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Artículos de revistas sobre el tema "Integer Partition Function"

Autor: Grafiati

Publicado: 3 de junio de 2025

Última modificación: 31 de julio de 2025

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1

Davis, Simon. "A recursion relation for the number of Goldbach partitions of an even integer." Journal of Discrete Mathematical Sciences and Cryptography 27, no. 1 (2024): 1–30. http://dx.doi.org/10.47974/jdmsc-1188.

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The contour integral representation of the number of Goldbach partitions of an even integer, G(n), is extended to an integral with a support function that equals a linear combination of integers {G(m)}. A support function is found such that there is a nontrivial integral relation relating number of Goldbach partitions of n and m < n. The proof of the existence of a partition of any even integer greater than or equal to four into the sum of two primes follows from a recursion relation, resulting from an integral identity, that yields a non-zero lower bound for G(n). A partition of n, given a

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2

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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3

Andrews, George. "Separable integer partition classes." Transactions of the American Mathematical Society, Series B 9, no. 21 (2022): 619–47. http://dx.doi.org/10.1090/btran/87.

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A classical method for partition generating function is developed into a tool with wide applications. New expansions of well-known theorems are derived, and new results for partitions with n n copies of n n are presented.

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4

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

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

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5

Matte, M. L. "Some Special Integer Partitions Generated by a Family of Functions." Trends in Computational and Applied Mathematics 24, no. 4 (2023): 717–44. http://dx.doi.org/10.5540/tcam.2023.024.04.00717.

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In this work, inspired by Ramanujan’s fifth order Mock Theta function f1(q), we define acollection of functions and look at them as generating functions for partitions of some integer n containing at least m parts equal to each one of the numbers from 1 to its greatest part s, with no gaps.We set a two-line matrix representation for these partitions for any m ≥ 2 and collect the values of the sum of the entries in the second line of those matrices. These sums contain information about some parts of the partitions, which lead us to closed formulas for the number of partitions generated by our f

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6

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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7

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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8

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

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9

Kim, Jun Kyo, and Sang Guen Hahn. "Recursive formulae for the multiplicative partition function." International Journal of Mathematics and Mathematical Sciences 22, no. 1 (1999): 213–16. http://dx.doi.org/10.1155/s0161171299222132.

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For a positive integern, letf(n)be the number of essentially different ways of writingnas a product of factors greater than1, where two factorizations of a positive integer are said to be essentially the same if they differ only in the order of the factors. This paper gives a recursive formula for the multiplicative partition functionf(n).

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10

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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11

Hansen, Jennie C. "A functional central limit theorem for the Ewens sampling formula." Journal of Applied Probability 27, no. 01 (1990): 28–43. http://dx.doi.org/10.1017/s0021900200038407.

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For each n &gt; 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 st

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12

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

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

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13

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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14

Christopher, A. David. "Remainder sum and quotient sum function." Discrete Mathematics, Algorithms and Applications 07, no. 01 (2015): 1550001. http://dx.doi.org/10.1142/s1793830915500019.

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This paper is concerned with two arithmetical functions namely remainder sum function and quotient sum function which are respectively the sequences A004125 and A006218 in Online Encyclopedia of Integer Sequences. The remainder sum function is defined by [Formula: see text] for every positive integer n, and quotient sum function is defined by [Formula: see text] where q(n, i) is the quotient obtained when n is divided by i. We establish few divisibility properties these functions enjoy and we found their bounds. Furthermore, we define restricted remainder sum function by RA(n) = ∑k∈A n mod k w

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15

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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16

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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17

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 tw

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18

Buragohain, P., and N. Saikia. "Some new congruences for simultaneously s-regular and t-distinct partition function." Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, no. 10 (November 9, 2024): 18–21. http://dx.doi.org/10.26907/0021-3446-2024-10-18-21.

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A partition of a positive integer n is said to be simultaneously s-regular and t-distinct partition if none of the parts is divisible by s and parts appear fewer than t times. In this paper, we present some new congruences for simultaneously s-regular and t-distinct partition function denoted by M d (n) with (s, t) (2, 5), (3, 4), (4, 9), (5\alpha, 5\beta ), (7\alpha, 7\beta ), (p, p), where\alpha and \beta are any positive integers and p is any prime.

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19

Blecher, Aubrey, Arnold Knopfmacher, and Michael Mays. "Shade in partitions." Online Journal of Analytic Combinatorics, no. 18 (December 31, 2023): 1–15. https://doi.org/10.61091/ojac-1807.

