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Journal articles on the topic 'Integer partitions'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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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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11

Merca, Mircea. "Rank partition functions and truncated theta identities." Applicable Analysis and Discrete Mathematics, no. 00 (2021): 23. http://dx.doi.org/10.2298/aadm190401023m.

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In 1944, Freeman Dyson defined the concept of rank of an integer partition and introduced without definition the term of crank of an integer partition. A definition for the crank satisfying the properties hypothesized for it by Dyson was discovered in 1988 by G.E. Andrews and F.G. Garvan. In this paper, we introduce truncated forms for two theta identities involving the generating functions for partitions with non-negative rank and non-negative crank. As corollaries we derive new infinite families of linear inequalities for the partition function p(n). The number of Garden of Eden partitions are also considered in this context in order to provide other infinite families of linear inequalities for p(n).

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12

Tutaş, Nesrin. "On Partitions and Arf Semigroups." Open Mathematics 17, no. 1 (2019): 343–55. http://dx.doi.org/10.1515/math-2019-0025.

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Abstract In this study we examine some combinatorial properties of the Arf semigroup. In previous work, the author and Karakaş, Gümüşbaş defined an Arf partition of a positive integer n. Here, we continue this work and give new results on Arf partitions. In particular, we analyze the relation among an Arf partition, its Young dual diagram, and the corresponding rational Young diagram. Additionally, this study contains some results that present the relations between partitions and Arf semigroup polynomials.

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13

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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14

Pąk, Karol. "Euler’s Partition Theorem." Formalized Mathematics 23, no. 2 (2015): 93–99. http://dx.doi.org/10.1515/forma-2015-0009.

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Abstract In this article we prove the Euler’s Partition Theorem which states that the number of integer partitions with odd parts equals the number of partitions with distinct parts. The formalization follows H.S. Wilf’s lecture notes [28] (see also [1]). Euler’s Partition Theorem is listed as item #45 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/ [27].

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15

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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16

Harris, V. C., and M. V. Subbarao. "On Product Partitions of Integers." Canadian Mathematical Bulletin 34, no. 4 (1991): 474–79. http://dx.doi.org/10.4153/cmb-1991-076-4.

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AbstractLet p*(n) denote the number of product partitions, that is, the number of ways of expressing a natural number n > 1 as the product of positive integers ≥ 2, the order of the factors in the product being irrelevant, with p*(1) = 1. For any integer if d is an ith power, and = 1, otherwise, and let . Using a suitable generating function for p*(n) we prove that

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17

Merca, Mircea. "Stirling numbers and integer partitions." Quaestiones Mathematicae 39, no. 4 (2015): 457–69. http://dx.doi.org/10.2989/16073606.2015.1096859.

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18

Bencherif, F., N. Benyahia-Tani, S. Bouroubi, O. Kihel, and Z. Yahi. "Integer partitions into Diophantine pairs." Quaestiones Mathematicae 40, no. 4 (2017): 435–42. http://dx.doi.org/10.2989/16073606.2017.1296903.

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19

Comtet, Alain, Satya N. Majumdar, and Stéphane Ouvry. "Integer partitions and exclusion statistics." Journal of Physics A: Mathematical and Theoretical 40, no. 37 (2007): 11255–69. http://dx.doi.org/10.1088/1751-8113/40/37/004.

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20

Yang, Winston C. "Derivatives are essentially integer partitions." Discrete Mathematics 222, no. 1-3 (2000): 235–45. http://dx.doi.org/10.1016/s0012-365x(99)00412-4.

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21

Benjamin, Arthur T., Jennifer J. Quinn, John J. Quinn, and Arkadiusz Wójs. "Composite Fermions and Integer Partitions." Journal of Combinatorial Theory, Series A 95, no. 2 (2001): 390–97. http://dx.doi.org/10.1006/jcta.2001.3182.

