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Journal articles on the topic 'P-partitions'

Author: Grafiati

Published: 4 June 2021

Last updated: 7 February 2022

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1

CUI, SU-PING, and NANCY SHAN SHAN GU. "CONGRUENCES FOR k DOTS BRACELET PARTITION FUNCTIONS." International Journal of Number Theory 09, no. 08 (December 2013): 1885–94. http://dx.doi.org/10.1142/s1793042113500644.

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Andrews and Paule introduced broken k-diamond partitions by using MacMahon's partition analysis. Recently, Fu found a generalization which he called k dots bracelet partitions and investigated some congruences for this kind of partitions. In this paper, by finding congruence relations between the generating function for 5 dots bracelet partitions and that for 5-regular partitions, we get some new congruences modulo 2 for the 5 dots bracelet partition function. Moreover, for a given prime p, we study arithmetic properties modulo p of k dots bracelet partitions.

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2

Féray, Valentin, and Victor Reiner. "$p$-partitions revisited." Journal of Commutative Algebra 4, no. 1 (March 2012): 101–52. http://dx.doi.org/10.1216/jca-2012-4-1-101.

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3

Stembridge, John R. "Enriched $P$-Partitions." Transactions of the American Mathematical Society 349, no. 2 (1997): 763–88. http://dx.doi.org/10.1090/s0002-9947-97-01804-7.

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4

KAAVYA, S. J. "CRANK 0 PARTITIONS AND THE PARITY OF THE PARTITION FUNCTION." International Journal of Number Theory 07, no. 03 (May 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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5

Hu, Z. X. "On ([n], P)-partitions." Advances in Applied Probability 24, no. 4 (December 1992): 775. http://dx.doi.org/10.1017/s000186780002485x.

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6

Brändén, Petter, and Madeleine Leander. "Lecture hall $P$-partitions." Journal of Combinatorics 11, no. 2 (2020): 391–412. http://dx.doi.org/10.4310/joc.2020.v11.n2.a9.

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Assaf, Sami, and Nantel Bergeron. "Flagged (P,ρ)-partitions." European Journal of Combinatorics 86 (May 2020): 103085. http://dx.doi.org/10.1016/j.ejc.2020.103085.

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8

Pesovic, Marko, and Tanja Stojadinovic. "Weighted P-partitions enumerator." Applicable Analysis and Discrete Mathematics, no. 00 (2021): 13. http://dx.doi.org/10.2298/aadm200525013p.

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To an extended generalized permutohedron we associate the weighted integer points enumerator, whose principal specialization is the f-polynomial. In the case of poset cones it refines Gessel's P-partitions enumerator. We show that this enumerator is a quasisymmetric function obtained by universal morphism from the Hopf algebra of posets.

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MULZER, WOLFGANG, and YANNIK STEIN. "ALGORITHMS FOR TOLERANT TVERBERG PARTITIONS." International Journal of Computational Geometry & Applications 24, no. 04 (December 2014): 261–73. http://dx.doi.org/10.1142/s0218195914600073.

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Let P be a d-dimensional n-point set. A partition [Formula: see text] of P is called a Tverberg partition if the convex hulls of all sets in [Formula: see text] intersect in at least one point. We say that [Formula: see text] is t-tolerant if it remains a Tverberg partition after deleting any t points from P. Soberón and Strausz proved that there is always a t-tolerant Tverberg partition with ⌈n/(d + 1)(t + 1)⌉ sets. However, no nontrivial algorithms for computing or approximating such partitions have been presented so far. For d ≤ 2, we show that the Soberón-Strausz bound can be improved, and we show how the corresponding partitions can be found in polynomial time. For d ≥ 3, we give the first polynomial-time approximation algorithm by presenting a reduction to the regular Tverberg problem (with no tolerance). Finally, we show that it is coNP-complete to determine whether a given Tverberg partition is t-tolerant.

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10

Hoshikawa, Kyosuke, Takuma Yuri, Hugo Giambini, and Yoshiro Kiyoshige. "Shoulder scaption is dependent on the behavior of the different partitions of the infraspinatus muscle." Surgical and Radiologic Anatomy 43, no. 5 (January 19, 2021): 653–59. http://dx.doi.org/10.1007/s00276-020-02674-6.

