An application of Pólya’s enumeration theorem to partitions of subsets of positive integers

Xiao Jun Wu; Chong-Yun Chao

Displaying similar documents to “An application of Pólya’s enumeration theorem to partitions of subsets of positive integers”

Some Parity Statistics in Integer Partitions

Aubrey Blecher, Toufik Mansour, Augustine O. Munagi (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

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We study integer partitions with respect to the classical word statistics of levels and descents subject to prescribed parity conditions. For instance, a partition with summands $λ ₁ \geq \dots \geq λ_{k}$ may be enumerated according to descents $λ_{i} > λ_{i + 1}$ while tracking the individual parities of $λ_{i}$ and $λ_{i + 1}$ . There are two types of parity levels, E = E and O = O, and four types of parity-descents, E > E, E > O, O > E and O > O, where E and O represent arbitrary even and odd summands. We obtain functional equations...

On the $q$ -Pell sequences and sums of tails

Alexander E. Patkowski (2017)

Czechoslovak Mathematical Journal

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We examine the $q$ -Pell sequences and their applications to weighted partition theorems and values of $L$ -functions. We also put them into perspective with sums of tails. It is shown that there is a deeper structure between two-variable generalizations of Rogers-Ramanujan identities and sums of tails, by offering examples of an operator equation considered in a paper published by the present author. The paper starts with the classical example offered by Ramanujan and studied by previous...

On odd and semi-odd linear partitions of cubic graphs

Jean-Luc Fouquet, Henri Thuillier, Jean-Marie Vanherpe, Adam P. Wojda (2009)

Discussiones Mathematicae Graph Theory

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A linear forest is a graph whose connected components are chordless paths. A linear partition of a graph G is a partition of its edge set into linear forests and la(G) is the minimum number of linear forests in a linear partition. In this paper we consider linear partitions of cubic simple graphs for which it is well known that la(G) = 2. A linear partition $L = (L_{B}, L_{R})$ is said to be odd whenever each path of $L_{B} \cup L_{R}$ has odd length and semi-odd whenever each path of $L_{B}$ (or each path of $L_{R}$ ) has odd length. In...

Subspaces of $L_{p}$ , p > 2, determined by partitions and weights

Dale E. Alspach, Simei Tong (2003)

Studia Mathematica

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Many of the known complemented subspaces of $L_{p}$ have realizations as sequence spaces. In this paper a systematic approach to defining these spaces which uses partitions and weights is introduced. This approach gives a unified description of many well known complemented subspaces of $L_{p}$ . It is proved that the class of spaces with such norms is stable under (p,2) sums. By introducing the notion of an envelope norm, we obtain a necessary condition for a Banach sequence space with norm given...

A note on representing dowling geometries by partitions

František Matúš, Aner Ben-Efraim (2020)

Kybernetika

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We prove that a rank $\geq 3$ Dowling geometry of a group $H$ is partition representable if and only if $H$ is a Frobenius complement. This implies that Dowling group geometries are secret-sharing if and only if they are multilinearly representable.

On simple partitions of ${[κ]}^{κ}$

David Asperó (2003)

Fundamenta Mathematicae

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For every uncountable regular cardinal κ, every κ-Borel partition of the space of all members of ${[κ]}^{κ}$ whose enumerating function does not have fixed points has a homogeneous club.

Ramsey partitions and proximity data structures

Manor Mendel, Assaf Naor (2007)

Journal of the European Mathematical Society

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This paper addresses two problems lying at the intersection of geometric analysis and theoretical computer science: The non-linear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion.We introduce the notion of Ramsey partitions of a finite metric space, and show that the existence of good Ramsey partitions implies a solution to the metric Ramsey problem for large distortion (also known as the non-linear version of the isomorphic Dvoretzky...

Linear combinations of partitions of unity with restricted supports

Christian Richter (2002)

Studia Mathematica

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Given a locally finite open covering of a normal space X and a Hausdorff topological vector space E, we characterize all continuous functions f: X → E which admit a representation $f = \sum_{C \in} a_{C} φ_{C}$ with $a_{C} \in E$ and a partition of unity $φ_{C} : C \in$ subordinate to . As an application, we determine the class of all functions f ∈ C(||) on the underlying space || of a Euclidean complex such that, for each polytope P ∈ , the restriction ${f |}_{P}$ attains its extrema at vertices of P. Finally, a class of extremal functions on the...

The spt-crank for overpartitions

Frank G. Garvan, Chris Jennings-Shaffer (2014)

Acta Arithmetica

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Bringmann, Lovejoy, and Osburn (2009, 2010) showed that the generating functions of the spt-overpartition functions $\bar{s p t} (n)$ , ${\bar{s p t}}_{1} (n)$ , ${\bar{s p t}}_{2} (n)$ , and M2spt(n) are quasimock theta functions, and satisfy a number of simple Ramanujan-like congruences. Andrews, Garvan, and Liang (2012) defined an spt-crank in terms of weighted vector partitions which combinatorially explain simple congruences modulo 5 and 7 for spt(n). Chen, Ji, and Zang (2013) were able to define this spt-crank in terms of ordinary partitions....