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Integer partitions of \( n \) are viewed as bargraphs (i.e., Ferrers diagrams rotated anticlockwise by 90 degrees) in which the \( i \)-th part of the partition \( x_i \) is given by the \( i \)-th column of the bargraph with \( x_i \) cells. The sun is at infinity in the northwest of our two-dimensional model, and each partition casts a shadow in accordance with the rules of physics. The number of unit squares in this shadow but not being part of the partition is found through a bivariate generating function in \( q \) tracking partition size and \( u \) tracking shadow. To do this, we define

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20

GRABNER, PETER J., ARNOLD KNOPFMACHER, and STEPHAN WAGNER. "A General Asymptotic Scheme for the Analysis of Partition Statistics." Combinatorics, Probability and Computing 23, no. 6 (2014): 1057–86. http://dx.doi.org/10.1017/s0963548314000418.

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We consider statistical properties of random integer partitions. In order to compute means, variances and higher moments of various partition statistics, one often has to study generating functions of the form P(x)F(x), where P(x) is the generating function for the number of partitions. In this paper, we show how asymptotic expansions can be obtained in a quasi-automatic way from expansions of F(x) around x = 1, which parallels the classical singularity analysis of Flajolet and Odlyzko in many ways. Numerous examples from the literature, as well as some new statistics, are treated via this met

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21

Ndegwa, Duncan, Loyford Njagi, Stephen Luketero, Benard Nzimbi, and Kikwai Benjamin. "Recursive Partitioning of Odd Integers into Primes and Semiprimes: A Novel Framework Toward Validating Lemoine’s Conjecture." IOSR Journal of Mathematics 21, no. 3 (2025): 36–43. https://doi.org/10.9790/0661-2103033643.

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This paper introduces a novel Recursive Partitioning Framework that builds upon additive number theory, with specific application to Lemoine's Conjecture, which asserts that every odd integer greater than 5 can be expressed as the sum of a prime and a semiprime. Inspired by recent developments in algorithmic formulations of Goldbachtype conjectures, we adapt the framework proposed by Sankei et al. (2023), originally used to partition even integers via expressions of the form; 𝐸 = (𝑃1 + 𝑃2 )+ (𝑃2 −𝑃1 ) 𝑛 with 𝑃1 , 𝑃2 ∈ ℙ, 𝑃2 > 𝑃1 , and 𝑛 ∈ ℕ, to develop a systematic method for generating and

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22

CARNEY, ALEXANDER, ANASTASSIA ETROPOLSKI, and SARAH PITMAN. "POWERS OF THE ETA-FUNCTION AND HECKE OPERATORS." International Journal of Number Theory 08, no. 03 (2012): 599–611. http://dx.doi.org/10.1142/s1793042112500339.

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Half-integer weight Hecke operators and their distinct properties play a major role in the theory surrounding partition numbers and Dedekind's eta-function. Generalizing the work of Ono in [K. Ono, The partition function and Hecke operators, Adv. Math.228 (2011) 527–534], here we obtain closed formulas for the Hecke images of all negative powers of the eta-function. These formulas are generated through the use of Faber polynomials. In addition, congruences for a large class of powers of Ramanujan's Delta-function are obtained in a corollary. We further exhibit a fast calculation for many large

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23

Acharyya, Amrita. "Type B set partitions, an analogue of restricted growth functions." Ars Combinatoria 162 (March 30, 2025): 191–204. https://doi.org/10.61091/ars162-14.

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<p> In this work, we study type B set partitions for a given specific positive integer \(k\) defined over \(\langle n \rangle = \{-n, -(n-1), \cdots, -1, 0, 1, \cdots, n-1, n\}\). We found a few generating functions of type B analogues for some of the set partition statistics defined by Wachs, White and Steingr&iacute;msson for partitions over positive integers \([n] = \{1, 2, \cdots, n\}\), both for standard and ordered set partitions respectively. We extended the idea of restricted growth functions utilized by Wachs and White for set partitions over \([n]\), in the scenario of \(\l

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24

Mayer, Daniel C. "Sharp bounds for the partition function of integer sequences." BIT 27, no. 1 (1987): 98–110. http://dx.doi.org/10.1007/bf01937358.

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25

Du, Julia Q. D., Edward Y. S. Liu, and Jack C. D. Zhao. "Congruence properties of pk(n)." International Journal of Number Theory 15, no. 06 (2019): 1267–90. http://dx.doi.org/10.1142/s1793042119500714.

