scholarly journals Structure of spaces of rhombus tilings in the lexicograhic case

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Éric Rémila
Keyword(s):  
Maximal Element ◽  
Unitarity Condition ◽  
Graded Poset ◽  
International Audience ◽  
Positive Integers

International audience Rhombus tilings are tilings of zonotopes with rhombohedra. We study a class of \emphlexicographic rhombus tilings of zonotopes, which are deduced from higher Bruhat orders relaxing the unitarity condition. Precisely, we fix a sequence $(v_1, v_2,\dots, v_D)$ of vectors of $ℝ^d$ and a sequence $(m_1, m_2,\dots, m_D)$ of positive integers. We assume (lexicographic hypothesis) that for each subsequence $(v_{i1}, v_{i2},\dots, v_{id})$ of length $d$, we have $det(v_{i1}, v_{i2},\dots, v_{id}) > 0$. The zonotope $Z$ is the set $\{ Σα _iv_i 0 ≤α _i ≤m_i \}$. Each prototile used in a tiling of $Z$ is a rhombohedron constructed from a subsequence of d vectors. We prove that the space of tilings of $Z$ is a graded poset, with minimal and maximal element.

scholarly journals Congruence successions in compositions

2014 ◽  
Vol Vol. 16 no. 1 (Combinatorics) ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck ◽  
Mark Wilson
Keyword(s):  
General Formula ◽  
Generating Function ◽  
Asymptotic Estimate ◽  
International Audience ◽  
Limiting Case ◽  
Positive Integers

Combinatorics International audience A composition is a sequence of positive integers, called parts, having a fixed sum. By an m-congruence succession, we will mean a pair of adjacent parts x and y within a composition such that x=y(modm). Here, we consider the problem of counting the compositions of size n according to the number of m-congruence successions, extending recent results concerning successions on subsets and permutations. A general formula is obtained, which reduces in the limiting case to the known generating function formula for the number of Carlitz compositions. Special attention is paid to the case m=2, where further enumerative results may be obtained by means of combinatorial arguments. Finally, an asymptotic estimate is provided for the number of compositions of size n having no m-congruence successions.

scholarly journals On the 2-adic order of Stirling numbers of the second kind and their differences

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Tamás Lengyel
Keyword(s):  
Positive Integer ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Binary Representation ◽  
Exact Order ◽  
Special Cases ◽  
International Audience ◽  
The Difference ◽  
Divisibility Properties ◽  
Positive Integers

International audience Let $n$ and $k$ be positive integers, $d(k)$ and $\nu_2(k)$ denote the number of ones in the binary representation of $k$ and the highest power of two dividing $k$, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that $\nu_2(S(2^n,k))=d(k)-1, 1\leq k \leq 2^n$. Here we prove that $\nu_2(S(c2^n,k))=d(k)-1, 1\leq k \leq 2^n$, for any positive integer $c$. We improve and extend this statement in some special cases. For the difference, we obtain lower bounds on $\nu_2(S(c2^{n+1}+u,k)-S(c2^n+u,k))$ for any nonnegative integer $u$, make a conjecture on the exact order and, for $u=0$, prove part of it when $k \leq 6$, or $k \geq 5$ and $d(k) \leq 2$. The proofs rely on congruential identities for power series and polynomials related to the Stirling numbers and Bell polynomials, and some divisibility properties.

scholarly journals Statistics on Lattice Walks and q-Lassalle Numbers

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Lenny Tevlin
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Integer Coefficients ◽  
Major Index ◽  
Lattice Walks ◽  
International Audience ◽  
The Difference ◽  
Positive Integers

International audience This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan. Cet document contient deux résultats. Tout d’abord, je vous propose un $q$-generalization d’une certaine séquence de nombres entiers positifs, liés à nombres de Catalan, introduites par Zeilberger (Lassalle, 2010). Ces $q$-integers sont des polynômes palindromiques à $q$ à coefficients entiers positifs. La positivité dépend de la positivité d’une certaine différence de produits de $q$-coefficients binomial.Pour ce faire, je vous présente une nouvelle inversion/major index sur les chemins du réseau. La différence de $q$-binomial coefficients est alors considérée comme une fonction de génération de trajets pondérés qui restent dans le demi-plan supérieur.

scholarly journals Paths of specified length in random k-partite graphs

2001 ◽  
Vol Vol. 4 no. 2 ◽  
Author(s):  
C.R. Subramanian
Keyword(s):  
Random Graphs ◽  
Expected Number ◽  
Vertex Set ◽  
International Audience ◽  
Simple Paths ◽  
Positive Integers

International audience Fix positive integers k and l. Consider a random k-partite graph on n vertices obtained by partitioning the vertex set into V_i, (i=1, \ldots,k) each having size Ω (n) and choosing each possible edge with probability p. Consider any vertex x in any V_i and any vertex y. We show that the expected number of simple paths of even length l between x and y differ significantly depending on whether y belongs to the same V_i (as x does) or not. A similar phenomenon occurs when l is odd. This result holds even when k,l vary slowly with n. This fact has implications to coloring random graphs. The proof is based on establishing bijections between sets of paths.

