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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"

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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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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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Das, Sabuj. "PARTITION CONGRUENCES AND DYSON’S RANK." International Journal of Research -GRANTHAALAYAH 2, no. 2 (November 30, 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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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 (July 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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Yan, Xiao-Hui. "Partitions of the set of nonnegative integers with identical representation functions." International Journal of Number Theory 15, no. 10 (November 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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Knopfmacher, Arnold, and Augustine O. Munagi. "Successions in integer partitions." Ramanujan Journal 18, no. 3 (December 30, 2008): 239–55. http://dx.doi.org/10.1007/s11139-008-9140-2.

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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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KIM, BYUNGCHAN, and EUNMI KIM. "BIASES IN INTEGER PARTITIONS." Bulletin of the Australian Mathematical Society 104, no. 2 (January 14, 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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RØDSETH, ØYSTEIN J., and JAMES A. SELLERS. "PARTITIONS WITH PARTS IN A FINITE SET." International Journal of Number Theory 02, no. 03 (September 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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Ballantine, Cristina, and Mircea Merca. "Combinatorial proof of the minimal excludant theorem." International Journal of Number Theory 17, no. 08 (February 26, 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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Dissertations / Theses on the topic "Integer partitions"

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Chen, Xujin, and 陳旭瑾. "Graph partitions and integer flows." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2004. http://hub.hku.hk/bib/B30286256.

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Zoghbi, Antoine C. "Algorithms for generating integer partitions." Thesis, University of Ottawa (Canada), 1993. http://hdl.handle.net/10393/6506.

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In this thesis we consider the problem of generating integer partitions. We provide an overview of all known algorithms for the sequential generation of partitions of an integer. The performance is measured and compared separately for the standard and multiplicity representation of integer partitions. We present two new algorithms for generating integer partitions in the standard representation which generate partitions in lexicographic and antilexicographic order respectively. We prove that both algorithms generate partitions with constant average delay (exclusive of the output; output is generated and not printed). Historically, all existing algorithms for generating integer partitions in the multiplicity representation showed better performance than all the existing algorithms for generating integer partitions in the standard representation. An empirical test shows that both new algorithms are a few times faster than any previously known algorithms for generating unrestricted integer partitions in the standard representation. Moreover, they are faster than any known algorithm for generating integer partitions in the multiplicity representation (exclusive of the output). We describe several modifications to existing algorithms, and a transformation of one algorithm from the standard to the multiplicity representation. Finally, we provide a brief overview of sequential and parallel algorithms that generate partitions at random, and an analysis of a parallel algorithm for generating all partitions.

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French, Jennifer. "Vector Partitions." Digital Commons @ East Tennessee State University, 2018. https://dc.etsu.edu/etd/3392.

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Integer partitions have been studied by many mathematicians over hundreds of years. Many identities exist between integer partitions, such as Euler’s discovery that every number has the same amount of partitions into distinct parts as into odd parts. These identities can be proven using methods such as conjugation or generating functions. Over the years, mathematicians have worked to expand partition identities to vectors. In 1963, M. S. Cheema proved that every vector has the same number of partitions into distinct vectors as into vectors with at least one component odd. This parallels Euler’s result for integer partitions. The primary purpose of this paper is to use generating functions to prove other vector partition identities that parallel results of integer partitions.

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Ralaivaosaona, Dimbinaina. "Limit theorems for integer partitions and their generalisations." Thesis, Stellenbosch : Stellenbosch University, 2012. http://hdl.handle.net/10019.1/20019.