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We present a unified approach to establish infinite families of congruences for [Formula: see text] for arbitrary positive integer [Formula: see text], where [Formula: see text] is given by the [Formula: see text]th power of the Euler product [Formula: see text]. For [Formula: see text], define [Formula: see text] to be the least positive integer such that [Formula: see text] and [Formula: see text] the least non-negative integer satisfying [Formula: see text]. Using the Atkin [Formula: see text]-operator, we find that the generating function of [Formula: see text] (respectively, [Formula: see

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26

Acosta Diaz, Róbinson J., Christian D. Rodríguez-Camargo, and Nami F. Svaiter. "Directed Polymers and Interfaces in Disordered Media." Polymers 12, no. 5 (2020): 1066. http://dx.doi.org/10.3390/polym12051066.

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We consider field theory formulation for directed polymers and interfaces in the presence of quenched disorder. We write a series representation for the averaged free energy, where all the integer moments of the partition function of the model contribute. The structure of field space is analysed for polymers and interfaces at finite temperature using the saddle-point equations derived from each integer moments of the partition function. For the case of an interface we obtain the wandering exponent ξ = ( 4 − d ) / 2 , also obtained by the conventional replica method for the replica symmetric sc

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27

Song, Jijian, and Bin Xu. "On Rational Functions with More than Three Branch Points." Algebra Colloquium 27, no. 02 (2020): 231–46. http://dx.doi.org/10.1142/s100538672000019x.

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Let d be a positive integer and Λ be a collection of partitions of d of the form (a1, …, ap), (b1, …, bq), (m1 + 1, 1, …, 1), …, (ml + 1, 1, …, 1), where (m1, …, ml) is a partition of p + q − 2 > 0. We prove that there exists a rational function on the Riemann sphere with branch data Λ if and only if max(m1, …, ml) < d/GCD(a1, …, ap, b1, …, bq). As an application, we give a new class of branch data which can be realized by Belyi functions on the Riemann sphere.

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28

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 r

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29

MERCA, MIRCEA. "Distinct partitions and overpartitions." Carpathian Journal of Mathematics 38, no. 1 (2021): 149–58. http://dx.doi.org/10.37193/cjm.2022.01.12.

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In 1963, Peter Hagis, Jr. provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the partition function $Q(n)$ which counts partitions of $n$ into distinct parts. Computing $Q(n)$ by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we investigate new connections between partitions into distinct parts and overpartitions and obtain a surprising recurrence relation for the number of partitions of $n$ into distinct parts. By particularization of this relation, we derive two

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30

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

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

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31

Chen, Na, Shane Chern, Yan Fan, and Ernest X. W. Xia. "Some generating functions and inequalities for the andrews–stanley partition functions." Proceedings of the Edinburgh Mathematical Society 65, no. 1 (2021): 120–35. http://dx.doi.org/10.1017/s0013091521000833.

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AbstractLet $\mathcal {O}(\pi )$ denote the number of odd parts in an integer partition $\pi$. In 2005, Stanley introduced a new statistic $\operatorname {srank}(\pi )=\mathcal {O}(\pi )-\mathcal {O}(\pi ')$, where $\pi '$ is the conjugate of $\pi$. Let $p(r,\,m;n)$ denote the number of partitions of $n$ with srank congruent to $r$ modulo $m$. Generating function identities, congruences and inequalities for $p(0,\,4;n)$ and $p(2,\,4;n)$ were then established by a number of mathematicians, including Stanley, Andrews, Swisher, Berkovich and Garvan. Motivated by these works, we deduce some genera

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32

Gisonni, Massimo, Tamara Grava, and Giulio Ruzza. "Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals." Annales Henri Poincaré 21, no. 10 (2020): 3285–339. http://dx.doi.org/10.1007/s00023-020-00922-4.

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Abstract We consider the Laguerre partition function and derive explicit generating functions for connected correlators with arbitrary integer powers of traces in terms of products of Hahn polynomials. It was recently proven in Cunden et al. (Ann. Inst. Henri Poincaré D, to appear) that correlators have a topological expansion in terms of weakly or strictly monotone Hurwitz numbers that can be explicitly computed from our formulae. As a second result, we identify the Laguerre partition function with only positive couplings and a special value of the parameter $$\alpha =-1/2$$ α = - 1 / 2 with

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33

A, Manjusree, and Srinivasa Sai Panuganti. "The number of smallest parts of Partitions of n." World Journal of Advanced Research and Reviews 17, no. 1 (2023): 119–25. https://doi.org/10.5281/zenodo.8053768.