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Abstract Purpose The purpose of this study was to investigate if the three partitions (superior, middle, and inferior partitions) of the infraspinatus muscle previously described in anatomical studies will present different behavior during scapular plane abduction (scaption) as described using shear-wave elastography, especially during initial range of motion. Methods Eight volunteers held their arm against gravity 15° intervals from 30° to 150° in scaption. Shear-wave elastography was implemented at each position to measure shear modulus at rest and during muscle contraction, as a surrogate for muscle stiffness, of each partition. Muscle activity was defined as the difference in stiffness values between the resting positions and those during muscle contraction (ΔE = stiffness at contraction—stiffness at rest). Results The activity value for the middle partition was 25.1 ± 10.8 kPa at 30° and increased up to 105° (52.2 ± 10.8 kPa), with a subsequent decrease at larger angle positions (p < .001). The superior partition showed a flatter and constant behavior with smaller activity values except at higher angles (p < .001). Peak activity values for the superior partition were observed at 135° (23.0 ± 12.0 kPa). Increase activity for inferior partition began at 60° and showed a peak at 135° (p < .001; 32.9 ± 13.8 kPa). Conclusion Stiffness measured using shear-wave elastography in each partition of the infraspinatus muscle demonstrated different behavior between these partitions during scaption. The middle partition generated force throughout scaption, while the superior and inferior partitions exerted force at end range.

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11

Bellissima, Fabio, and Massimo Mirolli. "A general treatment of equivalent modalities." Journal of Symbolic Logic 54, no. 4 (December 1989): 1460–71. http://dx.doi.org/10.1017/s0022481200041207.

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The problem of the nonequivalent modalities available in certain systems is a classical problem of modal logic. In this paper we deal with this problem without referring to particular logics, but considering the whole class of normal propositional logics. Given a logic L let P(L) (the m-partition generated by L) denote the set of the classes of L-equivalent modalities. Obviously, different logics may generate the same m-partition; the first problem arising from this general point of view is therefore to determine the cardinality of the set of all m-partitions. Since, as is well known, there exist normal logics, and since one immediately realizes that there are infinitely many m-partitions, the problem consists in choosing (assuming the continuum hypothesis) between ℵ0 and . In Theorem 1.2 we show that there are m-partitions, as many as the logics.The next problem which naturally arises consists in determining, given an m-partition P(L), the number of logics generating P(L) (in symbols, μ(P(L))). In Theorem 2.1(ii) we show that ∣{P(L): μ(P(L)) = }∣ = . Now, the set {L′) = P(L)} has a natural minimal element; that is, the logic L* axiomatized by K ∪ {φ(p) ↔ ψ(p): φ, ψ are L-equivalent modalities}; P(L) and L* can be, in some sense, identified, thus making the set of m-partitions a subset of the set of logics.

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12

Pusadan, Mohammad Yazdi, Joko Lianto Buliali, and Raden Venantius Hari Ginardi. "Optimum partition in flight route anomaly detection." Indonesian Journal of Electrical Engineering and Computer Science 14, no. 3 (June 1, 2019): 1315. http://dx.doi.org/10.11591/ijeecs.v14.i3.pp1315-1329.

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Anomaly detection of flight route can be analyzed with the availability of flight data set. Automatic Dependent Surveillance (ADS-B) is the data set used. The parameters used are timestamp, latitude, longitude, and speed. The purpose of the research is to determine the optimum area for anomaly detection through real time approach. The methods used are: a) clustering and cluster validity analysis; and b) False Identification Rate (FIR). The results archieved are four steps, i.e: a) Build segments based on waypoints; b) Partition area based on 3-Dimension features P<sub>1</sub> and P<sub>2</sub>; c) grouping; and d) Measurement of cluster validity. The optimum partition is generated by calculating the minimum percentage of FIR. The results achieved are: i) there are five partitions, i.e: (n/2, n/3, n/4, n/5) and ii) optimal partition of each 3D, that is: for P<sub>1</sub> was five partitions and the P<sub>2</sub> feature was four partitions

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13

Petersen, T. Kyle. "Cyclic Descents and P-Partitions." Journal of Algebraic Combinatorics 22, no. 3 (November 2005): 343–75. http://dx.doi.org/10.1007/s10801-005-4532-5.