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Thesis (PhD)--Stellenbosch University, 2012.
ENGLISH ABSTRACT: Various properties of integer partitions are studied in this work, in particular the number of summands, the number of ascents and the multiplicities of parts. We work on random partitions, where all partitions from a certain family are equally likely, and determine moments and limiting distributions of the different parameters. The thesis focuses on three main problems: the first of these problems is concerned with the length of prime partitions (i.e., partitions whose parts are all prime numbers), in particular restricted partitions (i.e., partitions where all parts are distinct). We prove a central limit theorem for this parameter and obtain very precise asymptotic formulas for the mean and variance. The second main focus is on the distribution of the number of parts of a given multiplicity, where we obtain a very interesting phase transition from a Gaussian distribution to a Poisson distribution and further to a degenerate distribution, not only in the classical case, but in the more general context of ⋋-partitions: partitions where all the summands have to be elements of a given sequence ⋋ of integers. Finally, we look into another phase transition from restricted to unrestricted partitions (and from Gaussian to Gumbel-distribution) as we study the number of summands in partitions with bounded multiplicities.
AFRIKAANSE OPSOMMING: Verskillende eienskappe van heelgetal-partisies word in hierdie tesis bestudeer, in die besonder die aantal terme, die aantal stygings en die veelvoudighede van terme. Ons werk met stogastiese partisies, waar al die partisies in ’n sekere familie ewekansig is, en ons bepaal momente en limietverdelings van die verskillende parameters. Die teses fokusseer op drie hoofprobleme: die eerste van hierdie probleme gaan oor die lengte van priemgetal-partisies (d.w.s., partisies waar al die terme priemgetalle is), in die besonder beperkte partisies (d.w.s., partisies waar al die terme verskillend is). Ons bewys ’n sentrale limietstelling vir hierdie parameter en verkry baie presiese asimptotiese formules vir die gemiddelde en die variansie. Die tweede hooffokus is op die verdeling van die aantal terme van ’n gegewe veelvoudigheid, waar ons ’n baie interessante fase-oorgang van ’n normaalverdeling na ’n Poisson-verdeling en verder na ’n ontaarde verdeling verkry, nie net in die klassieke geval nie, maar ook in die meer algemene konteks van sogenaamde ⋋-partities: partisies waar al die terme elemente van ’n gegewe ry ⋋ van heelgetalle moet wees.

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Jonsson, Markus. "Processes on Integer Partitions and Their Limit Shapes." Doctoral thesis, Mälardalens högskola, Utbildningsvetenskap och Matematik, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-35023.

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This thesis deals with processes on integer partitions and their limit shapes, with focus on deterministic and stochastic variants on one such process called Bulgarian solitaire. The main scientific contributions are the following. Paper I: Bulgarian solitaire is a dynamical system on integer partitions of n which converges to a unique fixed point if n=1+2+...+k is a triangular number. There are few results about the structure of the game tree, but when k tends to infinity the game tree itself converges to a structure that we are able to analyze. Its level sizes turns out to be a bisection of the Fibonacci numbers. The leaves in this tree structure are enumerated using Fibonacci numbers as well. We also demonstrate to which extent these results apply to the case when k is finite. Paper II: Bulgarian solitaire is played on n cards divided into several piles; a move consists of picking one card from each pile to form a new pile. In a recent generalization, σ-Bulgarian solitaire, the number of cards you pick from a pile is some function σ of the pile size, such that you pick σ(h) < h cards from a pile of size h. Here we consider a special class of such functions. Let us call σ well-behaved if σ(1) = 1 and if both σ(h) and h − σ(h) are non-decreasing functions of h. Well-behaved σ-Bulgarian solitaire has a geometric interpretation in terms of layers at certain levels being picked in each move. It also satisfies that if a stable configuration of n cards exists it is unique. Moreover, if piles are sorted in order of decreasing size then a configuration is convex if and only if it is a stable configuration of some well-behaved σ-Bulgarian solitaire. If sorted configurations are represented by Young diagrams and scaled down to have unit height and unit area, the stable configurations corresponding to an infinite sequence of well-behaved functions (σ1, σ2, ...) may tend to a limit shape Φ. We show that every convex Φ with certain properties can arise as the limit shape of some sequence of well-behaved σn. For the special case when σn(h) = ceil(qnh) for 0 < qn ≤ 1 (where ceil is the ceiling function rounding upward to the nearest integer), these limit shapes are triangular (in case qn2n → 0), or exponential (in case qn2n → ∞), or interpolating between these shapes (in case qn2n → C > 0). Paper III: We introduce pn-random qn-proportion Bulgarian solitaire (0 < pn,qn ≤ 1), played on n cards distributed in piles. In each pile, a number of cards equal to the proportion qn of the pile size rounded upward to the nearest integer are candidates to be picked. Each candidate card is picked with probability pn, independently of other candidate cards. This generalizes Popov's random Bulgarian solitaire, in which there is a single candidate card in each pile. Popov showed that a triangular limit shape is obtained for a fixed p as n tends to infinity. Here we let both pn and qn vary with n. We show that under the conditions qn2pnn/log n → ∞ and pnqn → 0 as n → ∞, the pn-random qn-proportion Bulgarian solitaire has an exponential limit shape. Paper IV: We consider two types of discrete-time Markov chains where the state space is a graded poset and the transitions are taken along the covering relations in the poset. The first type of Markov chain goes only in one direction, either up or down in the poset (an up chain or down chain). The second type toggles between two adjacent rank levels (an up-and-down chain). We introduce two compatibility concepts between the up-directed transition probabilities (an up rule) and the down-directed (a down rule), and we relate these to compatibility between up-and-down chains. This framework is used to prove a conjecture about a limit shape for a process on Young's lattice. Finally, we settle the questions whether the reverse of an up chain is a down chain for some down rule and whether there exists an up or down chain at all if the rank function is not bounded.