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George E Andrews derived formula for the number of smallest parts of partitions of a positive integer <em>n</em>. In this paper we derived the generating function for the number of smallest parts of all partitions of <em>n</em> utilizing r-partitions of <em>n</em>. We also derive the generating function for Ac(<em>n</em>) , the number of smallest parts of the partitions of <em>n</em> which are multiples of c and also to evaluate the sum of smallest parts of partitions of <em>n</em> by applying the concept of r-partitions of <em>n</em>.

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34

MOROZ, ALEXANDER. "UPPER AND LOWER BOUNDS ON THE PARTITION FUNCTION OF THE HOFSTADTER MODEL." Modern Physics Letters B 10, no. 09 (1996): 409–16. http://dx.doi.org/10.1142/s0217984996000468.

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Using unitary equivalence of magnetic translation operators, explicit upper and lower convex bounds on the partition function of the Hofstadter model are given for any rational “flux” and any value of Bloch momenta. These bounds (i) generalise straightforwardly to the case of a general asymmetric hopping and to the case of hopping of the form [Formula: see text] with n arbitrary integer larger than or equal 2, and (ii) allow to derive bounds on the derivatives of the partition function.

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35

Matsoukas, Themis. "Stochastic Theory of Discrete Binary Fragmentation—Kinetics and Thermodynamics." Entropy 24, no. 2 (2022): 229. http://dx.doi.org/10.3390/e24020229.

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We formulate binary fragmentation as a discrete stochastic process in which an integer mass k splits into two integer fragments j, k−j, with rate proportional to the fragmentation kernel Fj,k−j. We construct the ensemble of all distributions that can form in fixed number of steps from initial mass M and obtain their probabilities in terms of the fragmentation kernel. We obtain its partition function, the mean distribution and its evolution in time, and determine its stability using standard thermodynamic tools. We show that shattering is a phase transition that takes place when the stability c

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36

Leinartene, A. B., and A. P. Lyapin. "Applying computer algebra systems to study Chaundy-Bullard identities for the vector partition function with weight." Программирование, no. 2 (April 15, 2024): 79–83. http://dx.doi.org/10.31857/s0132347424020105.

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An algorithm for obtaining the Chaundy-Bullard identity for a vector partition function with weight that uses computer algebra methods is proposed. To automate this process in Maple, an algorithm was developed and implemented that calculates the values of the vector partition function with weight by finding non-negative solutions of systems of linear Diophantine equations that are used to form the identities involved. The algorithm’s input data is represented by the set of integer vectors that form a pointed lattice cone and by some point from this cone, and the Chaundy-Bullard identity for th

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37

NIK HASMI, NIK AIZAD, SITI FATIMAH ZAKARIA, and FATIMAH ABDUL RAZAK. "ZEROS OF PARTITION FUNCTION FOR Z5-SYMMETRIC MODEL ON SQUARE LATTICE-PARTICULAR CASE." JOURNAL OF SUSTAINABILITY SCIENCE AND MANAGEMENT 19, no. 6 (2024): 1–14. http://dx.doi.org/10.46754/jssm.2024.06.001.

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We study a statistical mechanics model of multiple-phase transitions called the ZQ-symmetric model on the square lattice for five possible spin directions (Q = 5). This study aims to investigate the existence of phase transition(s) and discuss the distribution of partition function zeros on the complex-temperature plane. Some cases of energy list χ are considered where they are written in an arbitrary arrangement of integer numbers (usually in decreasing value due to the angle of separation). The partition functions are computed using a transfer matrix approach, and their zeros are found numer

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38

DAMGAARD, P. H., and J. LACKI. "PARTITION FUNCTION ZEROS OF AN ISING SPIN GLASS." International Journal of Modern Physics C 06, no. 06 (1996): 819–43. http://dx.doi.org/10.1142/s012918319500068x.

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We study the pattern of zeros emerging from exact partition function evaluations of Ising spin glasses on conventional finite lattices of varying sizes. A large number of random bond configurations are probed in the framework of quenched averages. This study is motivated by the relationship between hierarchical lattice models whose partition function zeros fall on Julia sets and chaotic renormalization group flows in such models with frustration, and by the possible connection of the latter with spin glass behavior. In any finite volume, the simultaneous distribution of the zeros of all partit

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39

Xia, Ernest X. W. "Newman's identities, lucas sequences and congruences for certain partition functions." Proceedings of the Edinburgh Mathematical Society 63, no. 3 (2020): 709–36. http://dx.doi.org/10.1017/s0013091520000115.