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14

Fulman, Jason, and T. Kyle Petersen. "Card shuffling and P-partitions." Discrete Mathematics 344, no. 8 (August 2021): 112448. http://dx.doi.org/10.1016/j.disc.2021.112448.

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15

Chandon, Jean-Louis, and Florence Dano. "Analyses typologiques confirmatoires. Evaluation d'une partition hypothétique issue d'une étude sémiotique." Recherche et Applications en Marketing (French Edition) 12, no. 2 (June 1997): 1–22. http://dx.doi.org/10.1177/076737019701200201.

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On constate qu'il est très important d'avoir des méthodes d'évaluation des partitions afin d'évaluer les résultats des analyses typologiques. Cet article développe une méthode fondée sur des critères internes et externes permettant d'estimer l'adéquation entre les partitions devant être évaluées et les données à partir desquelles elles ont été générées. La première partie présente plusieurs critères internes et externes de validation en typologie, et les solutions retenues par le logiciel Evalu-P, conçu par le premier auteur. La seconde partie illustre les étapes de validation d'une partition hypothétique en cinq classes de consommateurs, issue des résultats d'une analyse sémiotique du discours des consommateurs sur le packaging. 43 partitions pour le riz et 54 partitions pour le shampoing ont été générées en utilisant des méthodes de classification hiérarchique et de réallocation. Ces partitions empiriques ont été évaluées par rapport à la partition hypothétique en utilisant cinq critères internes et cinq critères externes de validation.

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16

Hummel, Tamara Lakins. "Effective versions of Ramsey's Theorem: Avoiding the cone above 0′." Journal of Symbolic Logic 59, no. 4 (December 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 set A. Applications to reverse mathematics and introreducible sets are discussed.

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17

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

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

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18

Kiderlen, Markus, and Florian Pausinger. "Discrepancy of stratified samples from partitions of the unit cube." Monatshefte für Mathematik 195, no. 2 (March 7, 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 prove that the expected $${{{\mathcal {L}}}_p}$$ L p -discrepancy, $${{\mathbb {E}}}{{{\mathcal {L}}}_p}({{\mathcal {P}}}_{\varvec{\Omega }})^p$$ E L p ( P Ω ) p , of a point set $${{\mathcal {P}}}_{\varvec{\Omega }}$$ P Ω generated from any equivolume partition $${\varvec{\Omega }}$$ Ω is always strictly smaller than the expected $${{{\mathcal {L}}}_p}$$ L p -discrepancy of a set of N uniform random samples for $$p>1$$ p > 1 . For fixed N we consider classes of stratified samples based on equivolume partitions of the unit cube into convex sets or into sets with a uniform positive lower bound on their reach. It is shown that these classes contain at least one minimizer of the expected $${{{\mathcal {L}}}_p}$$ L p -discrepancy. We illustrate our results with explicit constructions for small N. In addition, we present a family of partitions that seems to improve the expected discrepancy of Monte Carlo sampling by a factor of 2 for every N.

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19

Wu, Yunjian. "Parity results for broken 11-diamond partitions." Open Mathematics 17, no. 1 (May 16, 2019): 402–6. http://dx.doi.org/10.1515/math-2019-0031.

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Abstract Recently, Dai proved new infinite families of congruences modulo 2 for broken 11-diamond partition functions by using Hecke operators. In this note, we establish new parity results for broken 11-diamond partition functions. In particular, we generalize the congruences due to Dai by utilizing an identity due to Newman. Furthermore, we prove some strange congruences modulo 2 for broken 11-diamond partition functions. For example, we prove that if p is a prime with p ≠ 23 and Δ11(2p + 1) ≡ 1 (mod 2), then for any k ≥ 0, Δ11(2p3k+3 + 1) ≡ 1 (mod 2), where Δ11(n) is the number of broken 11-diamond partitions of n.

