共查询到20条相似文献,搜索用时 15 毫秒
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Éva Czabarka Rigoberto Flórez Leandro Junes José L. Ramírez 《Discrete Mathematics》2018,341(10):2789-2807
A valley in a Dyck path is a local minimum, and a peak is a local maximum. A Dyck path is non-decreasing if the -coordinates of the valleys of the path valley form anon-decreasing sequence. In this paper we provide some statistics about peaks and valleys in non-decreasing Dyck paths, such as their total number, the number of low and high valleys, low and high peaks, etc. Our methods include bijective proofs, recursive relations, and the symbolic method for generating functions. 相似文献
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《Discrete Mathematics》2020,343(12):112118
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Using the bijection between partitions and vacillating tableaux, we establish a correspondence between pairs of noncrossing free Dyck paths of length 2n and noncrossing partitions of [2n+1] with n+1 blocks. In terms of the number of up steps at odd positions, we find a characterization of Dyck paths constructed from pairs of noncrossing free Dyck paths by using the Labelle merging algorithm. 相似文献
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The Catalan numbers occur in various counting problems in combinatorics. This paper reveals a connection between the Catalan numbers and list colouring of graphs. Assume is a graph and is a mapping. For a nonnegative integer , let be the extension of to the graph for which for each vertex of . Let be the minimum such that is not -choosable and be the minimum such that is not -paintable. We study the parameter and for arbitrary mappings . For , an -dominated path ending at is a monotonic path of the grid from to such that each vertex on satisfies . Let be the number of -dominated paths ending at . By this definition, the Catalan number equals . This paper proves that if has vertices and , then , where and for . Therefore, if , then equals the Catalan number . We also show that if is the disjoint union of graphs and , then and . This generalizes a result in Carraher et al. (2014), where the case each is a copy of is considered. 相似文献
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Sergi Elizalde 《Journal of Combinatorial Theory, Series A》2007,114(8):1481-1503
A k-triangulation of a convex polygon is a maximal set of diagonals so that no k+1 of them mutually cross in their interiors. We present a bijection between 2-triangulations of a convex n-gon and pairs of non-crossing Dyck paths of length 2(n−4). This solves the problem of finding a bijective proof of a result of Jonsson for the case k=2. We obtain the bijection by constructing isomorphic generating trees for the sets of 2-triangulations and pairs of non-crossing Dyck paths. 相似文献
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For any pattern of length at most two, we enumerate equivalence classes of ?ukasiewicz paths of length where two paths are equivalent whenever the occurrence positions of are identical on these paths. As a byproduct, we give a constructive bijection between Motzkin paths and some equivalence classes of ?ukasiewicz paths. 相似文献
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Ira M. Gessel 《Discrete Mathematics》2010,310(23):3421-3425
We prove a conjecture of Drake and Kim: the number of 2-distant noncrossing partitions of {1,2,…,n} is equal to the sum of weights of Motzkin paths of length n, where the weight of a Motzkin path is a product of certain fractions involving Fibonacci numbers. We provide two proofs of their conjecture: one uses continued fractions and the other is combinatorial. 相似文献
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《Discrete Mathematics》2023,346(6):113372
We provide enumerating results for partial knight's paths of a given size. We prove algebraically that zigzag knight's paths of a given size ending on the x-axis are enumerated by the generalized Catalan numbers, and we give a constructive bijection with peakless Motzkin paths of a given length. After enumerating partial knight's paths of a given length, we prove that zigzag knight's paths of a given length ending on the x-axis are counted by the Catalan numbers. Finally, we give a constructive bijection with Dyck paths of a given length. 相似文献
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In the two disjoint shortest paths problem ( 2-DSPP), the input is a graph (or a digraph) and its vertex pairs and , and the objective is to find two vertex-disjoint paths and such that is a shortest path from to for , if they exist. In this paper, we give a first polynomial-time algorithm for the undirected version of the 2-DSPP with an arbitrary non-negative edge length function. 相似文献
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Szu-En Cheng Sergi Elizalde Anisse Kasraoui Bruce E. Sagan 《Discrete Mathematics》2013,313(22):2552-2565
We prove a generalization of a conjecture of Dokos, Dwyer, Johnson, Sagan, and Selsor giving a recursion for the inversion polynomial of 321-avoiding permutations. We also answer a question they posed about finding a recursive formula for the major index polynomial of 321-avoiding permutations. Other properties of these polynomials are investigated as well. Our tools include Dyck and 2-Motzkin paths, polyominoes, and continued fractions. 相似文献
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Motivated by Ramsey-type questions, we consider edge-colorings of complete graphs and complete bipartite graphs without rainbow path. Given two graphs and , the -colored Gallai–Ramsey number is defined to be the minimum integer such that and for every , every rainbow -free coloring (using all colors) of the complete graph contains a monochromatic copy of . In this paper, we first provide some exact values and bounds of . Moreover, we define the -colored bipartite Gallai–Ramsey number as the minimum integer such that and for every , every rainbow -free coloring (using all colors) of the complete bipartite graph contains a monochromatic copy of . Furthermore, we describe the structures of complete bipartite graph with no rainbow and , respectively. Finally, we find the exact values of (), (where is a subgraph of ), and by using the structural results. 相似文献
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Mattia Fogagnolo Lorenzo Mazzieri Andrea Pinamonti 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2019,36(4):1151-1179
We provide monotonicity formulas for solutions to the p-Laplace equation defined in the exterior of a convex domain. A number of analytic and geometric consequences are derived, including the classical Minkowski inequality as well as new characterizations of rotationally symmetric solutions and domains. The proofs rely on the conformal splitting technique introduced by the second author in collaboration with V. Agostiniani. 相似文献
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To determine the size of -graphs with given graph parameters is an interesting problem. Chvátal and Hanson (JCTB, 1976) gave a tight upper bound of the size of 2-graphs with restricted maximum degree and matching number; Khare (DM, 2014) studied the same problem for linear 3-graphs with restricted matching number and maximum degree. In this paper, we give a tight upper bound of the size of 3-graphs with bounded codegree and matching number. 相似文献
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The number of Borel orbits in the symmetric space is analyzed, various (bivariate) generating functions are found. Relations to lattice path combinatorics are explored. 相似文献
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Let be the -color Ramsey number of an odd cycle of length . It is shown that for each fixed , for all sufficiently large , where is a constant. This improves an old result by Bondy and Erd?s (1973). 相似文献
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We consider the structure of -free subgraphs of graphs with high minimal degree. We prove that for every there exists an so that the following holds. For every graph with chromatic number from which one can delete an edge and reduce the chromatic number, and for every graph on vertices in which all degrees are at least , any subgraph of which is -free and contains the maximum number of copies of the complete graph is -colorable.We also consider several extensions for the case of a general forbidden graph of a given chromatic number, and for subgraphs maximizing the number of copies of balanced blowups of complete graphs. 相似文献