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Matte, Marília Luiza. "Matrix representations for integer partitions : some consequences and a new approach." Universidade Federal do Rio Grande do Sul, 2018. http://hdl.handle.net/10183/178603.

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O presente trabalho dedica-se ao estudo de algumas consequências da representação matricial para conjuntos de partições de inteiros e funções mock theta. Na primeira parte do texto, classificamos as partições geradas por seis diferentes funções mock theta, de acordo com a soma das entradas da segunda linha das matrizes associadas, e apresentamos algumas fórmulas fechadas e identidades para essas partições. De nimos também a família ffm (q)gm 1 de funções mock theta, inspiradas pelo que chamamos de versão sem sinal da função f1(q). Fornecemos uma representação matricial análoga para as funções fm (q), o que leva a resultados interessantes a respeito das partições geradas por elas. A parte II do texto trata de uma nova abordagem que gera um conjunto diferente de partições de inteiros. A definição desse conjunto baseia-se na construção de um caminho sobre o reticulado Z2, determinado pela representao matricial para diferentes conjuntos de partições de n, e que liga a reta x + y = n a origem. As novas partições possuem apenas partes mpares distintas, com algumas restições particulares. Esse processo de construção de novas partições, chamado de Path Procedure, e aplicado a partições irrestritas, bem como para partições contadas pelas 1a e 2a Identidades de Rogers-Ramanujan e funções mock theta f5 (q) e T1(q).

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Davis, Simon. "On the existence of a non-zero lower bound for the number of Goldbach partitions of an even integer." Universität Potsdam, 2002. http://opus.kobv.de/ubp/volltexte/2008/2647/.

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The Goldbach partitions of an even number greater than 2, given by the sums of two prime addends, form the non-empty set for all integers 2n with 2 ≤ n ≤ 2 × 1014. It will be shown how to determine by the method of induction the existence of a non-zero lower bound for the number of Goldbach partitions of all even integers greater than or equal to 4. The proof depends on contour arguments for complex functions in the unit disk.

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Pétréolle, Mathias. "Quelques développements combinatoires autour des groupes de Coxeter et des partitions d'entiers." Thesis, Lyon 1, 2015. http://www.theses.fr/2015LYO10237/document.

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Cette thèse porte sur l'étude de la combinatoire énumérative, plus particulièrement autour des partitions d'entiers et des groupes de Coxeter. Dans une première partie, à l'instar de Han et de Nekrasov-Okounkov, nous étudions des développements combinatoires des puissances de la fonction êta de Dedekind, en termes de longueurs d'équerres de partitions d'entiers. Notre approche, bijective, utilise notamment les identités de Macdonald en types affines (en particulier le type C), généralisant l'approche de Han en type A. Nous étendons ensuite avec de nouveaux paramètres ces développements, grâce à de nouvelles propriétés de la décomposition de Littlewood vis-à-vis des partitions et statistiques considérées. Cela nous permet de déduire des formules des équerres symplectiques, ainsi qu'une connexion avec la théorie des représentations. Dans une seconde partie, nous étudions les éléments cycliquement pleinement commutatifs dans les groupes de Coxeter introduits par Boothby et al., qui forment une sous famille des éléments pleinement commutatifs. Nous commençons par développer une construction, la clôture cylindrique, donnant un cadre théorique qui est aux éléments CPC ce que les empilements de Viennot sont aux éléments PC. Nous donnons une caractérisation des éléments CPC en terme de clôtures cylindriques pour n'importe quel système de Coxeter. Celle-ci nous permet de déterminer en termes d'expressions réduites les éléments CPC dans tous les groupes de Coxeter finis ou affines, et d'en déduire dans tous ces groupes l'énumération de ces éléments. En utilisant la théorie des automates finis, nous montrons aussi que la série génératrice de ces éléments est une fraction rationnelle
This thesis focuses on enumerative combinatorics, particularly on integer partitions and Coxeter groups. In the first part, like Han and Nekrasov-Okounkov, we study the combinatorial expansion of power of the Dedekind's eta function, in terms of hook lengths of integer partitions. Our approach, bijective, use the Macdonald identities in affine types, generalizing the study of Han in the case of type A. We extend with new parameters the expansions that we obtained through new properties of the Littlewood decomposition. This enables us to deduce symplectic hook length formulas and a connexion with representation theory. In the second part, we study the cyclically fully commutative elements in Coxeter groups, introduced by Boothby et al., which are a sub family of the fully commutative elements. We start by introducing a new construction, the cylindrical closure, which give a theoretical framework for the CPC elements analogous to the Viennot's heaps for fully commutative elements. We give a characterization of CPC elements in terms of cylindrical closures in any Coxeter groups. This allows to deduce a characterization of these elements in terms of reduced decompositions in all finite and affine Coxeter and their enumerations in those groups. By using the theory of finite state automata, we show that the generating function of these elements is always rational, in all Coxeter groups

