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Relevant bibliographies by topics / Integer partitions

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Journal articles

Academic literature on the topic 'Integer partitions'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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Journal articles on the topic "Integer partitions"

1

Engel, Konrad, Tadeusz Radzik, and Jan-Christoph Schlage-Puchta. "Optimal integer partitions." European Journal of Combinatorics 36 (February 2014): 425–36. http://dx.doi.org/10.1016/j.ejc.2013.09.004.

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2

Bisi, C., G. Chiaselotti, and P. A. Oliverio. "Sand Piles Models of Signed Partitions with Piles." ISRN Combinatorics 2013 (January 13, 2013): 1–7. http://dx.doi.org/10.1155/2013/615703.

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Let be nonnegative integers. In this paper we study the basic properties of a discrete dynamical model of signed integer partitions that we denote by . A generic element of this model is a signed integer partition with exactly all distinct nonzero parts, whose maximum positive summand is not exceeding and whose minimum negative summand is not less than . In particular, we determine the covering relations, the rank function, and the parallel convergence time from the bottom to the top of by using an abstract Sand Piles Model with three evolution rules. The lattice was introduced by the first two authors in order to study some combinatorial extremal sum problems.

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3

Das, Sabuj. "PARTITION CONGRUENCES AND DYSON’S RANK." International Journal of Research -GRANTHAALAYAH 2, no. 2 (2014): 49–60. http://dx.doi.org/10.29121/granthaalayah.v2.i2.2014.3066.

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In this article the rank of a partition of an integer is a certain integer associated with the partition. The term has first introduced by freeman Dyson in a paper published in Eureka in 1944. In 1944, F.S. Dyson discussed his conjectures related to the partitions empirically some Ramanujan’s famous partition congruences. In 1921, S. Ramanujan proved his famous partition congruences: The number of partitions of numbers 5n+4, 7n+5 and 11n +6 are divisible by 5, 7 and 11 respectively in another way. In 1944, Dyson defined the relations related to the rank of partitions. These are later proved by Atkin and Swinnerton-Dyer in 1954. The proofs are analytic relying heavily on the properties of modular functions. This paper shows how to generate the generating functions for In this paper, we show how to prove the Dyson’s conjectures with rank of partitions.

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4

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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5

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] does not hold.

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6

Knopfmacher, Arnold, and Augustine O. Munagi. "Successions in integer partitions." Ramanujan Journal 18, no. 3 (2008): 239–55. http://dx.doi.org/10.1007/s11139-008-9140-2.

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7

Borg, Peter. "Strongly intersecting integer partitions." Discrete Mathematics 336 (December 2014): 80–84. http://dx.doi.org/10.1016/j.disc.2014.07.018.

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8

KIM, BYUNGCHAN, and EUNMI KIM. "BIASES IN INTEGER PARTITIONS." Bulletin of the Australian Mathematical Society 104, no. 2 (2021): 177–86. http://dx.doi.org/10.1017/s0004972720001495.

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AbstractWe show that there are biases in the number of appearances of the parts in two residue classes in the set of ordinary partitions. More precisely, let $p_{j,k,m} (n)$ be the number of partitions of n such that there are more parts congruent to j modulo m than parts congruent to k modulo m for $m \geq 2$ . We prove that $p_{1,0,m} (n)$ is in general larger than $p_{0,1,m} (n)$ . We also obtain asymptotic formulas for $p_{1,0,m}(n)$ and $p_{0,1,m}(n)$ for $m \geq 2$ .

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9

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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10

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 combinatorial proof of this result and new properties of the function [Formula: see text]. We generalize this combinatorial interpretation to [Formula: see text], the sum of least [Formula: see text]-gaps in all partitions of [Formula: see text]. The least [Formula: see text]-gap of a partition [Formula: see text] is the smallest positive integer that does not appear at least [Formula: see text] times as a part of [Formula: see text].

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