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20

Grippo, Luciano N., Martín Matamala, Martín D. Safe, and Maya J. Stein. "Convex p-partitions of bipartite graphs." Theoretical Computer Science 609 (January 2016): 511–14. http://dx.doi.org/10.1016/j.tcs.2015.11.014.

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21

Gessel, Ira M., Zhicong Lin, and Jiang Zeng. "Jacobi–Stirling polynomials and P-partitions." European Journal of Combinatorics 33, no. 8 (November 2012): 1987–2000. http://dx.doi.org/10.1016/j.ejc.2012.06.008.

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22

Petersen, T. Kyle. "Enriched P-partitions and peak algebras." Advances in Mathematics 209, no. 2 (March 2007): 561–610. http://dx.doi.org/10.1016/j.aim.2006.05.016.

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23

Park, SeungKyung. "P-partitions and q-stirling numbers." Journal of Combinatorial Theory, Series A 68, no. 1 (October 1994): 33–52. http://dx.doi.org/10.1016/0097-3165(94)90090-6.

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24

Malvenuto, Claudia. "P-Partitions and the plactic congruence." Graphs and Combinatorics 9, no. 1 (March 1993): 63–73. http://dx.doi.org/10.1007/bf01195328.

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25

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 (September 8, 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 methodology. In addition, we show how to compute further terms in the asymptotic expansions of previously studied partition statistics.

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Ulas, Maciej, and Błażej Żmija. "On p-adic valuations of colored p-ary partitions." Monatshefte für Mathematik 188, no. 2 (October 23, 2018): 351–68. http://dx.doi.org/10.1007/s00605-018-1227-2.

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27

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 (July 2, 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 Ramanujan-type congruences for b5(n), and we prove the striking result that the number of 5-regular partitions of the number n is even for at least 75% of the positive integers n.

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28

LENCZEWSKI, ROMUALD, and RAFAŁ SAŁAPATA. "NONCOMMUTATIVE BROWNIAN MOTIONS ASSOCIATED WITH KESTEN DISTRIBUTIONS AND RELATED POISSON PROCESSES." Infinite Dimensional Analysis, Quantum Probability and Related Topics 11, no. 03 (September 2008): 351–75. http://dx.doi.org/10.1142/s0219025708003154.

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We introduce and study a noncommutative two-parameter family of noncommutative Brownian motions in the free Fock space. They are associated with Kesten laws and give a continuous interpolation between Brownian motions in free probability and monotone probability. The combinatorics of our model is based on ordered non-crossing partitions, in which to each such partition P we assign the weight w(P) = pe(P)qe'(P), where e(P) and e'(P) are, respectively, the numbers of disorders and orders in P related to the natural partial order on the set of blocks of P implemented by the relation of being inner or outer. In particular, we obtain a simple relation between Delaney's numbers (related to inner blocks in non-crossing partitions) and generalized Euler's numbers (related to orders and disorders in ordered non-crossing partitions). An important feature of our interpolation is that the mixed moments of the corresponding creation and annihilation processes also reproduce their monotone and free counterparts, which does not take place in other interpolations. The same combinatorics is used to construct an interpolation between free and monotone Poisson processes.

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29

Bogosel, Beniamin, and Virginie Bonnaillie-Noël. "Minimal partitions for $p$-norms of eigenvalues." Interfaces and Free Boundaries 20, no. 1 (May 3, 2018): 129–63. http://dx.doi.org/10.4171/ifb/399.

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30

Huang, Yufei, and Bolian Liu. "On the identities of modulo-p partitions." Mathematical and Computer Modelling 54, no. 9-10 (November 2011): 2385–91. http://dx.doi.org/10.1016/j.mcm.2011.05.046.

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31

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

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

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32

Kong, Xu. "A concise proof to the spectral and nuclear norm bounds through tensor partitions." Open Mathematics 17, no. 1 (May 16, 2019): 365–73. http://dx.doi.org/10.1515/math-2019-0028.