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Silva, Eduardo Alves da. "Formas ponderadas do Teorema de Euler e partições com raiz : estabelecendo um tratamento combinatório para certas identidades de Ramanujan." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2018. http://hdl.handle.net/10183/183163.

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O artigo Weighted forms of Euler's theorem de William Y.C. Chen e Kathy Q. Ji, em resposta ao questionamento de George E. Andrews, matemático estadunidense, sobre encontrar demonstrações combinatórias de duas identidades no Caderno Perdido de Ramanujan, nos mostra algumas formas ponderadas do Teorema de Euler sobre partições com partes ímpares e partes distintas via a introdução do conceito de partição com raiz. A propositura deste trabalho é envolta à apresentação de resultados sobre partições com raiz de modo a posteriormente realizar formulações combinatórias das identidades de Ramanujan por meio deste conceito, procurando estabelecer conexões com formas ponderadas do Teorema de Euler. Em particular, a bijeção de Sylvester e a iteração de Pak da função de Dyson são elementos primordiais para obtê-las.
The article Weighted forms of Euler's theorem by William Y.C. Chen and Kathy Q. Ji in response to the questioning of George E. Andrews, American mathematician, about nding combinatorial proofs for two identities in Ramanujan's Lost Notebook shows us some weighted forms of Euler's Theorem on partitions with odd parts and distinct parts through the introduction of the concept of rooted partition. The purpose of this work involves the presentation of results on rooted partitions in order to make combinatorial formulations of Ramanujan's identities, seeking to establish connections with weighted forms of Euler's Theorem. In particular, the Sylvester's bijection and the Pak's iteration of the Dyson's map are primordial elements to obtain them.

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Konan, Isaac. "Rogers-Ramanujan type identities : bijective proofs and Lie-theoretic approach." Thesis, Université de Paris (2019-....), 2020. http://www.theses.fr/2020UNIP7087.

