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1.
The Ramanujan Journal - Let $${\bar{p}}(n)$$ denote the number of overpartitions of n. Recently, a number of congruences modulo powers of 2, 3 and 5 have been discovered. The moduli for these... 相似文献
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《Discrete Mathematics》2022,345(1):112672
Recently, Andrews proved two conjectures of Beck related to ranks of partitions. Very recently, Chern established some results on weighted rank and crank moments and proved many Andrews-Beck type congruences. Motivated by Andrews and Chern's work, Lin, Peng and Toh proved a number of Andrews-Beck type congruences for k-colored partitions. At the end of their paper, Lin, Peng and Toh posed several conjectures on Andrews-Beck type congruences. In this paper, we confirm one of those conjectures based on some q-series identities. 相似文献
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Let \(\bar{p}(n)\) denote the number of overpartitions of \(n\). Recently, Fortin–Jacob–Mathieu and Hirschhorn–Sellers independently obtained 2-, 3- and 4-dissections of the generating function for \(\bar{p}(n)\) and derived a number of congruences for \(\bar{p}(n)\) modulo 4, 8 and 64 including \(\bar{p}(8n+7)\equiv 0 \pmod {64}\) for \(n\ge 0\). In this paper, we give a 16-dissection of the generating function for \(\bar{p}(n)\) modulo 16 and show that \(\bar{p}(16n+14)\equiv 0\pmod {16}\) for \(n\ge 0\). Moreover, using the \(2\)-adic expansion of the generating function for \(\bar{p}(n)\) according to Mahlburg, we obtain that \(\bar{p}(\ell ^2n+r\ell )\equiv 0\pmod {16}\), where \(n\ge 0\), \(\ell \equiv -1\pmod {8}\) is an odd prime and \(r\) is a positive integer with \(\ell \not \mid r\). In particular, for \(\ell =7\) and \(n\ge 0\), we get \(\bar{p}(49n+7)\equiv 0\pmod {16}\) and \(\bar{p}(49n+14)\equiv 0\pmod {16}\). We also find four congruence relations: \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n) \pmod {16}\) for \(n\ge 0\), \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {32}\) where \(n\) is not a square of an odd positive integer, \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {64}\) for \(n\not \equiv 1,2,5\pmod {8}\) and \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {128}\) for \(n\equiv 0\pmod {4}\). 相似文献
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We present the transformation of several sums of positive integer powers of the sine and cosine into non-trigonometric combinatorial forms. The results are applied to the derivation of generating functions and to the number of the closed walks on a path and in a cycle. 相似文献
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We present two new Ramanujan-type congruences modulo 5 for overpartitions. We also give an affirmative answer to a conjecture of Dou and Lin, which includes four congruences modulo 25 for overpartition. 相似文献
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The Ramanujan Journal - In 2009, Corteel, Savelief and Vuleti? generalized the concept of overpartitions to a new object called plane overpartitions. In recent work, the author considered a... 相似文献
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A congruence on a conjecture of van Hamme is established. This result confirms a particular case of a congruence conjecture of Swisher. 相似文献
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O. Favaron H. Karami R. Khoeilar S. M. Sheikholeslami L. Volkmann 《Journal of Graph Theory》2010,64(4):323-329
Let G(V, E) be a simple, undirected graph where V is the set of vertices and E is the set of edges. A b‐dimensional cube is a Cartesian product I1×I2×···×Ib, where each Ii is a closed interval of unit length on the real line. The cubicity of G, denoted by cub(G), is the minimum positive integer b such that the vertices in G can be mapped to axis parallel b‐dimensional cubes in such a way that two vertices are adjacent in G if and only if their assigned cubes intersect. An interval graph is a graph that can be represented as the intersection of intervals on the real line—i.e. the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. Suppose S(m) denotes a star graph on m+1 nodes. We define claw number ψ(G) of the graph to be the largest positive integer m such that S(m) is an induced subgraph of G. It can be easily shown that the cubicity of any graph is at least ?log2ψ(G)?. In this article, we show that for an interval graph G ?log2ψ(G)??cub(G)??log2ψ(G)?+2. It is not clear whether the upper bound of ?log2ψ(G)?+2 is tight: till now we are unable to find any interval graph with cub(G)>?log2ψ(G)?. We also show that for an interval graph G, cub(G)??log2α?, where α is the independence number of G. Therefore, in the special case of ψ(G)=α, cub(G) is exactly ?log2α2?. The concept of cubicity can be generalized by considering boxes instead of cubes. A b‐dimensional box is a Cartesian product I1×I2×···×Ib, where each Ii is a closed interval on the real line. The boxicity of a graph, denoted box(G), is the minimum k such that G is the intersection graph of k‐dimensional boxes. It is clear that box(G)?cub(G). From the above result, it follows that for any graph G, cub(G)?box(G)?log2α?. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 323–333, 2010 相似文献
