, Jimena Davis, Kevin James, Elizabeth Perez, and Charles Swannack. "Computing the integer partition function." Mathematics of Computation 76, no. 259 (2007): 1619–39. http://dx.doi.org/10.1090/s0025-5718-07-01966-7.

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8

KAAVYA, S. J. "CRANK 0 PARTITIONS AND THE PARITY OF THE PARTITION FUNCTION." International Journal of Number Theory 07, no. 03 (2011): 793–801. http://dx.doi.org/10.1142/s1793042111004381.

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A well-known problem regarding the integer partition function p(n) is the parity problem, how often is p(n) even or odd? Motivated by this problem, we obtain the following results: (1) A generating function for the number of crank 0 partitions of n. (2) An involution on the crank 0 partitions whose fixed points are called invariant partitions. We then derive a generating function for the number of invariant partitions. (3) A generating function for the number of self-conjugate rank 0 partitions.

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9

Svaiter, B. F., and N. F. Svaiter. "The distributional zeta-function in disordered field theory." International Journal of Modern Physics A 31, no. 25 (2016): 1650144. http://dx.doi.org/10.1142/s0217751x1650144x.

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In this paper, we present a new mathematical rigorous technique for computing the average free energy of a disordered system with quenched randomness, using the replicas. The basic tool of this technique is a distributional zeta-function, a complex function whose derivative at the origin yields the average free energy of the system as the sum of two contributions: the first one is a series in which all the integer moments of the partition function of the model contribute; the second one, which cannot be written as a series of the integer moments, can be made as small as desired. This result su

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10

PENNISTON, DAVID. "ARITHMETIC OF ℓ-REGULAR PARTITION FUNCTIONS". International Journal of Number Theory 04, № 02 (2008): 295–302. http://dx.doi.org/10.1142/s1793042108001341.

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Let bℓ(n) denote the number of ℓ-regular partitions of n, where ℓ is prime and 3 ≤ ℓ ≤ 23. In this paper we prove results on the distribution of bℓ(n) modulo m for any odd integer m > 1 with 3 ∤ m if ℓ ≠ 3.

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