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Cette thèse relève de la théorie des partitions d’entiers, à l’intersection de la combinatoire et de la théorie de nombres. En particulier, nous étudions les identités de type Rogers-Ramanujan sous le spectre de la méthode des mots pondérés. Une révision de cette méthode nous permet d’introduire de nouveaux objets combinatoires au delà de la notion classique de partitions d’entiers: partitions colorées généralisées. À l’aide de ces nouveaux éléments, nous établissons de nouvelles identités de type Rogers-Ramanujanvia deux approches différentes. La première approche consiste en une preuve combinatoire, essentiellement bijective, des identités étudiées. Cette approche nous a ainsi permis d’établir des identités généralisant plusieurs identités importantes de la théorie: l’identité de Schur et l’identité Göllnitz, l’identité de Glaisher généralisant l’identité d’Euler, les identités de Siladić, de Primc et de Capparelli issues de la théorie des représentations de algèbres de Lie affines. La deuxième approche fait appel à la théorie des cristaux parfaits, issue de la théorie des représentations des algèbres de Lie affines. Nous interprétons ainsi le caractère des représentations standards comme des identités de partitions d’entiers colorées généralisées. En particulier, cette approche permet d’établir des formules assez simplifiées du caractère pour toutes les représentations standards de niveau 1 des types affines A(1) n-1, A(2) 2n , D(2) n+1, A(2) 2n-1, B(1) n , D(1) n
The topic of this thesis belongs to the theory of integer partitions, at the intersection of combinatorics and number theory. In particular, we study Rogers-Ramanujan type identities in the framework of the method of weighted words. This method revisited allows us to introduce new combinatorial objects beyond the classical notion of integer partitions: the generalized colored partitions. Using these combinatorial objects, we establish new Rogers-Ramanujan identities via two different approaches.The first approach consists of a combinatorial proof, essentially bijective, of the studied identities. This approach allowed us to establish some identities generalizing many important identities of the theory of integer partitions : Schur’s identity and Göllnitz’ identity, Glaisher’s identity generalizing Euler’s identity, the identities of Siladić, of Primc and of Capparelli coming from the representation theory of affine Lie algebras. The second approach uses the theory of perfect crystals, coming from the representation theory of affine Lie algebras. We view the characters of standard representations as some identities on the generalized colored partitions. In particular, this approach allows us to establish simple formulas for the characters of all the level one standard representations of type A(1) n-1, A(2) 2n , D(2) n+1, A(2) 2n-1, B(1) n , D(1) n

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Books on the topic "Integer partitions"

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Alladi, Krishnaswami, Frank Garvan, and Ae Ja Yee. Ramanujan 125: International conference to commemorate the 125th anniversary of Ramanujan's birth, Ramanujan 125, November 5--7, 2012, University of Florida, Gainesville, Florida. Providence, Rhode Island: American Mathematical Society, 2014.

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Andrews, George E. Integer Partitions. 2nd ed. Cambridge University Press, 2004.

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Andrews, George E., and Kimmo Eriksson. Integer Partitions. 2nd ed. Cambridge University Press, 2004.

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Faiz, Asma. In Search of Lost Glory. Oxford University Press, 2021. http://dx.doi.org/10.1093/oso/9780197567135.001.0001.

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This book traces the trajectory of Sindhi nationalism in its quest for lost glory. It examines the Sindhi nationalist movement through its various stages, ranging from pre-partition identity construction in pursuit of the separation of Sindh from Bombay, to the post-partition travails of a community which lost its identity and its capital as a result of the arrival of millions of migrants from India (Muhajirs) and of the actions of an over-bearing central government. Going beyond the state and its power play, the book examines the long history of Sindhi-Muhajir contestation for resources in the post-partition period. The book develops a comprehensive profile of the agency of nationalist parties in Sindh, including the Sindhudesh detour and the later fragmentation of the Jiye Sind movement, which was followed by the emergence of new parties. The author also analyzes the dual role of the Pakistan Peoples Party (PPP) as an ethnic entrepreneur inside the province while operating as a federal party outside Sindh. The book covers nationalist contention at three levels: the struggle for power between Sindh and a dominant Centre; the inter-ethnic conflict between Sindhis and Muhajirs; and the intra-ethnic contestation between the Sindhi nationalists themselves and the PPP.

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Eckert, Astrid M. West Germany and the Iron Curtain. Oxford University Press, 2019. http://dx.doi.org/10.1093/oso/9780190690052.001.0001.

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West Germany and the Iron Curtain takes a fresh look at the history of Cold War Germany and the German reunification process from the spatial perspective of the West German borderlands that emerged along the volatile inter-German border after 1945. These border regions constituted the Federal Republic’s most sensitive geographical space, in which it had to confront partition and engage its socialist neighbor, East Germany, in concrete ways. Each issue that arose in these borderlands—from economic deficiencies to border tourism, environmental pollution, landscape change, and the siting decision for a major nuclear facility—was magnified and mediated by the presence of what became the most militarized border of its day, the Iron Curtain. In topical chapters, the book traces each of these issues across the caesura of 1989–1990, thereby integrating the “long” postwar era with the postunification decades. At the heart of this deeply-researched study stands an environmental history of the Iron Curtain that explores transboundary pollution, landscape change, and a planned nuclear industrial site at Gorleben that was meant to bring jobs into the depressed border regions. As Eckert demonstrates, the borderlands that emerged with partition and disappeared with reunification did not merely mirror larger developments in the Federal Republic’s history but actually helped shape them.