scholarly journals A further correspondence between $(bc,\bar{b})$-parking functions and $(bc,\bar{b})$-forests

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Heesung Shin ◽  
Jiang Zeng
Keyword(s):  
The Other ◽  
Parking Functions ◽  
Fixed Sequence ◽  
Parking Function ◽  
International Audience ◽  
Positive Integers ◽  
Appl Math

International audience For a fixed sequence of $n$ positive integers $(a,\bar{b}) := (a, b, b,\ldots, b)$, an $(a,\bar{b})$-parking function of length $n$ is a sequence $(p_1, p_2, \ldots, p_n)$ of positive integers whose nondecreasing rearrangement $q_1 \leq q_2 \leq \cdots \leq q_n$ satisfies $q_i \leq a+(i-1)b$ for any $i=1,\ldots, n$. A $(a,\bar{b})$-forest on $n$-set is a rooted vertex-colored forests on $n$-set whose roots are colored with the colors $0, 1, \ldots, a-1$ and the other vertices are colored with the colors $0, 1, \ldots, b-1$. In this paper, we construct a bijection between $(bc,\bar{b})$-parking functions of length $n$ and $(bc,\bar{b})$-forests on $n$-set with some interesting properties. As applications, we obtain a generalization of Gessel and Seo's result about $(c,\bar{1})$-parking functions [Ira M. Gessel and Seunghyun Seo, Electron. J. Combin. $\textbf{11}$(2)R27, 2004] and a refinement of Yan's identity [Catherine H. Yan, Adv. Appl. Math. $\textbf{27}$(2―3):641―670, 2001] between an inversion enumerator for $(bc,\bar{b})$-forests and a complement enumerator for $(bc,\bar{b})$-parking functions. Soit $(a,\bar{b}) := (a, b, b,\ldots, b)$ une suite d'entiers positifs. Une $(a,\bar{b})$-fonction de parking est une suite $(p_1, p_2, \ldots, p_n)$ d'entiers positives telle que son réarrangement non décroissant $q_1 \leq q_2 \leq \cdots \leq q_n$ satisfait $q_i \leq a+(i-1)b$ pour tout $i=1,\ldots, n$. Une $(a,\bar{b})$-forêt enracinée sur un $n$-ensemble est une forêt enracinée dont les racines sont colorées avec les couleurs $0, 1, \ldots, a-1$ et les autres sommets sont colorés avec les couleurs $0, 1, \ldots, b-1$. Dans cet article, on construit une bijection entre $(bc,\bar{b})$-fonctions de parking et $(bc,\bar{b})$-forêts avec des des propriétés intéressantes. Comme applications, on obtient une généralisation d'un résultat de Gessel-Seo sur $(c,\bar{1})$-fonctions de parking [Ira M. Gessel and Seunghyun Seo, Electron. J. Combin. $\textbf{11}$(2)R27, 2004] et une extension de l'identité de Yan [Catherine H. Yan, Adv. Appl. Math. $\textbf{27}$(2―3):641―670, 2001] entre l'énumérateur d'inversion de $(bc,\bar{b})$-forêts et l'énumérateur complémentaire de $(bc,\bar{b})$-fonctions de parking.

scholarly journals Partition-theoretic formulas for arithmetic densities, II

2021 ◽  
Vol Volume 43 - Special... ◽  
Author(s):  
Ken Ono ◽  
Robert Schneider ◽  
Ian Wagner
Keyword(s):  
Dirichlet Series ◽  
First Principles ◽  
Prime Numbers ◽  
Integer Partitions ◽  
Root Of Unity ◽  
International Audience ◽  
Positive Integers ◽  
Limiting Values ◽  
Original Theorem ◽  
Do So

International audience In earlier work generalizing a 1977 theorem of Alladi, the authors proved a partition-theoretic formula to compute arithmetic densities of certain subsets of the positive integers N as limiting values of q-series as q → ζ a root of unity (instead of using the usual Dirichlet series to compute densities), replacing multiplicative structures of N by analogous structures in the integer partitions P. In recent work, Wang obtains a wide generalization of Alladi's original theorem, in which arithmetic densities of subsets of prime numbers are computed as values of Dirichlet series arising from Dirichlet convolutions. Here the authors prove that Wang's extension has a partition-theoretic analogue as well, yielding new q-series density formulas for any subset of N. To do so, we outline a theory of q-series density calculations from first principles, based on a statistic we call the "q-density" of a given subset. This theory in turn yields infinite families of further formulas for arithmetic densities.

scholarly journals Some new optimal and suboptimal infinite families of undirected double-loop networks

