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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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Let be an array of nonnegative numbers satisfying the recurrence relation with and unless . In this paper, we first prove that the array can be generated by some context-free Grammars, which gives a unified proof of many known results. Furthermore, we present criteria for real rootedness of row-generating functions and asymptotical normality of rows of . Applying the criteria to some arrays related to tree-like tableaux, interior and left peaks, alternating runs, flag descent numbers of group of type , and so on, we get many results in a unified manner. Additionally, we also obtain the continued fraction expansions for generating functions related to above examples. As results, we prove the strong -log-convexity of some generating functions. 相似文献
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A graph is -colorable if it admits a vertex partition into a graph with maximum degree at most and a graph with maximum degree at most . We show that every -free planar graph is -colorable. We also show that deciding whether a -free planar graph is -colorable is NP-complete. 相似文献
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For given graphs , , the -color Ramsey number, denoted by , is the smallest integer such that if we arbitrarily color the edges of a complete graph of order with colors, then it always contains a monochromatic copy of colored with , for some . Let be a cycle of length and a star of order . In this paper, firstly we give a general upper bound of . In particular, for the 3-color case, we have and this bound is tight in some sense. Furthermore, we prove that for all and , and if is a prime power, then the equality holds. 相似文献
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We give methods for constructing many self-dual -codes and Type II -codes of length 2n starting from a given self-dual -code and Type II -code of length 2n, respectively. As an application, we construct extremal Type II -codes of length 24 for and extremal Type II -codes of length 32 for . We also construct new extremal Type II -codes of lengths 56 and 64. 相似文献
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《Discrete Mathematics》2022,345(11):113021
In 2007, Andrews and Paule published the eleventh paper in their series on MacMahon's partition analysis, with a particular focus on broken k-diamond partitions. On the way to broken k-diamond partitions, Andrews and Paule introduced the idea of k-elongated partition diamonds. Recently, Andrews and Paule revisited the topic of k-elongated partition diamonds. Using partition analysis and the Omega operator, they proved that the generating function for the partition numbers produced by summing the links of k-elongated plane partition diamonds of length n is given by for each . A significant portion of their recent paper involves proving several congruence properties satisfied by and , using modular forms as their primary proof tool. In this work, our goal is to extend some of the results proven by Andrews and Paule in their recent paper by proving infinitely many congruence properties satisfied by the functions for an infinite set of values of k. The proof techniques employed are all elementary, relying on generating function manipulations and classical q-series results. 相似文献
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Let and be positive integers with . Given a permutation of integers , we consider -consecutive sums of , i.e., for , where we let . What we want to do in this paper is to know the exact value of where denotes the set of all permutations of . In this paper, we determine the exact values of for some particular cases of and . As a corollary of the results, we obtain , and for any . 相似文献