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AbstractLet r be an integer with 2 ≤ r ≤ 24 and let pr(n) be defined by $\sum _{n=0}^\infty p_r(n) q^n = \prod _{k=1}^\infty (1-q^k)^r$. In this paper, we provide uniform methods for discovering infinite families of congruences and strange congruences for pr(n) by using some identities on pr(n) due to Newman. As applications, we establish many infinite families of congruences and strange congruences for certain partition functions, such as Andrews's smallest parts function, the coefficients of Ramanujan's ϕ function and p-regular partition functions. For example, we prove that for n ≥ 0, \[ \t

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40

Alvarez, P. D. "Exact partition function of the Potts model on the Sierpinski gasket and the Hanoi lattice." Journal of Statistical Mechanics: Theory and Experiment 2024, no. 8 (2024): 083101. http://dx.doi.org/10.1088/1742-5468/ad64bc.

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Abstract We present an analytic study of the Potts model partition function on the Sierpinski and Hanoi lattices, which are self-similar lattices of triangular shape with non integer Hausdorff dimension. Both lattices are examples of non-trivial thermodynamics in less than two dimensions, where mean field theory does not apply. We used and explain a method based on ideas of graph theory and renormalization group theory to derive exact equations for appropriate variables that are similar to the restricted partition functions. We benchmark our method with Metropolis Monte Carlo simulations. The

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41

Hughes, Kim. "Ramanujan Congruences For p-k(n) Modulo Powers Of 17." Canadian Journal of Mathematics 43, no. 3 (1991): 506–25. http://dx.doi.org/10.4153/cjm-1991-031-0.

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For each integer r we define the sequence pr(n) by We note that p-1(n) = p(n), the ordinary partition function. On account of this some authors set r = — k to make positive values of k correspond to positive powers of the generating function for p(n): We follow this convention here. In [3], Atkin proved the following theorem.

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42

Wang, Liuquan. "Arithmetic identities and congruences for partition triples with 3-cores." International Journal of Number Theory 12, no. 04 (2016): 995–1010. http://dx.doi.org/10.1142/s1793042116500627.

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Let [Formula: see text] denote the number of partition triples of [Formula: see text] where each partition is 3-core. With the help of generating function manipulations, we find several infinite families of arithmetic identities and congruences for [Formula: see text]. Moreover, let [Formula: see text] denote the number of representations of a non-negative integer [Formula: see text] in the form [Formula: see text] with [Formula: see text] We find three arithmetic relations between [Formula: see text] and [Formula: see text], such as [Formula: see text].

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43

BACCAR, N. "SETS WITH EVEN PARTITION FUNCTIONS AND CYCLOTOMIC NUMBERS." Journal of the Australian Mathematical Society 100, no. 3 (2016): 289–302. http://dx.doi.org/10.1017/s1446788715000439.

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Let $P\in \mathbb{F}_{2}[z]$ be such that $P(0)=1$ and degree $(P)\geq 1$. Nicolas et al. [‘On the parity of additive representation functions’, J. Number Theory73 (1998), 292–317] proved that there exists a unique subset ${\mathcal{A}}={\mathcal{A}}(P)$ of $\mathbb{N}$ such that $\sum _{n\geq 0}p({\mathcal{A}},n)z^{n}\equiv P(z)~\text{mod}\,2$, where $p({\mathcal{A}},n)$ is the number of partitions of $n$ with parts in ${\mathcal{A}}$. Let $m$ be an odd positive integer and let ${\it\chi}({\mathcal{A}},.)$ be the characteristic function of the set ${\mathcal{A}}$. Finding the elements of the

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44

ZHOU, NIAN HONG. "ON THE DISTRIBUTION OF THE RANK STATISTIC FOR STRONGLY CONCAVE COMPOSITIONS." Bulletin of the Australian Mathematical Society 100, no. 2 (2019): 230–38. http://dx.doi.org/10.1017/s0004972719000169.

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A strongly concave composition of $n$ is an integer partition with strictly decreasing and then increasing parts. In this paper we give a uniform asymptotic formula for the rank statistic of a strongly concave composition introduced by Andrews et al. [‘Modularity of the concave composition generating function’, Algebra Number Theory7(9) (2013), 2103–2139].