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Bao-Xing Chen ◽  
Ji-Xiang Meng ◽  
Wen-Jun Xiao
Keyword(s):  
Necessary Conditions ◽  
Undirected Graph ◽  
Double Loop ◽  
Circulant Graph ◽  
Double Loop Network ◽  
International Audience ◽  
Sufficient And Necessary Conditions ◽  
Loop Network ◽  
Double Loop Networks ◽  
Positive Integers

International audience Let n, s be positive integers such that 2 ≤ s < n and s = n/2 . An undirected double-loop network G(n; 1, s) is an undirected graph (V,E), where V =Zn={0, 1, 2, . . . , n−1} and E={(i, i+1 (mod n)), (i, i+s (mod n)) | i ∈ Z}. It is a circulant graph with n nodes and degree 4. In this paper, the sufficient and necessary conditions for a class of undirected double-loop networks to be optimal are presented. By these conditions, 6 new optimal and 5 new suboptimal infinite families of undirected double-loop networks are given.

scholarly journals Ideal solutions of the Tarry-Escott problem of degree eleven with applications to sums of thirteenth powers.

2008 ◽  
Vol Volume 31 ◽  
Author(s):  
Ajai Choudhry ◽  
Jaroslaw Wroblewski
Keyword(s):  
Diophantine Equations ◽  
Numerical Solutions ◽  
Infinite Family ◽  
Simultaneous Equations ◽  
Simultaneous Diophantine Equations ◽  
Ideal Solutions ◽  
International Audience ◽  
Positive Integers ◽  
The Ideal

International audience This paper is concerned with the system of simultaneous diophantine equations $\sum_{i=1}^6A_i^k=\sum_{i=1}^6B_i^k$ for $k=2, 4, 6, 8, 10.$ Till now only two numerical solutions of the system are known. This paper provides an infinite family of solutions. It is well-known that solutions of the above system lead to ideal solutions of the Tarry-Escott Problem of degree $11$, that is, of the system of simultaneous equations, $\sum_{i=1}^{12}a_i^k=\sum_{i=1}^{12}b_i^k$ for $k=1, 2, 3,\ldots,11.$ We use one of the ideal solutions to prove new results on sums of $13^{th}$ powers. In particular, we prove that every integer can be expressed as a sum or difference of at most $27$ thirteenth powers of positive integers.

scholarly journals Lifted generalized permutahedra and composition polynomials

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Federico Ardila ◽  
Jeffrey Doker
Keyword(s):  
Exponential Function ◽  
Homotopy Theory ◽  
Polynomial Interpolation ◽  
Combinatorial Properties ◽  
Nous Montrons ◽  
International Audience ◽  
Positive Integers

International audience We introduce a "lifting'' construction for generalized permutohedra, which turns an $n$-dimensional generalized permutahedron into an $(n+1)$-dimensional one. We prove that this construction gives rise to Stasheff's multiplihedron from homotopy theory, and to the more general "nestomultiplihedra,'' answering two questions of Devadoss and Forcey. We construct a subdivision of any lifted generalized permutahedron whose pieces are indexed by compositions. The volume of each piece is given by a polynomial whose combinatorial properties we investigate. We show how this "composition polynomial'' arises naturally in the polynomial interpolation of an exponential function. We prove that its coefficients are positive integers, and conjecture that they are unimodal. Nous introduisons une construction de "lifting'' (redressement) pour permutaèdres généralisés, qui transforme un permutaèdre généralisé de dimension $n$ en un de dimension $n+1$. Nous démontrons que cette construction conduit au multiplièdre de Stasheff à partir de la théorie d'homotopie, et aux "nestomultiplièdres", ce qui répond à deux questions de Devadoss et Forcey. Nous construisons une subdivision de n'importe quel permutaèdre généralisé dont les pièces sont indexées par compositions. La volume de chaque pièce est donnée par un polynôme dont nous recherchons les propriétés combinatoires. Nous montrons comment ce "polynôme de composition'' surgit naturellement dans l'interpolation d'une fonction exponentielle. Nous démontrons que ses coefficients sont strictement positifs, et nous conjecturons qu'ils sont unimodaux.

scholarly journals Links in the complex of weakly separated collections

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Suho Oh ◽  
David Speyer
Keyword(s):  
Short Note ◽  
Cluster Algebras ◽  
Combinatorial Objects ◽  
The Face ◽  
International Audience ◽  
Plabic Graphs ◽  
Positive Integers ◽  
Do So

International audience Plabic graphs are combinatorial objects used to study the totally nonnegative Grassmannian. Faces of plabic graphs are labeled by k-element sets of positive integers, and a collection of such k-element sets are the face labels of a plabic graph if that collection forms a maximal weakly separated collection. There are moves that one can apply to plabic graphs, and thus to maximal weakly separated collections, analogous to mutations of seeds in cluster algebras. In this short note, we show if two maximal weakly separated collections can be mutated from one to another, then one can do so while freezing the face labels they have in common. In particular, this provides a new, and we think simpler, proof of Postnikov's result that any two reduced plabic graphs with the same decorated permutations can be mutated to each other.

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