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Abstract On estimations of the lower and upper bounds for the spectral and nuclear norm of a tensor, Li established neat bounds for the two norms based on regular tensor partitions, and proposed a conjecture for the same bounds to be hold based on general tensor partitions [Z. Li, Bounds on the spectral norm and the nuclear norm of a tensor based on tensor partition, SIAM J. Matrix Anal. Appl., 37 (2016), pp. 1440-1452]. Later, Chen and Li provided a solution to the conjecture [Chen B., Li Z., On the tensor spectral p-norm and its dual norm via partitions]. In this short paper, we present a concise and different proof for the validity of the conjecture, which also offers a new and simpler proof to the bounds of the spectral and nuclear norms established by Li for regular tensor partitions. Two numerical examples are provided to illustrate tightness of these bounds.

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Walker, Grant. "Horizontal partitions and Kleshchev's algorithm." Mathematical Proceedings of the Cambridge Philosophical Society 120, no. 1 (July 1996): 55–60. http://dx.doi.org/10.1017/s0305004100074661.

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We shall work with rational Mn(K)-modules, where K is an infinite field of prime characteristic p > 0, and Mn(K) is the full matrix semigroup of n × n matrices over K. Recall that equivalence classes of simple rational Mn(K)-modules are parametrized by the set of all partitions λ = (λ1, λ2, …, λl) of length l = l(λ) ≤ n, and that the socle L(λ) of the Schur module (or dual Weyl module) H0(λ) is a simple Mn(K)-module whose highest weight corresponds to the partition λ.

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34

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 (January 20, 1997): 121–26. http://dx.doi.org/10.1142/s0217979297000150.

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

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35

Archibald, Margaret, Aubrey Blecher, Charlotte Brennan, Arnold Knopfmacher, and Toufik Mansour. "Two by two squares in set partitions." Mathematica Slovaca 70, no. 1 (February 25, 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 partitions of [n] are precisely the words π = π1π2⋯πn such that π1 = 1, and if i < j then the first occurrence of the letter i precedes the first occurrence of j. Such words are known as restricted growth functions. In this paper we find the number of squares of side two in the bargraph representation of the restricted growth functions of set partitions of [n]. These squares can overlap and their bases are not necessarily on the x-axis. We determine the generating function P(x, y, q) for the number of set partitions of [n] with exactly k blocks according to the number of squares of size two. From this we derive exact and asymptotic formulae for the mean number of two by two squares over all set partitions of [n].

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36

Garvan, Frank G. "Some Congruences for Partitions that are p -Cores." Proceedings of the London Mathematical Society s3-66, no. 3 (May 1993): 449–78. http://dx.doi.org/10.1112/plms/s3-66.3.449.

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37

Khukhro, E. I. "Construction of finite p-groups that admit partitions." Siberian Mathematical Journal 30, no. 6 (1990): 1010–19. http://dx.doi.org/10.1007/bf00970924.

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38

Zhang, He. "A comment on “‘NP = P?’ and restricted partitions”." Information Sciences 38, no. 3 (June 1986): 299–301. http://dx.doi.org/10.1016/0020-0255(86)90029-0.

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39

Deuber, Walter, and Neil Hindman. "Partitions and sums of (m, p, c)-sets." Journal of Combinatorial Theory, Series A 45, no. 2 (July 1987): 300–302. http://dx.doi.org/10.1016/0097-3165(87)90020-3.

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Li, Lang, Silvana Borges, Robarge D. Jason, Changyu Shen, Zeruesenay Desta, and David Flockhart. "A Penalized Mixture Model Approach in Genotype/Phenotype Association Analysis for Quantitative Phenotypes." Cancer Informatics 9 (January 2010): CIN.S3493. http://dx.doi.org/10.4137/cin.s3493.

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A mixture normal model has been developed to partition genotypes in predicting quantitative phenotypes. Its estimation and inference are performed through an EM algorithm. This approach can conduct simultaneous genotype clustering and hypothesis testing. It is a valuable method for predicting the distribution of quantitative phenotypes among multi-locus genotypes across genes or within a gene. This mixture model's performance is evaluated in data analyses for two pharmacogenetics studies. In one example, thirty five CYP2D6 genotypes were partitioned into three groups to predict pharmacokinetics of a breast cancer drug, Tamoxifen, a CYP2D6 substrate (p-value = 0.04). In a second example, seventeen CYP2B6 genotypes were categorized into three clusters to predict CYP2B6 protein expression (p-value = 0.002). The biological validities of both partitions are examined using established function of CYP2D6 and CYP2B6 alleles. In both examples, we observed genotypes clustered in the same group to have high functional similarities. The power and recovery rate of the true partition for the mixture model approach are investigated in statistical simulation studies, where it outperforms another published method.