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22

Schmidt, Frank. "Integer Partitions and Binary Trees." Advances in Applied Mathematics 28, no. 3-4 (2002): 592–601. http://dx.doi.org/10.1006/aama.2001.0797.

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23

Hohloch, Sonja. "Optimal transport and integer partitions." Discrete Applied Mathematics 190-191 (August 2015): 75–85. http://dx.doi.org/10.1016/j.dam.2015.04.002.

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24

Eggleton. "Equisum Partitions of Sets of Positive Integers." Algorithms 12, no. 8 (2019): 164. http://dx.doi.org/10.3390/a12080164.

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Let V be a finite set of positive integers with sum equal to a multiple of the integer b. When does V have a partition into b parts so that all parts have equal sums? We develop algorithmic constructions which yield positive, albeit incomplete, answers for the following classes of set V, where n is a given positive integer: (1) an initial interval a∈Z+:a≤n; (2) an initial interval of primes p∈P:p≤n, where P is the set of primes; (3) a divisor set d∈Z+:d|n; (4) an aliquot set d∈Z+:d|n, d<n. Open general questions and conjectures are included for each of these classes.

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25

DE MENESES, CLÁUDIO N., and CID C. DE SOUZA. "EXACT SOLUTIONS OF RECTANGULAR PARTITIONS VIA INTEGER PROGRAMMING." International Journal of Computational Geometry & Applications 10, no. 05 (2000): 477–522. http://dx.doi.org/10.1142/s0218195900000280.

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Given a rectangle R in the plane and a finite set P of points in its interior, consider the partitions of the surface of R into smaller rectangles. A partition is feasible with respect to P if each point in P lie on the boundary of some rectangle of the partition. The length of a partition is computed as the sum of the lengths of the line segments defining the boundary of its rectangles. The goal is to find a feasible partition with minimum length. This problem, denoted by RGP, belongs to [Formula: see text]-hard and has application in VLSI design. In this paper we investigate how to obtain exact solutions for the RGP. We introduce two different Integer Programming formulations and carry out a theoretical study to evaluate and compare the strength of their bounds. Computational experiments are reported for Branch-and-Cut and Branch-and-Price algorithms we have implemented for the first and the second formulation, respectively. Randomly generated instances with |P|≤200 are solved exactly. The tests indicate that the size of the instances solved with our algorithms decrease by an order of magnitude in the absence of corectilinear points in P, a special case of RGP whose complexity is still open.

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26

Rovenchak, Andrij. "Partition Function Formalism in the Problem of Multidimensional Integer Partitions." Computational Methods in Science and Technology 16, no. 2 (2010): 187–90. http://dx.doi.org/10.12921/cmst.2010.16.02.187-190.

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27

BERNDT, BRUCE C., AE JA YEE, and ALEXANDRU ZAHARESCU. "ON THE PARITY OF PARTITION FUNCTIONS." International Journal of Mathematics 14, no. 04 (2003): 437–59. http://dx.doi.org/10.1142/s0129167x03001740.

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Let S denote a subset of the positive integers, and let pS(n) be the associated partition function, that is, pS(n) denotes the number of partitions of the positive integer n into parts taken from S. Thus, if S is the set of positive integers, then pS(n) is the ordinary partition function p(n). In this paper, working in the ring of formal power series in one variable over the field of two elements Z/2Z, we develop new methods for deriving lower bounds for both the number of even values and the number of odd values taken by pS(n), for n ≤ N. New very general theorems are obtained, and applications are made to several partition functions.

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28

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 could be proved using the theory of modular functions. Modular transformation of certain shifted identities leads to shiftless partition identities. Again let a be a fixed positive integer, and S, T be distinct sets of positive integers. A shiftless partition identity has the form [Formula: see text] In this paper, we show, except in one case, how all known 1-shifted and shiftless identities follow from a four-parameter theta-function identity due to Jacobi. New shifted and shiftless partition identities are proved.