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Book chapters on the topic "Integer partitions"

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Arndt, Jörg. "Integer partitions." In Matters Computational, 339–53. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-14764-7_16.

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Borgs, Christian, Jennifer T. Chayes, Stephan Mertens, and Boris Pittel. "Constrained Integer Partitions." In LATIN 2004: Theoretical Informatics, 59–68. Berlin, Heidelberg: Springer Berlin Heidelberg, 2004. http://dx.doi.org/10.1007/978-3-540-24698-5_10.

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Canfield, E. Rodney, and Herbert S. Wilf. "On the Growth of Restricted Integer Partition Functions." In Partitions, q-Series, and Modular Forms, 39–46. New York, NY: Springer New York, 2011. http://dx.doi.org/10.1007/978-1-4614-0028-8_4.

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Sebő, András. "Path Partitions, Cycle Covers and Integer Decomposition." In Graph Theory, Computational Intelligence and Thought, 183–99. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-02029-2_18.

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Kolokolov, A. A. "Regular Partitions and Cuts in Integer Programming." In Discrete Analysis and Operations Research, 59–79. Dordrecht: Springer Netherlands, 1996. http://dx.doi.org/10.1007/978-94-009-1606-7_6.

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Latapy, Matthieu. "Generalized Integer Partitions, Tilings of Zonotopes and Lattices." In Formal Power Series and Algebraic Combinatorics, 256–67. Berlin, Heidelberg: Springer Berlin Heidelberg, 2000. http://dx.doi.org/10.1007/978-3-662-04166-6_23.

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Constantin, Hannah, Ben Houston-Edwards, and Nathan Kaplan. "Numerical Sets, Core Partitions, and Integer Points in Polytopes." In Springer Proceedings in Mathematics & Statistics, 99–127. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-68032-3_7.

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Ferrari, Luca, Renzo Pinzani, and Simone Rinaldi. "Enumerative Results on Integer Partitions Using the ECO Method." In Mathematics and Computer Science III, 25–36. Basel: Birkhäuser Basel, 2004. http://dx.doi.org/10.1007/978-3-0348-7915-6_3.

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Spircu, Tiberiu, and Stefan V. Pantazi. "Catalan Numbers Associated to Integer Partitions and the Super-Exponential." In Soft Computing Applications, 412–25. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-51992-6_33.

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Goles, Eric, Michel Morvan, and Ha Duong Phan. "About the Dynamics of Some Systems Based on Integer Partitions and Compositions." In Formal Power Series and Algebraic Combinatorics, 214–25. Berlin, Heidelberg: Springer Berlin Heidelberg, 2000. http://dx.doi.org/10.1007/978-3-662-04166-6_19.

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Conference papers on the topic "Integer partitions"

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Krishnamachari, Ramprasad S., and Panos Y. Papalambros. "Optimal Hierarchical Decomposition Synthesis Using Integer Programming." In ASME 1996 Design Engineering Technical Conferences and Computers in Engineering Conference. American Society of Mechanical Engineers, 1996. http://dx.doi.org/10.1115/96-detc/dac-1088.

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Abstract 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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Sambinelli, M., C. N. Lintzmayer, C. N. Da Silva, and O. Lee. "Vertex partition problems in digraphs ⇤." In III Encontro de Teoria da Computação. Sociedade Brasileira de Computação - SBC, 2018. http://dx.doi.org/10.5753/etc.2018.3174.

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Let D be a digraph and k be a positive integer. Linial (1981) conjectured that the k-norm of a k-minimum path partition of a digraph D is at most max{PC2 C |C| : C is a partial k-coloring of D}. Berge (1982) conjectured that every k-minimum path partition contains a partial k-coloring orthogonal to it. It is well known that Berge's Conjecture implies Linial's Conjecture. In this work, we verify Berge's Conjecture, and consequently Linial's Conjecture, for locally in-semicomplete digraphs and k-minimum path partitions containing only two paths. Moreover, we verify a conjecture related to Berge's and Linial's Conjectures for locally in-semicomplete digraphs.

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Chen, Ning-yu. "Some Properties of Partitions of Positive Integer as the Sum of Distinct Numbers." In 2012 Fourth International Conference on Computational and Information Sciences (ICCIS). IEEE, 2012. http://dx.doi.org/10.1109/iccis.2012.287.