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Deco, Gustavo, Christian Schittenkopf, and Bernd Schürmann. "Information Flow in Chaotic Symbolic Dynamics for Finite and Infinitesimal Resolution." International Journal of Bifurcation and Chaos 07, no. 01 (January 1997): 97–105. http://dx.doi.org/10.1142/s0218127497000078.

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We introduce an information-theory-based concept for the characterization of the information flow in chaotic systems in the framework of symbolic dynamics for finite and infinitesimal measurement resolutions. The information flow characterizes the loss of information about the initial conditions, i.e. the decay of statistical correlations (i.e. nonlinear and non-Gaussian) between the entire past and a point p steps into the future as a function of p. In the case where the partition generating the symbolic dynamics is finite, the information loss is measured by the mutual information that measures the statistical correlations between the entire past and a point p steps into the future. When the partition used is a generator and only one step ahead is observed (p = 1), our definition includes the Kolmogorov–Sinai entropy concept. The profiles in p of the mutual information describe the short- and long-range forecasting possibilities for the given partition resolution. For chaos it is more relevant to study the information loss for the case of infinitesimal partitions which characterizes the intrinsic behavior of the dynamics on an extremely fine scale. Due to the divergence of the mutual information for infinitesimal partitions, the "intrinsic" information flow is characterized by the conditional entropy which generalizes the Kolmogorov–Sinai entropy for the case of observing the uncertainty more than one step into the future. The intrinsic information flow offers an instrument for characterizing deterministic chaos by the transmission of information from the past to the future.

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de Paula, A. L., and Dagoberto S. Freitas. "Dynamics of entanglement among the environment oscillators of a many-body system." Modern Physics Letters B 30, no. 17 (June 30, 2016): 1650222. http://dx.doi.org/10.1142/s0217984916502225.

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In this work, we extend the discussion that began in Ref. 16 [A. L. de Paula, Jr., J. G. G. de Oliveira, Jr., J. G. P. de Faria, D. S. Freitas and M. C. Nemes, Phys. Rev. A 89 (2014) 022303] to deal with the dynamics of the concurrence of a many-body system. In that previous paper, the discussion was focused on the residual entanglement between the partitions of the system. The purpose of the present contribution is to shed some light on the dynamical properties of entanglement among the environment oscillators. We consider a system consisting of a harmonic oscillator linearly coupled to N others and solve the corresponding dynamical problem analytically. We divide the environment into two arbitrary partitions and the entanglement dynamics between any of these partitions is quantified and it shows that in the case when excitations in each partition are equal, the concurrence reaches the value 1 and the two partitions of the environment are maximally entangled. For long times, the excitations of the main oscillator are completely transferred to environment and the environment oscillators are found entangled.

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GILL, CHRISTOPHER C. "YOUNG MODULE MULTIPLICITIES, DECOMPOSITION NUMBERS AND THE INDECOMPOSABLE YOUNG PERMUTATION MODULES." Journal of Algebra and Its Applications 13, no. 05 (February 25, 2014): 1350147. http://dx.doi.org/10.1142/s0219498813501478.

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We study the multiplicities of Young modules as direct summands of permutation modules on cosets of Young subgroups. Such multiplicities have become known as the p-Kostka numbers. We classify the indecomposable Young permutation modules, and, applying the Brauer construction for p-permutation modules, we give some new reductions for p-Kostka numbers. In particular, we prove that p-Kostka numbers are preserved under multiplying partitions by p, and strengthen a known reduction corresponding to adding multiples of a p-power to the first row of a partition.

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Xiao, Wang, and Aihua Li. "Counting certain quadratic partitions of zero modulo a prime number." Open Mathematics 19, no. 1 (January 1, 2021): 198–211. http://dx.doi.org/10.1515/math-2021-0032.