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45

Kim, Seung-Yeon, Richard J. Creswick, Chi-Ning Chen, and Chin-Kun Hu. "Partition function zeros of the Q-state Potts model for non-integer Q." Physica A: Statistical Mechanics and its Applications 281, no. 1-4 (2000): 262–67. http://dx.doi.org/10.1016/s0378-4371(00)00036-4.

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46

Delplace, Franck. "An Analytic Form for Riemann Zeta Function at Integer Values." Journal of Mathematics Research 14, no. 6 (2022): 1. http://dx.doi.org/10.5539/jmr.v14n6p1.

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An original definition of the generalized Euler-Mascheroni constants allowed us to prove that their infinite sum converges to the number (1-Ln2) . By considering this number is the Lebesgue measure of a set defined as the difference between the area of the square unit and the area under the curve y=1/x 1&le;x&le;2 ; we introduced a partition of this set such that each Lebesgue measure of the subsets can be related to values of Riemann zeta function at integers. From this relationship, we proved that the Lambert W function can produce all &zeta;(s)&nbsp; values whatever is the p

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47

Penniston, David. "11-Regular partitions and a Hecke eigenform." International Journal of Number Theory 15, no. 06 (2019): 1251–59. http://dx.doi.org/10.1142/s1793042119500696.

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A partition of a positive integer [Formula: see text] is called [Formula: see text]-regular if none of its parts is divisible by [Formula: see text]. Let [Formula: see text] denote the number of 11-regular partitions of [Formula: see text]. In this paper we give a complete description of the behavior of [Formula: see text] modulo [Formula: see text] when [Formula: see text] in terms of the arithmetic of the ring [Formula: see text]. This description is obtained by relating the generating function for these values of [Formula: see text] to a Hecke eigenform, and as a byproduct we find exact cri

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48

Pires, Eduardo J. Solteiro, Adelaide Cerveira, and José Baptista. "Wind Farm Cable Connection Layout Optimization Using a Genetic Algorithm and Integer Linear Programming." Computation 11, no. 12 (2023): 241. http://dx.doi.org/10.3390/computation11120241.

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This work addresses the wind farm (WF) optimization layout considering several substations. It is given a set of wind turbines jointly with a set of substations, and the goal is to obtain the optimal design to minimize the infrastructure cost and the cost of electrical energy losses during the wind farm lifetime. The turbine set is partitioned into subsets to assign to each substation. The cable type and the connections to collect wind turbine-produced energy, forwarding to the corresponding substation, are selected in each subset. The technique proposed uses a genetic algorithm (GA) and an in

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49

GARVAN, F. G. "CONGRUENCES FOR ANDREWS' SMALLEST PARTS PARTITION FUNCTION AND NEW CONGRUENCES FOR DYSON'S RANK." International Journal of Number Theory 06, no. 02 (2010): 281–309. http://dx.doi.org/10.1142/s179304211000296x.

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Let spt (n) denote the total number of appearances of smallest parts in the partitions of n. Recently, Andrews showed how spt (n) is related to the second rank moment, and proved some surprising Ramanujan-type congruences mod 5, 7 and 13. We prove a generalization of these congruences using known relations between rank and crank moments. We obtain explicit Ramanujan-type congruences for spt (n) mod ℓ for ℓ = 11, 17, 19, 29, 31 and 37. Recently, Bringmann and Ono proved that Dyson's rank function has infinitely many Ramanujan-type congruences. Their proof is non-constructive and utilizes the th

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50

DAS, SUMIT R., AVINASH DHAR, GAUTAM MANDAL, and SPENTA R. WADIA. "W-INFINITY WARD IDENTITIES AND CORRELATION FUNCTIONS IN THE c=1 MATRIX MODEL." Modern Physics Letters A 07, no. 11 (1992): 937–53. http://dx.doi.org/10.1142/s0217732392000835.

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We explore consequences of W-infinity symmetry in the fermionic field theory of the c=1 matrix model. We derive exact Ward identities relating correlation functions of the bilocal operator. These identities can be expressed as equations satisfied by the effective action of a three-dimensional theory and contain non-perturbative information about the model. We use these identities to calculate the two-point function of the bilocal operator in the double scaling limit. We extract the operator whose two-point correlator has a single pole at an (imaginary) integer value of the energy. We then rewr

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