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Journal articles on the topic 'Partities (Mathematics)'

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Published: 4 June 2021
Last updated: 1 February 2022

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1

Moyer, Patricia Seray. "Links to Literature: A Remainder of One: Exploring Partitive Division." Teaching Children Mathematics 6, no. 8 (2000): 517–21. http://dx.doi.org/10.5951/tcm.6.8.0517.

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Children's literature can be a springboard for conversations about mathematical concepts. Austin (1998) suggests that good children's literature with a mathematical theme provides a context for both exploring and extending mathematics problems embedded in stories. In the context of discussing a story, children connect their everyday experiences with mathematics and have opportunities to make conjectures about quantities, equalities, or other mathematical ideas; negotiate their understanding of mathematical concepts; and verbalize their thinking. Children's books that prompt mathematical conver
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2

Andrews, George E., and Peter Paule. "MacMahon's partition analysis XII: Plane partitions." Journal of the London Mathematical Society 76, no. 3 (2007): 647–66. http://dx.doi.org/10.1112/jlms/jdm079.

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3

Hummel, Tamara Lakins. "Effective versions of Ramsey's Theorem: Avoiding the cone above 0′." Journal of Symbolic Logic 59, no. 4 (1994): 1301–25. http://dx.doi.org/10.2307/2275707.

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AbstractRamsey's Theorem states that if P is a partition of [ω]k into finitely many partition classes, then there exists an infinite set of natural numbers which is homogeneous for P. We consider the degrees of unsolvability and arithmetical definability properties of infinite homogeneous sets for recursive partitions. We give Jockusch's proof of Seetapun's recent theorem that for all recursive partitions of [ω]2 into finitely many pieces, there exists an infinite homogeneous set A such that ∅′ ≰TA. Two technical extensions of this result are given, establishing arithmetical bounds for such a
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4

HEDEN, OLOF. "A SURVEY OF THE DIFFERENT TYPES OF VECTOR SPACE PARTITIONS." Discrete Mathematics, Algorithms and Applications 04, no. 01 (2012): 1250001. http://dx.doi.org/10.1142/s1793830912500012.

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A vector space partition is here a collection [Formula: see text] of subspaces of a finite vector space V(n, q), of dimension n over a finite field with q elements, with the property that every non-zero vector is contained in a unique member of [Formula: see text]. Vector space partitions relate to finite projective planes, design theory and error correcting codes. In the first part of the paper I will discuss some relations between vector space partitions and other branches of mathematics. The other part of the paper contains a survey of known results on the type of a vector space partition,
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5

Caicedo, Andrés Eduardo, and Brittany Shelton. "Of Puzzles and Partitions: Introducing Partiti." Mathematics Magazine 91, no. 1 (2018): 20–23. http://dx.doi.org/10.1080/0025570x.2018.1403233.

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6

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 cou
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7

Józefiak, Tadeusz, and Jerzy Weyman. "Representation-theoretic interpretation of a formula of D. E. Littlewood." Mathematical Proceedings of the Cambridge Philosophical Society 103, no. 2 (1988): 193–96. http://dx.doi.org/10.1017/s0305004100064768.

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This note is a continuation of our attempts (see [3]) to give a satisfactory representation-theoretic justification of the following formula of D. E. Littlewood:where sI is the Schur symmetric function corresponding to a partition I, |I| is the weight of I, r(I) is the rank of I, and the summation ranges over all self-conjugate partitions (i.e. partitions I such that I = I where I is the partition conjugate to I).
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8

Goulden, I. P. "Exact Values for Degree Sums Over Strips of Young Diagrams." Canadian Journal of Mathematics 42, no. 5 (1990): 763–75. http://dx.doi.org/10.4153/cjm-1990-040-4.

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If λ = (λ1,…, λm) where λ1,…,λm are nonnegative integers with λ1 ≥…≥ λm, then λ is a partition of |λ| = λ1 + …+λm, and we write λ⊢ |λ|. The non-zero λi's are the parts of λ, so λ1 is the largest part, and ℓ(λ) is the number of parts of λ. Two partitions with the same parts, so they differ only in number of zeros, are the same. The set of all partitions, including the partition of 0 (with 0 parts) is denoted by The conjugate of λ, denoted by , is the partition (μ1,…, μk), in which μi is the number of λ's that are ≥i , for i = 1,…, k, where k=λ1.
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9

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 a
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10

Bérard, P., and B. Helffer. "Remarks on the boundary set of spectral equipartitions." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 372, no. 2007 (2014): 20120492. http://dx.doi.org/10.1098/rsta.2012.0492.