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Let e be a positive integer, p be an odd prime, , and be the finite field of q elements. Let . The graph is a bipartite graph with vertex partitions and , and edges defined as follows: a vertex is adjacent to a vertex if and only if and . If and , the graph contains no cycles of length less than eight and is edge-transitive. Motivated by certain questions in extremal graph theory and finite geometry, people search for examples of graphs containing no cycles of length less than eight and not isomorphic to the graph , even without requiring them to be edge-transitive. So far, no such graphs have been found. It was conjectured that if both f and g are monomials, then no such graphs exist. In this paper we prove the conjecture. 相似文献
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The Ramanujan Journal - In this paper, we obtain inequalities on $$M_2$$-ranks of overpartitions modulo 6. Let $$\overline{N}_2(s,m,n)$$ be the number of overpartitions of n whose $$M_2$$-rank is... 相似文献
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In this paper we prove a conjecture of Metsch about the maximum number of lines intersecting a pointset in PG(2,q), presented at the conference “Combinatorics 2002”. As a consequence, we give a short proof of the famous Jamison, Brouwer and Schrijver bound on the size of the smallest affine blocking set in AG(2,q). 相似文献
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We investigate the amenability of the semigroup algebras \({\ell^1(S/\rho)}\) , where \({\rho}\) is a group congruence (not necessarily minimal) on a semigroup S. We relate this to a new notion of amenability of Banach algebras modulo an ideal, to prove a version of Johnson’s theorem for a large class of semigroups, including inverse semigroups, E-inversive semigroup and E-inversive E-semigroups. 相似文献
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Let ir(G) and γ(G) be the irredundance number and the domination number of a graph G, respectively. A graph G is called irredundance perfect if ir(H)=γ(H), for every induced subgraph H of G. In this article we present a result which immediately implies three known conjectures on irredundance perfect graphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 292–306, 2002 相似文献
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Jacobson, Levin, and Scheinerman introduced the fractional Ramsey function rf (a1, a2, …, ak) as an extension of the classical definition for Ramsey numbers. They determined an exact formula for the fractional Ramsey function for the case k=2. In this article, we answer an open problem by determining an explicit formula for the general case k>2 by constructing an infinite family of circulant graphs for which the independence numbers can be computed explicitly. This construction gives us two further results: a new (infinite) family of star extremal graphs which are a superset of many of the families currently known in the literature, and a broad generalization of known results on the chromatic number of integer distance graphs. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 164–178, 2010 相似文献
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James G. Oxley 《Combinatorica》1985,5(4):343-345
This note proves a conjecture of Kahn by showing that ifX is a 3-element independent set in a 3-connected non-binary matroid M, thenM has a connected non-binary minor havingX as a basis.
This research was partially supported by an LSU Summer Research Grant. 相似文献
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Luís Sequeira 《Algebra Universalis》2006,55(4):495-508
A new condition of compatibility with projections, applicable to some Maltsev filters, is defined and shown to hold, among
others, for the filter of congruence-modular varieties. As a consequence, it is shown that there exist no simple counterexamples
(in a specified sense) to the modularity conjecture.
This paper is dedicated to Walter Taylor.
Received November 5, 2005; accepted in final form April 3, 2006. 相似文献
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A tree T is said to be bad, if it is the vertex‐disjoint union of two stars plus an edge joining the center of the first star to an end‐vertex of the second star. A tree T is good, if it is not bad. In this article, we prove a conjecture of Alan Hartman that, for any spanning tree T of K2m, where m ≥ 4, there exists a (2m − 1)‐edge‐coloring of K2m such that all the edges of T receive distinct colors if and only if T is good. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 7–17, 1999 相似文献