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Abstract Consider an odd prime number p ≡ 2 ( mod 3 ) p\equiv 2\hspace{0.3em}\left(\mathrm{mod}\hspace{0.3em}3) . In this paper, the number of certain type of partitions of zero in Z / p Z {\mathbb{Z}}\hspace{-0.1em}\text{/}\hspace{-0.1em}p{\mathbb{Z}} is calculated using a combination of elementary combinatorics and number theory. The focus is on the three-part partitions of 0 in Z / p Z {\mathbb{Z}}\hspace{-0.1em}\text{/}\hspace{-0.1em}p{\mathbb{Z}} with all three parts chosen from the set of non-zero quadratic residues mod p p . Such partitions are divided into two types. Those with exactly two of the three parts identical are classified as type I. The type II partitions are those with all three parts being distinct. The number of partitions of each type is given. The problem of counting such partitions is well related to that of counting the number of non-trivial solutions to the Diophantine equation x 2 + y 2 + z 2 = 0 {x}^{2}+{y}^{2}+{z}^{2}=0 in the ring Z / p Z {\mathbb{Z}}\hspace{-0.1em}\text{/}\hspace{-0.1em}p{\mathbb{Z}} . Correspondingly, solutions to this equation are also classified as type I or type II. We give the number of solutions to the equation corresponding to each type.

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Johnson, C. A. "Distributive ideals and partition relations." Journal of Symbolic Logic 51, no. 3 (September 1986): 617–25. http://dx.doi.org/10.2307/2274018.

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It is a theorem of Rowbottom [12] that ifκis measurable andIis a normal prime ideal onκ, then for eachλ<κ,In this paper a natural structural property of ideals, distributivity, is considered and shown to be related to this and other ideal theoretic partition relations.The set theoretical terminology is standard (see [7]) and background results on the theory of ideals may be found in [5] and [8]. Throughoutκwill denote an uncountable regular cardinal, andIa proper, nonprincipal,κ-complete ideal onκ.NSκis the ideal of nonstationary subsets ofκ, andIκ= {X⊆κ∣∣X∣<κ}. IfA∈I+(=P(κ) −I), then anI-partitionofAis a maximal collectionW⊆,P(A) ∩I+so thatX∩ Y ∈IwheneverX, Y∈W, X≠Y. TheI-partitionWis said to be disjoint if distinct members ofWare disjoint, and in this case, fordenotes the unique member ofWcontainingξ. A sequence 〈Wα∣α<η} ofI-partitions ofAis said to be decreasing if wheneverα<β<ηandX∈Wβthere is aY∈Wαsuch thatX⊆Y. (i.e.,WβrefinesWα).

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Arima, Isao, and Hiroyuki Tagawa. "Generalized (P,ω)-partitions and generating functions for trees." Journal of Combinatorial Theory, Series A 103, no. 1 (July 2003): 137–50. http://dx.doi.org/10.1016/s0097-3165(03)00091-8.

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Kim, Byungchan, and Jeremy Rouse. "Explicit bounds for the number of $p$-core partitions." Transactions of the American Mathematical Society 366, no. 2 (August 19, 2013): 875–902. http://dx.doi.org/10.1090/s0002-9947-2013-05883-7.

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Allen, Edward E. "Descent Monomials, P-Partitions and Dense Garsia-Haiman Modules." Journal of Algebraic Combinatorics 20, no. 2 (September 2004): 173–93. http://dx.doi.org/10.1023/b:jaco.0000047281.84115.b7.

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Chen, William Y. C., Alan J. X. Guo, Peter L. Guo, Harry H. Y. Huang, and Thomas Y. H. Liu. "$s$-Inversion Sequences and $P$-Partitions of Type $B$." SIAM Journal on Discrete Mathematics 30, no. 3 (January 2016): 1632–43. http://dx.doi.org/10.1137/130942140.

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Gao, Wei, Qing-Hu Hou, and Guoce Xin. "On P-partitions related to ordinal sums of posets." European Journal of Combinatorics 30, no. 5 (July 2009): 1370–81. http://dx.doi.org/10.1016/j.ejc.2008.10.007.

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