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Given a bounded open set in (or in a Riemannian manifold), and a partition of Ω by k open sets ω j , we consider the quantity , where λ ( ω j ) is the ground state energy of the Dirichlet realization of the Laplacian in ω j . We denote by ℒ k ( Ω ) the infimum of over all k -partitions. A minimal k -partition is a partition that realizes the infimum. Although the analysis of minimal k -partitions is rather standard when k =2 (we find the nodal domains of a second eigenfunction), the analysis for higher values of k becomes non-trivial and quite interesting. Minimal partitions are in particular
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11

Subbarao, M. V. "Product partitions and recursion formulae." International Journal of Mathematics and Mathematical Sciences 2004, no. 33 (2004): 1725–35. http://dx.doi.org/10.1155/s0161171204307258.

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Utilizing a method briefly hinted in the author's paper written in 1991 jointly with V. C. Harris, we derive here a number of unpublished recursion formulae for a variety of product partition functions which we believe have not been considered before in the literature. These include the functionsp*(n;k,h)(which stands for the number of product partitions ofn>1intokparts of whichhare distinct), andp(d)*(n;m)(which stands for the number of product partitions ofninto exactlymparts with at mostdrepetitions of any part). We also derive recursion formulae for certain product partition functions w
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12

Al, Busra, and Mustafa Alkan. "On relations for the partitions of numbers." Filomat 34, no. 2 (2020): 567–74. http://dx.doi.org/10.2298/fil2002567a.

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In this paper, we present some interrelations among some restricted and unrestricted partitions of integers. Mainly, we derive new effective formulas for the partition function and compare our partition formula with well known recurrence formulas. Moreover, as the number increase, we observe how effective the new recurance formulas are.
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13

Crane, Harry, and Peter McCullagh. "Reversible Markov structures on divisible set partitions." Journal of Applied Probability 52, no. 03 (2015): 622–35. http://dx.doi.org/10.1017/s0021900200113336.

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We studyk-divisible partition structures, which are families of random set partitions whose block sizes are divisible by an integerk= 1, 2, …. In this setting, exchangeability corresponds to the usual invariance under relabeling by arbitrary permutations; however, fork> 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 exchangeablek-divisible partitions that are consistent under random deletion. We
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14

DIGBY, DAVID, WILLIAM SEFFENS, and FISSEHA ABEBE. "RUNS OF AMINO ACIDS ARE LONGER THAN EXPECTED IN PROTEINS BASED ON A GRAPH THEORY REPRESENTATION OF THE GENETIC CODE." Journal of Biological Systems 10, no. 04 (2002): 319–35. http://dx.doi.org/10.1142/s0218339002000718.

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An in silico study of mRNA secondary structure has found a bias within the coding sequences of genes that favors "in-frame" pairing of nucleotides. This pairing of codons, each with its reverse-complement, partitions the 20 amino acids into three subsets. The genetic code can therefore be represented by a three-component graph. The composition of proteins in terms of amino acid membership in the three subgroups has been measured, and sequence runs of members within the same subgroup have been analyzed using a runs statistic based on Z-scores. In a GENBANK database of over 416,000 protein seque
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15

Garvan, F. G. "More cranks and t-cores." Bulletin of the Australian Mathematical Society 63, no. 3 (2001): 379–91. http://dx.doi.org/10.1017/s0004972700019481.

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Dedicated to George Szekeres on the occasion of his 90th BirthdayIn 1990, new statistics on partitions (called cransk) were found which combinatorially prove Ramanujan's congruences for the partition function modulo 5, 7, 11 and 25. The methods are extended to find cranks for Ramanujan's partition congruence modulo 49. A more explicit form of the crank is given for the modulo 25 congruence.
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16

Mileti, Joseph R. "Partition Theorems and Computability Theory." Bulletin of Symbolic Logic 11, no. 3 (2005): 411–27. http://dx.doi.org/10.2178/bsl/1122038995.