On diffeomorphisms deleting weak compacta in Banach spaces

Daniel Azagra, Alejandro Montesinos (2004)

Studia Mathematica

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We prove that if X is an infinite-dimensional Banach space with $C^{p}$ smooth partitions of unity then X and X∖ K are $C^{p}$ diffeomorphic for every weakly compact set K ⊂ X.

The covering number for category and partition relations on $P_{ω} (λ)$

Pierre Matet (2002)

Fundamenta Mathematicae

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We show that cov(M) is the least infinite cardinal λ such that $P_{ω} (λ)$ (the set of all finite subsets of λ ) fails to satisfy a certain natural generalization of Ramsey’s Theorem.

On general Franklin systems

Gevorkyan Gegham, Kamont Anna

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AbstractWe study general Franklin systems, i.e. systems of orthonormal piecewise linear functions corresponding to quasi-dyadic sequences of partitions of [0,1]. The following problems are treated: unconditionality of the general Franklin basis in $L^{p}$ , 1 < p < ∞, and $H^{p}$ , 1/2 < p ≤ 1; equivalent conditions for the unconditional convergence of the Franklin series in $L^{p}$ for 0< p ≤ 1; relation between Haar and Franklin series with identical coefficients; characterization of the spaces...

Open and solved problems concerning polarized partition relations

Shimon Garti, Saharon Shelah (2016)

Fundamenta Mathematicae

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We list some open problems concerning the polarized partition relation. We solve a couple of them, by showing that for every limit non-inaccessible ordinal α there exists a forcing notion ℙ such that the strong polarized relation $(\binom{ℵ_{α + 1}}{ℵ_{α}}) \to {(\binom{ℵ_{α + 1}}{ℵ_{α}})}_{2}^{1, 1}$ holds in $V^{ℙ}$ .

Fragments of strong compactness, families of partitions and ideal extensions

Laura Fontanella, Pierre Matet (2016)

Fundamenta Mathematicae

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We investigate some natural combinatorial principles related to the notion of mild ineffability, and use them to obtain new characterizations of mild ineffable and weakly compact cardinals. We also show that one of these principles may be satisfied by a successor cardinal. Finally, we establish a version for $_{κ} (λ)$ of the canonical Ramsey theorem for pairs.

A partition of the Catalan numbers and enumeration of genealogical trees

Rainer Schimming (1996)

Discussiones Mathematicae Graph Theory

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A special relational structure, called genealogical tree, is introduced; its social interpretation and geometrical realizations are discussed. The numbers $C_{n, k}$ of all abstract genealogical trees with exactly n+1 nodes and k leaves is found by means of enumeration of code words. For each n, the $C_{n, k}$ form a partition of the n-th Catalan numer Cₙ, that means $C_{n, 1} + C_{n, 2} + . . . + C_{n, n} = C ₙ$ .

Pressure and recurrence

Véronique Maume-Deschamps, Bernard Schmitt, Mariusz Urbański, Anna Zdunik (2003)

Fundamenta Mathematicae

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We deal with a subshift of finite type and an equilibrium state μ for a Hölder continuous function. Let αⁿ be the partition into cylinders of length n. We compute (in particular we show the existence of the limit) $l i m_{n \to \infty} n^{- 1} l o g \sum_{j = 0}^{τ ₙ (x)} μ (α ⁿ (T^{j} (x)))$ , where $α ⁿ (T^{j} (x))$ is the element of the partition containing $T^{j} (x)$ and τₙ(x) is the return time of the trajectory of x to the cylinder αⁿ(x).

On a magnetic characterization of spectral minimal partitions

Bernard Helffer, Thomas Hoffmann-Ostenhof (2013)

Journal of the European Mathematical Society

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Given a bounded open set $Ω$ in $ℝ^{n}$ (or in a Riemannian manifold) and a partition of $Ω$ by $k$ open sets $D_{j}$ , we consider the quantity ${𝚖𝚊𝚡}_{j} λ (D_{j})$ where $λ (D_{j})$ is the ground state energy of the Dirichlet realization of the Laplacian in $D_{j}$ . If we denote by $ℒ_{k} (Ω)$ the infimum over all the $k$ -partitions of ${𝚖𝚊𝚡}_{j} λ (D_{j})$ , a minimal $k$ -partition is then a partition which realizes the infimum. When $k = 2$ , we find the two nodal domains of a second eigenfunction, but the analysis of higher $k$ ’s is non trivial and quite interesting. In this...

Operator theorems on $L^{P}$ -convergence to zero $(1 ⩽ p < + \infty)$

R. Zaharopol (1984)

Annales scientifiques de l'Université de Clermont-Ferrand 2. Série Probabilités et applications

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Partition ideals below $ℵ_{ω}$

P. Dodos, J. Lopez-Abad, S. Todorcevic (2012)

Fundamenta Mathematicae

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Motivated by an application to the unconditional basic sequence problem appearing in our previous paper, we introduce analogues of the Laver ideal on ℵ₂ living on index sets of the form ${[ℵ_{k}]}^{ω}$ and use this to refine the well-known high-dimensional polarized partition relation for $ℵ_{ω}$ of Shelah.