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Cruz, Jadder Bismarck de Sousa, Cândida Nunes da Silva, and Orlando Lee. "Some Partial Results on Linial's Conjecture for Matching-Spine Digraphs." In Encontro de Teoria da Computação. Sociedade Brasileira de Computação - SBC, 2021. http://dx.doi.org/10.5753/etc.2021.16386.

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Let $k$ be a positive integer. A \emph{partial $k$-coloring} of a digraph $D$ is a set $\calC$ of $k$ disjoint stable sets and has \emph{weight} defined as $\sum_{C \in \calC} |C|$. An \emph{optimal} $k$-coloring is a $k$-coloring of maximum weight. A \emph{path partition} of a digraph $D$ is a set $\calP$ of disjoint paths of $D$ that covers its vertex set and has \emph{$k$-norm} defined as $\sum_{P \in \mathcal{P}} \min\{|P|,k\}$. A path partition $\calP$ is \emph{$k$-optimal} if it has minimum $k$-norm. A digraph $D$ is \emph{matching-spine} if its vertex set can be partitioned into sets $X$ and $Y$, such that $D[X]$ has a Hamilton path and the arc set of $D[Y]$ is a matching. Linial (1981) conjectured that the $k$-norm of a $k$-optimal path partition of a digraph is at most the weight of an optimal partial $k$-coloring. We present some partial results on this conjecture for matching-spine digraphs.

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Hanebutte, Ulf, and Jacob Hemstad. "ISx: A Scalable Integer Sort for Co-design in the Exascale Era." In 2015 9th International Conference on Partitioned Global Address Space Programming Models (PGAS). IEEE, 2015. http://dx.doi.org/10.1109/pgas.2015.21.

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Boonjing, Veera, and Santit Narabin. "An Integer Partition Based Algorithm for Coalition Structure Generation." In 7th IEEE International Conference on Computer and Information Technology (CIT 2007). IEEE, 2007. http://dx.doi.org/10.1109/cit.2007.157.

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Song, Zhihang, Bruce T. Murray, and Bahgat Sammakia. "Prediction of Hot Aisle Partition Airflow Boundary Conditions." In ASME 2013 International Technical Conference and Exhibition on Packaging and Integration of Electronic and Photonic Microsystems. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/ipack2013-73049.

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The integration of a simulation-based Artificial Neural Network (ANN) with a Genetic Algorithm (GA) has been explored as a real-time design tool for data center thermal management. The computation time for the ANN-GA approach is significantly smaller compared to a fully CFD-based optimization methodology for predicting data center operating conditions. However, difficulties remain when applying the ANN model for predicting operating conditions for configurations outside of the geometry used for the training set. One potential remedy is to partition the room layout into a finite number of characteristic zones, for which the ANN-GA model readily applies. Here, a multiple hot aisle/cold aisle data center configuration was analyzed using the commercial software FloTHERM. The CFD results are used to characterize the flow rates at the inter-zonal partitions. Based on specific reduced subsets of desired treatment quantities from the CFD results, such as CRAC and server rack air flow rates, the approach was applied for two different CRAC configurations and various levels of CRAC and server rack flow rates. Utilizing the compact inter-zonal boundary conditions, good agreement for the airflow and temperature distributions is achieved between predictions from the CFD computations for the entire room configuration and the reduced order zone-level model for different operating conditions and room layouts.

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Yan, Ning, Bin Li, Jizheng Xu, Houqiang Li, and Feng Wu. "Diagonal motion partitions for inter prediction in HEVC." In 2016 Visual Communications and Image Processing (VCIP). IEEE, 2016. http://dx.doi.org/10.1109/vcip.2016.7805575.

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Li Xia, Yanjia Zhao, Ming Xie, Jinyan Shao, and Jin Dong. "Mixed integer programming based nested partition algorithm for facility location optimization problems." In 2008 IEEE International Conference on Service Operations and Logistics, and Informatics (SOLI). IEEE, 2008. http://dx.doi.org/10.1109/soli.2008.4682933.

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Malyuta, Danylo, Behcet Acikmese, Martin Cacan, and David S. Bayard. "Partition-based Feasible Integer Solution Pre-computation for Hybrid Model Predictive Control." In 2019 18th European Control Conference (ECC). IEEE, 2019. http://dx.doi.org/10.23919/ecc.2019.8795790.

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