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The connections between mathematical logic and combinatorics have a rich history. This paper focuses on one aspect of this relationship: understanding the strength, measured using the tools of computability theory and reverse mathematics, of various partition theorems. To set the stage, recall two of the most fundamental combinatorial principles, König's Lemma and Ramsey's Theorem. We denote the set of natural numbers by ω and the set of finite sequences of natural numbers by ω<ω. We also identify each n ∈ ω with its set of predecessors, so n = {0, 1, 2, …, n − 1}.
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17

Haiman, Mark. "On Realization of Bjorner's 'Continuous Partition Latice' by Measurable Partitions." Transactions of the American Mathematical Society 343, no. 2 (1994): 695. http://dx.doi.org/10.2307/2154737.

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18

BOYARSKY, A., and Y. S. LOU. "CHAOTIC BEHAVIOR OF HIGHER DIMENSIONAL TRANSFORMATIONS DEFINED ON COUNTABLE PARTITIONS." International Journal of Bifurcation and Chaos 03, no. 04 (1993): 1045–49. http://dx.doi.org/10.1142/s0218127493000866.

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Jablonski maps are higher dimensional maps defined on rectangular partitions with each component a function of only one variable. It is well known that expanding Jablonski maps have absolutely continuous invariant measures. In this note we consider Jablonski maps defined on countable partitions. Such maps occur, for example, in multivariable number theoretic problems. The main result establishes the existence of an absolutely continuous invariant measure for Jablonski maps on a countable partition with the additional condition that the images of all the partition elements form a finite collect
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19

Tomescu, Mihaela Aurelia, Lorentz Jäntschi, and Doina Iulia Rotaru. "Figures of Graph Partitioning by Counting, Sequence and Layer Matrices." Mathematics 9, no. 12 (2021): 1419. http://dx.doi.org/10.3390/math9121419.

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A series of counting, sequence and layer matrices are considered precursors of classifiers capable of providing the partitions of the vertices of graphs. Classifiers are given to provide different degrees of distinctiveness for the vertices of the graphs. Any partition can be represented with colors. Following this fundamental idea, it was proposed to color the graphs according to the partitions of the graph vertices. Two alternative cases were identified: when the order of the sets in the partition is relevant (the sets are distinguished by their positions) and when the order of the sets in t
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20

KENNY, BRIAN G., and TONY W. DIXON. "AMBIGUITY IN THE DETERMINATION OF THE FREE ENERGY ASSOCIATED WITH THE CRITICAL CIRCLE MAP." ANZIAM Journal 50, no. 2 (2008): 177–84. http://dx.doi.org/10.1017/s1446181108000291.

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AbstractWe consider a simple model to describe the widths of the mode-locked intervals for the critical circle map. By using two different partitions of the rational numbers based on Farey series and Farey tree levels, respectively, we calculate the free energy analytically at selected points for each partition. It emerges that the result of the calculation depends on the method of partition. An implication of this finding is that the generalized dimensions Dq are different for the two types of partition except when q=0; that is, only the Hausdorff dimension is the same in both cases.
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21

Mansour, Toufik, Mark Shattuck, and Stephan Wagner. "Enumerating set partitions by the number of positions between adjacent occurrences of a letter." Applicable Analysis and Discrete Mathematics 4, no. 2 (2010): 284–308. http://dx.doi.org/10.2298/aadm100425019m.

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A partition ? of the set [n] = {1, 2,...,n} is a collection {B1,...,Bk} of nonempty disjoint subsets of [n] (called blocks) whose union equals [n]. Suppose that the subsets Bi are listed in increasing order of their minimal elements and ? = ?1, ?2...?n denotes the canonical sequential form of a partition of [n] in which iEB?i for each i. In this paper, we study the generating functions corresponding to statistics on the set of partitions of [n] with k blocks which record the total number of positions of ? between adjacent occurrences of a letter. Among our results are explicit formulas for the
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22

Chouraqui, Fabienne. "The Herzog–Schönheim conjecture for finitely generated groups." International Journal of Algebra and Computation 29, no. 06 (2019): 1083–112. http://dx.doi.org/10.1142/s0218196719500425.