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29

Andrews, George E., Peter Paule, and Axel Riese. "Macmahon's partition analysis IX: K-gon partitions." Bulletin of the Australian Mathematical Society 64, no. 2 (2001): 321–29. http://dx.doi.org/10.1017/s0004972700039988.

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Dedicated to George Szekeres on the occasion of his 90th birthdayMacMahon devoted a significant portion of Volume II of his famous book Combinatory Analysis to the introduction of Partition Analysis as a computational method for solving combinatorial problems in connection with systems of linear diophantine inequalities and equations. In a series of papers we have shown that MacMahon's method turns into an extremely powerful tool when implemented in computer algebra. In this note we explain how the use of the package Omega developed by the authors has led to a generalisation of a classical counting problem related to triangles with sides of integer length.

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30

Crane, Harry, and Peter McCullagh. "Reversible Markov structures on divisible set partitions." Journal of Applied Probability 52, no. 3 (2015): 622–35. http://dx.doi.org/10.1239/jap/1445543836.

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We study k-divisible partition structures, which are families of random set partitions whose block sizes are divisible by an integer k = 1, 2, …. In this setting, exchangeability corresponds to the usual invariance under relabeling by arbitrary permutations; however, for k > 1, the ordinary deletion maps on partitions no longer preserve divisibility, and so a random deletion procedure is needed to obtain a partition structure. We describe explicit Chinese restaurant-type seating rules for generating families of exchangeable k-divisible partitions that are consistent under random deletion. We further introduce the notion of Markovian partition structures, which are ensembles of exchangeable Markov chains on k-divisible partitions that are consistent under a random process of Markovian deletion. The Markov chains we study are reversible and refine the class of Markov chains introduced in Crane (2011).

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31

Cui, Su-Ping, and Nancy S. S. Gu. "Congruences modulo powers of 2 for generalized Frobenius partitions with six colors." International Journal of Number Theory 15, no. 06 (2019): 1173–81. http://dx.doi.org/10.1142/s1793042119500647.

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A generalized Frobenius partition of [Formula: see text] with [Formula: see text] colors is a two-rowed array [Formula: see text] where [Formula: see text], and the integer entries are taken from [Formula: see text] distinct copies of the non-negative integers distinguished by color, and the rows are ordered first by size and then by color with no two consecutive like entries in any row. Let [Formula: see text] denote the number of this kind of partitions of [Formula: see text] with [Formula: see text] colors. In this paper, we establish some congruences modulo powers of 2 for [Formula: see text].

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32

Canfield, E. Rodney, Carla D. Savage, and Herbert S. Wilf. "Regularly spaced subsums of integer partitions." Acta Arithmetica 115, no. 3 (2004): 205–16. http://dx.doi.org/10.4064/aa115-3-1.

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33

Blecher, Aubrey, Toufik Mansour, and Augustine O. Munagi. "Some Parity Statistics in Integer Partitions." Bulletin Polish Acad. Sci. Math. 63, no. 2 (2015): 123–40. http://dx.doi.org/10.4064/ba63-2-3.

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34

Canfield, E. Rodney. "Integer partitions and the Sperner property." Theoretical Computer Science 307, no. 3 (2003): 515–29. http://dx.doi.org/10.1016/s0304-3975(03)00235-4.

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35

Sudarshan, E. Sampathkumar S. R., B. D. Acharya, and Suresh Nayak. "Super magical Partitions of an Integer." Electronic Notes in Discrete Mathematics 15 (May 2003): 161–63. http://dx.doi.org/10.1016/s1571-0653(04)00569-4.

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36

Songvilay, Sonexay, Areerak Chaiworn, and Detchat Samart. "STUDY OF SOME RESTRICTED INTEGER PARTITIONS." Far East Journal of Mathematical Sciences (FJMS) 119, no. 2 (2019): 195–210. http://dx.doi.org/10.17654/ms119020195.

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37

Archibald, Margaret, Aubrey Blecher, and Arnold Knopfmacher. "Distinct r-tuples in integer partitions." Ramanujan Journal 50, no. 2 (2019): 237–52. http://dx.doi.org/10.1007/s11139-019-00180-x.

