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We study classes of set partitions determined by the avoidance of multiple patterns, applying a natural notion of partition containment that has been introduced by Sagan. We say that two sets S and T of patterns are equivalent if for each n the number of partitions of size n avoiding all the members of S is the same as the number of those that avoid all the members of T. 相似文献
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Let Π = B_1/B_2/… /B_k be any set partition of[n]= {1,2,...,n} satisfying that entries are increasing in each block and blocks are arranged in increasing order of their first entries.Then Callan defined the flattened Π to be the permutation of[n]obtained by erasing the divers between its blocks,and Callan also enumerated the number of set partitions of[n]whose flattening avoids a single3-letter pattern.Mansour posed the question of counting set partitions of[n]whose flattening avoids a pattern of length 4.In this paper,we present the number of set partitions of[n]whose flattening avoids one of the patterns:1234,1243,1324,1342,1423,1432,3142 and 4132. 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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David Callan 《Discrete Mathematics》2009,309(12):4187-4191
To flatten a set partition (with apologies to Mathematica®) means to form a permutation by erasing the dividers between its blocks. Of course, the result depends on how the blocks are listed. For the usual listing—increasing entries in each block and blocks arranged in increasing order of their first entries—we count the partitions of [n] whose flattening avoids a single 3-letter pattern. Five counting sequences arise: a null sequence, the powers of 2, the Fibonacci numbers, the Catalan numbers, and the binomial transform of the Catalan numbers. 相似文献
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N. M. Bogoliubov 《Journal of Mathematical Sciences》2009,158(6):771-786
An exactly solvable boson model, the so-called “phase model,” is considered. A relation between certain transition matrix
elements of this model and boxed plane partitions, three-dimensional Young diagrams placed into a box of finite size, is established,
It is shown that the natural model describing the behavior of friendly walkers, ones that can share the same lattice sites,
is the “phase model.” An expression for the number of all admissible nests of lattice paths made by a fixed number of friendly
walkers for a certain number of steps is obtained. Bibliography 35 titles.
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 5–30. 相似文献
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Yidong Sun 《Discrete Mathematics》2009,309(9):2640-2648
A Motzkin path of length n is a lattice path from (0,0) to (n,0) in the plane integer lattice Z×Z consisting of horizontal-steps (1,0), up-steps (1,1), and down-steps (1,−1), which never passes below the x-axis. A u-segment (resp. h-segment) of a Motzkin path is a maximal sequence of consecutive up-steps (resp. horizontal-steps). The present paper studies two kinds of statistics on Motzkin paths: “number of u-segments” and “number of h-segments”. The Lagrange inversion formula is utilized to represent the weighted generating function for the number of Motzkin paths according to the two statistics as a sum of the partial Bell polynomials or the potential polynomials. As an application, a general framework for studying compositions are also provided. 相似文献
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Sen-Peng Eu Tung-Shan Fu Justin T. Hou Te-Wei Hsu 《Journal of Combinatorial Theory, Series A》2013,120(7):1786-1803
In this paper, we propose a notion of colored Motzkin paths and establish a bijection between the n-cell standard Young tableaux (SYT) of bounded height and the colored Motzkin paths of length n. This result not only gives a lattice path interpretation of the standard Young tableaux but also reveals an unexpected intrinsic relation between the set of SYTs with at most rows and the set of SYTs with at most 2d rows. 相似文献
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The Ramanujan Journal - In recent work, M. Schneider and the first author studied a curious class of integer partitions called “sequentiallyc congruent” partitions: the mth part is... 相似文献
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In this paper, we compute the congruences of Catalan and Motzkin numbers modulo 4 and 8. In particular, we prove the conjecture proposed by Deutsch and Sagan that no Motzkin number is a multiple of 8. 相似文献
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Let us call a lattice path in from to using , , and steps and never going below the -axis, a -path (of order ). A super -path is a -path which is permitted to go below the -axis. We relate the total number of humps in all of the -paths of order to the number of super -paths, where a hump is defined to be a sequence of steps of the form , . This generalizes recent results concerning the cases when and or . A similar relation may be given involving peaks (consecutive steps of the form ). 相似文献
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We prove various congruences for Catalan and Motzkin numbers as well as related sequences. The common thread is that all these sequences can be expressed in terms of binomial coefficients. Our techniques are combinatorial and algebraic: group actions, induction, and Lucas’ congruence for binomial coefficients come into play. A number of our results settle conjectures of Cloitre and Zumkeller. The Thue-Morse sequence appears in several contexts. 相似文献
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Stevo Todorcevic 《Israel Journal of Mathematics》1985,52(1-2):53-58
A kind ofk
+-Aronszajn tree is used to construct some strong negative partition relations onk
+. 相似文献
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《Journal of Pure and Applied Algebra》2022,226(9):107056
In this paper, we relate combinatorial conditions for polarizations of powers of the graded maximal ideal with rank conditions on submodules generated by collections of Young tableaux. We apply discrete Morse theory to the hypersimplex resolution introduced by Batzies–Welker to show that the L-complex of Buchsbaum and Eisenbud for powers of the graded maximal ideal is supported on a CW-complex. We then translate the “spanning tree condition” of Almousa–Fløystad–Lohne characterizing polarizations of powers of the graded maximal ideal into a condition about which sets of hook tableaux span a certain Schur module. As an application, we give a complete combinatorial characterization of polarizations of so-called “restricted powers” of the graded maximal ideal. 相似文献
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