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Let [Formula: see text] be a group and [Formula: see text] be subgroups of [Formula: see text] of indices [Formula: see text], respectively. In 1974, Herzog and Schönheim conjectured that if [Formula: see text], [Formula: see text], is a coset partition of [Formula: see text], then [Formula: see text] cannot be distinct. We consider the Herzog–Schönheim conjecture for free groups of finite rank and develop a new combinatorial approach, using covering spaces. We define [Formula: see text] the space of coset partitions of [Formula: see text] and show [Formula: see text] is a metric space with in
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23

DAVISON, BEN, JARED ONGARO, and BALÁZS SZENDRŐI. "Enumerating coloured partitions in 2 and 3 dimensions." Mathematical Proceedings of the Cambridge Philosophical Society 169, no. 3 (2019): 479–505. http://dx.doi.org/10.1017/s0305004119000252.

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AbstractWe study generating functions of ordinary and plane partitions coloured by the action of a finite subgroup of the corresponding special linear group. After reviewing known results for the case of ordinary partitions, we formulate a conjecture concerning a basic factorisation property of the generating function of coloured plane partitions that can be thought of as an orbifold analogue of a conjecture of Maulik et al., now a theorem, in three-dimensional Donaldson–Thomas theory. We study natural quantisations of the generating functions arising from geometry, discuss a quantised version
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24

Gordon, A. D., and M. Vichi. "Fuzzy partition models for fitting a set of partitions." Psychometrika 66, no. 2 (2001): 229–47. http://dx.doi.org/10.1007/bf02294837.

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25

HIRSCHHORN, MICHAEL D., and JAMES A. SELLERS. "ELEMENTARY PROOFS OF PARITY RESULTS FOR 5-REGULAR PARTITIONS." Bulletin of the Australian Mathematical Society 81, no. 1 (2009): 58–63. http://dx.doi.org/10.1017/s0004972709000525.

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AbstractIn a recent paper, Calkin et al. [N. Calkin, N. Drake, K. James, S. Law, P. Lee, D. Penniston and J. Radder, ‘Divisibility properties of the 5-regular and 13-regular partition functions’, Integers8 (2008), #A60] used the theory of modular forms to examine 5-regular partitions modulo 2 and 13-regular partitions modulo 2 and 3; they obtained and conjectured various results. In this note, we use nothing more than Jacobi’s triple product identity to obtain results for 5-regular partitions that are stronger than those obtained by Calkin and his collaborators. We find infinitely many Ramanuj
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26

Jakimczuk, Rafael. "Restricted partitions." International Journal of Mathematics and Mathematical Sciences 2004, no. 36 (2004): 1893–96. http://dx.doi.org/10.1155/s0161171204306502.

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We prove a known partitions theorem by Bell in an elementary and constructive way. Our proof yields a simple recursive method to compute the corresponding Sylvester polynomials for the partition. The previous known methods to obtain these polynomials are in general not elementary.
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27

Kiderlen, Markus, and Florian Pausinger. "Discrepancy of stratified samples from partitions of the unit cube." Monatshefte für Mathematik 195, no. 2 (2021): 267–306. http://dx.doi.org/10.1007/s00605-021-01538-4.

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AbstractWe extend the notion of jittered sampling to arbitrary partitions and study the discrepancy of the related point sets. Let $${\varvec{\Omega }}=(\Omega _1,\ldots ,\Omega _N)$$ Ω = ( Ω 1 , … , Ω N ) be a partition of $$[0,1]^d$$ [ 0 , 1 ] d and let the ith point in $${{\mathcal {P}}}$$ P be chosen uniformly in the ith set of the partition (and stochastically independent of the other points), $$i=1,\ldots ,N$$ i = 1 , … , N . For the study of such sets we introduce the concept of a uniformly distributed triangular array and compare this notion to related notions in the literature. We pro
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28

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 Besse
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29

Kestenband, Barbu C. "Partitioning projective planes into arcs." Mathematical Proceedings of the Cambridge Philosophical Society 104, no. 3 (1988): 435–40. http://dx.doi.org/10.1017/s0305004100065634.