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38

Stojmenović, Ivan, and Antoine Zoghbi. "Fast algorithms for genegrating integer partitions." International Journal of Computer Mathematics 70, no. 2 (1998): 319–32. http://dx.doi.org/10.1080/00207169808804755.

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39

YAMANAKA, K., S. i. KAWANO, Y. KIKUCHI, and S. i. NAKANO. "Constant Time Generation of Integer Partitions." IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E90-A, no. 5 (2007): 888–95. http://dx.doi.org/10.1093/ietfec/e90-a.5.888.

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40

Santos, J. P. O., and M. L. Matte. "A New Approach to Integer Partitions." Bulletin of the Brazilian Mathematical Society, New Series 49, no. 4 (2018): 811–47. http://dx.doi.org/10.1007/s00574-018-0082-z.

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41

Eliahou, Shalom, and Martin J. Erickson. "Mutually describing multisets and integer partitions." Discrete Mathematics 313, no. 4 (2013): 422–33. http://dx.doi.org/10.1016/j.disc.2012.11.014.

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42

YAN, XIAO-HUI. "ON PARTITIONS OF NONNEGATIVE INTEGERS AND REPRESENTATION FUNCTIONS." Bulletin of the Australian Mathematical Society 99, no. 03 (2018): 385–87. http://dx.doi.org/10.1017/s0004972718001223.

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Let $\mathbb{N}$ be the set of all nonnegative integers. For any set $A\subset \mathbb{N}$ , let $R(A,n)$ denote the number of representations of $n$ as $n=a+a^{\prime }$ with $a,a^{\prime }\in A$ . There is no partition $\mathbb{N}=A\cup B$ such that $R(A,n)=R(B,n)$ for all sufficiently large integers $n$ . We prove that a partition $\mathbb{N}=A\cup B$ satisfies $|R(A,n)-R(B,n)|\leq 1$ for all nonnegative integers $n$ if and only if, for each nonnegative integer $m$ , exactly one of $2m+1$ and $2m$ is in $A$ .

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43

Chen, Shi-Chao. "Arithmetic properties of a partition pair function." International Journal of Number Theory 10, no. 06 (2014): 1583–94. http://dx.doi.org/10.1142/s1793042114500468.

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For a positive integer n, let ped (n) be the number of partitions of n where the even parts are distinct, and [Formula: see text] be the number of overpartitions of n into odd parts. Moreover, let Q(n) denote the number of the partition pairs of n into two colors (say, red and blue), where the parts colored red satisfy restrictions of partitions counted by ped (n), while the parts colored blue satisfy restrictions of partitions counted by [Formula: see text]. We establish several congruences for Q(n). We also obtain an asymptotic formula for Q(n).

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44

Ahmed, Chwas, Paul Martin, and Volodymyr Mazorchuk. "On the number of principal ideals in d-tonal partition monoids." Annals of Combinatorics 25, no. 1 (2021): 79–113. http://dx.doi.org/10.1007/s00026-020-00518-z.

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AbstractFor a positive integer d, a non-negative integer n and a non-negative integer $$h\le n$$ h ≤ n , we study the number $$C_{n}^{(d)}$$ C n ( d ) of principal ideals; and the number $$C_{n,h}^{(d)}$$ C n , h ( d ) of principal ideals generated by an element of rank h, in the d-tonal partition monoid on n elements. We compute closed forms for the first family, as partial cumulative sums of known sequences. The second gives an infinite family of new integral sequences. We discuss their connections to certain integral lattices as well as to combinatorics of partitions.

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45

Gramain, Jean-Baptiste, and Jørn B. Olsson. "On bar lengths in partitions." Proceedings of the Edinburgh Mathematical Society 56, no. 2 (2013): 535–50. http://dx.doi.org/10.1017/s0013091512000387.