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We show how to partition certain classes of finite projective planes into equicardinal arcs. Several partitions of this kind are to be found in the recent literature and they have aroused a certain amount of interest on two counts, as we shall shortly see.
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Archibald, Margaret, Aubrey Blecher, Charlotte Brennan, Arnold Knopfmacher, and Toufik Mansour. "Two by two squares in set partitions." Mathematica Slovaca 70, no. 1 (2020): 29–40. http://dx.doi.org/10.1515/ms-2017-0328.

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AbstractA partition π of a set S is a collection B1, B2, …, Bk of non-empty disjoint subsets, alled blocks, of S such that $\begin{array}{} \displaystyle \bigcup _{i=1}^kB_i=S. \end{array}$ We assume that B1, B2, …, Bk are listed in canonical order; that is in increasing order of their minimal elements; so min B1 < min B2 < ⋯ < min Bk. A partition into k blocks can be represented by a word π = π1π2⋯πn, where for 1 ≤ j ≤ n, πj ∈ [k] and $\begin{array}{} \displaystyle \bigcup _{i=1}^n \{\pi_i\}=[k], \end{array}$ and πj indicates that j ∈ Bπj. The canonical representations of all set par
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31

Levi, Inessa, and Steve Seif. "Permutational labelling of constant weight Gray codes." Bulletin of the Australian Mathematical Society 65, no. 3 (2002): 399–406. http://dx.doi.org/10.1017/s000497270002044x.

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We prove that for positive integers n and r satisfying 1 < r < n, with the single exception of n = 4 and r = 2, there exists a constant weight Gray code of r-sets of Xn = {1, 2, …, n} that admits an orthogonal labelling by distinct partitions, with each subsequent partition obtained from the previous one by an application of a permutation of the underlying set. Specifically, an r-set A and a partition π of Xn are said to be orthogonal if every class of π meets A in exactly one element. We prove that for all n and r as stated, and taken modulo , there exists a list of the distinct r-sets
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32

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 applicatio
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Smith, Gregory L. "Partitions and (m and n) Sums of Products---Two Cell Partition." SIAM Journal on Discrete Mathematics 14, no. 3 (2001): 356–69. http://dx.doi.org/10.1137/s0895480198349014.

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34

Berger, Marc A., Alexander Felzenbaum, and Aviezri Fraenkel. "The Herzog-Schönheim Conjecture for Finite Nilpotent Groups." Canadian Mathematical Bulletin 29, no. 3 (1986): 329–33. http://dx.doi.org/10.4153/cmb-1986-050-0.

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AbstractThe purpose of this note is to prove the Herzog-Schônheim [3] conjecture for finite nilpotent groups. This conjecture states that any nontrivial partition of a group into cosets must contain two cosets of the same index (Corollary IV below). See Porubský [4, Section 8] for a perspective on coset partitions.
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35

MASRI, RIAD. "Singular moduli and the distribution of partition ranks modulo 2." Mathematical Proceedings of the Cambridge Philosophical Society 160, no. 2 (2015): 209–32. http://dx.doi.org/10.1017/s0305004115000675.

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AbstractIn this paper, we prove an asymptotic formula with a power saving error term for traces of weight zero weakly holomorphic modular forms of level N along Galois orbits of Heegner points on the modular curve X0(N). We use this result to study the distribution of partition ranks modulo 2. In particular, we give an asymptotic formula with a power saving error term for the number of partitions of a positive integer n with even (respectively, odd) rank. We use these results to deduce a strong quantitative form of equidistribution of partition ranks modulo 2.
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36

Burkett, Shawn T. "An algorithm for computing a supercharacter theory generated from a given partition." International Journal of Algebra and Computation 31, no. 05 (2021): 819–30. http://dx.doi.org/10.1142/s0218196721500387.

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Let [Formula: see text] be a finite group. The set of all supercharacter theories of [Formula: see text] forms a lattice, where the join operation coincides with the join operation on the lattice of partitions of [Formula: see text], with partial order given by refinement. The meet operation is more complicated however, and seems difficult to describe. In this paper, we outline algorithms for determining the coarsest supercharacter theory whose associated partition is finer than a given partition. One of the primary applications is to compute the supercharacters and superclasses for the meet o
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37

Last, Günter. "Stationary partitions and Palm probabilities." Advances in Applied Probability 38, no. 3 (2006): 602–20. http://dx.doi.org/10.1239/aap/1158684994.

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A stationary partition based on a stationary point process N in ℝd is an ℝd-valued random field π={π(x): x∈ℝd} such that both π(y)∈N for each y∈ℝd and the random partition {{y∈ℝd: π(y)=x}: x∈N} is stationary jointly with N. Stationary partitions may be considered as general versions of the stationary random tessellations studied in stochastic geometry. As in the special case of the Voronoi tessellation, a stationary partition can be used to relate the underlying stationary probability measure to the associated Palm probability measure of N. In doing so, we will develop some basic theory for st
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KRYSTEK, ANNA, and HIROAKI YOSHIDA. "THE COMBINATORICS OF THE r-FREE CONVOLUTION." Infinite Dimensional Analysis, Quantum Probability and Related Topics 06, no. 04 (2003): 619–27. http://dx.doi.org/10.1142/s0219025703001419.

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In this paper we shall give combinatorial remarks on the r-free convolution. In particular, we shall introduce the set partition statistic on non-crossing partitions, which gives the r-free deformed moment-cumulant formula. We shall also give the probability measure of the r-free Poisson law and its moments, exactly.
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39

Debnath, Lokenath. "Srinivasa Ramanujan (1887-1920) and the theory of partitions of numbers and statistical mechanics a centennial tribute." International Journal of Mathematics and Mathematical Sciences 10, no. 4 (1987): 625–40. http://dx.doi.org/10.1155/s0161171287000772.

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This centennial tribute commemorates Ramanujan the Mathematician and Ramanujan the Man. A brief account of his life, career, and remarkable mathematical contributions is given to describe the gifted talent of Srinivasa Ramanujan. As an example of his creativity in mathematics, some of his work on the theory of partition of numbers has been presented with its application to statistical mechanics.
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JING, NAIHUAN, KAILASH C. MISRA, and CARLA D. SAVAGE. "ON MULTI-COLOR PARTITIONS AND THE GENERALIZED ROGERS–RAMANUJAN IDENTITIES." Communications in Contemporary Mathematics 03, no. 04 (2001): 533–48. http://dx.doi.org/10.1142/s0219199701000482.

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Basil Gordon, in the sixties, and George Andrews, in the seventies, generalized the Rogers–Ramanujan identities to higher moduli. These identities arise in many areas of mathematics and mathematical physics. One of these areas is representation theory of infinite dimensional Lie algebras, where various known interpretations of these identities have led to interesting applications. Motivated by their connections with Lie algebra representation theory, we give a new interpretation of a sum related to generalized Rogers–Ramanujan identities in terms of multi-color partitions.
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41

Cawley, Elise. "Smooth Markov partitions and toral automorphisms." Ergodic Theory and Dynamical Systems 11, no. 4 (1991): 633–51. http://dx.doi.org/10.1017/s0143385700006404.

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AbstractWe show that the only hyperbolic toral automorphisms f for which there exist Markov partitions with piecewise smooth boundary are those for which a power fk is linearly covered by a direct product of automorphisms of the 2-torus. Only a finite number of shapes occur in a certain natural set of cross-sections of the partition boundary. The behavior of the stratified structure of a piecewise smooth boundary under the mapping forces these shapes to be self-similar. This, together with expanding properties of the mapping, means that a piecewise smooth partition is in fact piecewise linear.
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42

Pauliac, Christian H. Salas. "Group2: Generating the Finest Partition that is Coarser than Two Given Partitions." Stata Journal: Promoting communications on statistics and Stata 13, no. 4 (2013): 867–75. http://dx.doi.org/10.1177/1536867x1301300411.

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43

GNEDIN, ALEXANDER, and JIM PITMAN. "Poisson Representation of a Ewens Fragmentation Process." Combinatorics, Probability and Computing 16, no. 6 (2007): 819–27. http://dx.doi.org/10.1017/s0963548306008352.

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A simple explicit construction is provided of a partition-valued fragmentation process whose distribution on partitions of [n] = 1,. . .,nat time θ ≥ 0 is governed by the Ewens sampling formula with parameter θ. These partition-valued processes are exchangeable and consistent, asnvaries. They can be derived by uniform sampling from a corresponding mass fragmentation process defined by cutting a unit interval at the points of a Poisson process with intensity θx−1dx on/mathbbR+, arranged to beintensifying as θ increases.
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44

Savage, Carla D. "The mathematics of lecture hall partitions." Journal of Combinatorial Theory, Series A 144 (November 2016): 443–75. http://dx.doi.org/10.1016/j.jcta.2016.06.006.