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AbstractWe present, given an odd integer d, a decomposition of the multiset of bar lengths of a bar partition λ as the union of two multisets, one consisting of the bar lengths in its d-core partition cd(λ) and the other consisting of modified bar lengths in its d-quotient partition. In particular, we obtain that the multiset of bar lengths in cd(λ) is a sub-multiset of the multiset of bar lengths in λ. Also, we obtain a relative bar formula for the degrees of spin characters of the Schur extensions of $\mathfrak{S}_n$. The proof involves a recent similar result for partitions, proved by Bessenrodt and the authors.

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46

Fergusson, Kevin John. "Partitions into large unequal parts from a general sequence." Journal of the Australian Mathematical Society 80, no. 1 (2006): 13–44. http://dx.doi.org/10.1017/s1446788700011368.

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AbstractAn asymptotic estimate is obtained for the number of partitions of the positive integer n into unequal parts coming from a sequence u, with each part greater than m, under suitable conditions on the sequence u. The estimate holds uniformly with respect to integers m such that 0 ≤ m ≤ n1−δ, as n → ∞, where δ is a given real number, such that 0 < δ < 1.

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47

Huang, H. Y., and F. Y. Wu. "The Infinite-State Potts Model and Solid Partitions of an Integer." International Journal of Modern Physics B 11, no. 01n02 (1997): 121–26. http://dx.doi.org/10.1142/s0217979297000150.

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It has been established that the infinite-state Potts model in d dimensions generates restricted partitions of integers in d-1 dimensions, the latter a well-known intractable problem in number theory for d>3. Here we consider the d=4 problem. We consider a Potts model on an L × M × N × P hypercubic lattice whose partition function GLMNP(t) generates restricted solid partitions on an L × M × N lattice with each part no greater than P. Closed-form expressions are obtained for G222P(t) and we evaluated its zeroes in the complex t plane for different values of P. On the basis of our numerical results we conjecture that all zeroes of the enumeration generating function GLMNP(t) lie on the unit circle |t|=1 in the limit that any of the indices L, M, N, P becomes infinite.

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Szekeres, G. "Asymptotic distribution of the number and size of parts in unequal partitions." Bulletin of the Australian Mathematical Society 36, no. 1 (1987): 89–97. http://dx.doi.org/10.1017/s0004972700026320.

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An asymptotic formula is derived for the number of partitions of a large positive integer n into r unequal positive integer parts and maximal summand k. The number of parts has a normal distribution about its maximum, the largest summand an extreme-value distribution. For unrestricted partitions the two distributions coincide and both are extreme-valued. The problem of joint distribution of unrestricted partitions with r parts and largest summand k remains unsolved.

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Krishnamachari, R. S., and P. Y. Papalambros. "Optimal Hierarchical Decomposition Synthesis Using Integer Programming." Journal of Mechanical Design 119, no. 4 (1997): 440–47. http://dx.doi.org/10.1115/1.2826388.

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Decomposition synthesis in optimal design is the process of creating an optimal design model by selecting objectives and constraints so that it can be directly partitioned into an appropriate decomposed form. Such synthesis results are not unique since there may be many partitions that satisfy the decomposition requirements. Introducing suitable criteria an optimal decomposition synthesis process can be defined in a manner analogous to optimal partitioning formulations. The article presents an integer programming formulation and solution techniques for synthesizing hierarchically decomposed optimal design problems. Examples for designing a pressure vessel, an automotive caliper disc brake and a speed reducer are also presented.

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Joyce, Paul, and Simon Tavaré. "A convergence theorem for symmetric functionals of random partitions." Journal of Applied Probability 29, no. 2 (1992): 280–90. http://dx.doi.org/10.2307/3214566.

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This paper gives general conditions under which symmetric functionals of random partitions of the integer m converge in distribution as m → ∞. The main result is used to settle a conjecture of Donnelly et al. (1991) to the effect that the mean of the sum of the square roots of the relative sizes of the components of a random mapping of m integers converges to π/2 as m → ∞.

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