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45

Hammond, Paul, and Richard Lewis. "Congruences in ordered pairs of partitions." International Journal of Mathematics and Mathematical Sciences 2004, no. 47 (2004): 2509–12. http://dx.doi.org/10.1155/s0161171204311439.

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Dyson defined the rank of a partition (as the first part minus the number of parts) whilst investigating certain congruences in the sequencep−1(n). The rank has been widely studied as have been other statistics, such as the crank. In this paper a “birank” is defined which relates to ordered pairs of partitions, and is used in an elementary proof of a congruence inp−2(n).
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46

Hansen, Jennie C. "A functional central limit theorem for the Ewens sampling formula." Journal of Applied Probability 27, no. 1 (1990): 28–43. http://dx.doi.org/10.2307/3214593.

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For each n > 0, the Ewens sampling formula from population genetics is a measure on the set of all partitions of the integer n. To determine the limiting distributions for the part sizes of a partition with respect to the measures given by this formula, we associate to each partition a step function on [0, 1]. Each jump in the function equals the number of parts in the partition of a certain size. We normalize these functions and show that the induced measures on D[0, 1] converge to Wiener measure. This result complements Kingman's frequency limit theorem [10] for the Ewens partition struct
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47

MUTAFCHIEV, LJUBEN. "The Size of the Largest Part of Random Weighted Partitions of Large Integers." Combinatorics, Probability and Computing 22, no. 3 (2013): 433–54. http://dx.doi.org/10.1017/s0963548313000047.

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We consider partitions of the positive integernwhose parts satisfy the following condition. For a given sequence of non-negative numbers {bk}k≥1, a part of sizekappears in exactlybkpossible types. Assuming that a weighted partition is selected uniformly at random from the set of all such partitions, we study the asymptotic behaviour of the largest partXn. LetD(s)=∑k=1∞bkk−s,s=σ+iy, be the Dirichlet generating series of the weightsbk. Under certain fairly general assumptions, Meinardus (1954) obtained the asymptotic of the total number of such partitions asn→∞. Using the Meinardus scheme of con
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48

WENCHANG, CHU. "COMMON SOURCE OF NUMEROUS THETA FUNCTION IDENTITIES." Glasgow Mathematical Journal 49, no. 1 (2007): 61–79. http://dx.doi.org/10.1017/s0017089507003424.

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Abstract.Motivated by the recent work due to Warnaar (2005), two new and elementary proofs are presented for a very useful q-difference equation on eight shifted factorials of infinite order. As the common source of theta function identities, this q-difference equation is systematically explored to review old and establish new identities on Ramanujan's partition functions. Most of the identities obtained can be interpreted in terms of theorems on classical partitions.
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49

Subbarao, M. V., and A. K. Agarwal. "Further Theorems of the Rogers-Ramanujan Type Theorems*." Canadian Mathematical Bulletin 31, no. 2 (1988): 210–14. http://dx.doi.org/10.4153/cmb-1988-032-3.

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AbstractWe give three new partition theorems of the classical Rogers-Ramanujan type which are very much in the style of MacMahon. These are a continuation of four theorems of the same kind given recently by the second author. One of these new theorems, very similar to one of the original Rogers-Ramanuj an - MacMahon type theorems is as follows: Let C(n) denote the number of partitions of n into parts congruent to ±2, ± 3, ±4, ± 5, ±6, ±7 (mod 20). Let D(n) denote the number of partitions of n of the form n = b1 + b2 + … + bt, where bt ≧ 2, bt ≧ bi + 1 and if 1 ≦ i ≦ [(t - 2)/2], bi - bi + 1 ≧
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50

Landman, Bruce M., Ezra A. Brown, and Frederick J. Portier. "Partitions of bi-partite numbers into at mostj parts." Graphs and Combinatorics 8, no. 1 (1992): 65–73. http://dx.doi.org/10.1007/bf01271709.

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