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1.
We define the down sets (lower covers, respectively) sequence of an ordered set. We show that the number of down set sequences of an n-ordered set is equal to the n-th Catalan Number. We give a characterization of down sets sequences of an ordered set and another characterization of lower covers sequences of an ordered set. 相似文献
2.
《Discrete Mathematics》2022,345(9):112891
We calculate moments of the so-called Kesten distribution by means of the expansion of the denominator of the density of this distribution and then integrate all summands with respect to the semicircle distribution. By comparing this expression with the formulae for the moments of Kesten's distribution obtained by other means, we find identities involving polynomials whose power coefficients are closely related to Catalan numbers, Catalan triangles, binomial coefficients. Finally, as applications of these identities we obtain various interesting relations between the aforementioned numbers, also concerning Lucas, Fibonacci and Fine numbers. 相似文献
3.
Stefan Stanimirovi? 《Applied mathematics and computation》2011,217(22):9122-9132
In this paper we use the Catalan matrix power as a tool for deriving identities involving Catalan numbers and hypergeometric functions. For that purpose, we extend earlier investigated relations between the Catalan matrix and the Pascal matrix by inserting the Catalan matrix power and particulary the squared Catalan matrix in those relations. We also pay attention to some relations between Catalan matrix powers of different degrees, which allows us to derive the simplification formula for hypergeometric function 3F2, as well as the simplification formula for the product of the Catalan number and the hypergeometric function 3F2. Some identities involving Catalan numbers, proved by the non-matrix approach, are also given. 相似文献
5.
We prove two conjectures on sums of products of Catalan triangle numbers, which were originally conjectured by Miana et al. [Discrete Math. 340 (2017), 2388–2397]. The first one is proved by using Zeilberger's algorithm, and the second one is proved by establishing its q-analogue. 相似文献
6.
In this paper, we consider combinatorial numbers , mentioned as Catalan triangle numbers where . These numbers unify the entries of the Catalan triangles and for appropriate values of parameters and , i.e., and . In fact, these numbers are suitable rearrangements of the known ballot numbers and some of these numbers are the well-known Catalan numbers that is .We present identities for sums (and alternating sums) of , squares and cubes of and, consequently, for and . In particular, one of these identities solves an open problem posed in Gutiérrez et al. (2008). We also give some identities between and harmonic numbers . Finally, in the last section, new open problems and identities involving are conjectured. 相似文献
7.
Xiaojing Chen 《Journal of Mathematical Analysis and Applications》2009,349(2):311-316
By combining inverse series relations with binomial convolutions and telescoping method, moments of Catalan numbers are evaluated, which resolves a problem recently proposed by Gutiérrez et al. [J.M. Gutiérrez, M.A. Hernández, P.J. Miana, N. Romero, New identities in the Catalan triangle, J. Math. Anal. Appl. 341 (1) (2008) 52-61]. 相似文献
8.
In this paper we obtain the moments {Φm}m?0 defined by
9.
《Discrete Mathematics》2022,345(11):113065
10.
Thierry Dana-Picard 《International Journal of Mathematical Education in Science & Technology》2013,44(1):63-69
We compute in three different ways the same definite parametric integral. By-products are the derivation of a combinatorial identity and two integral presentations of Catalan numbers. One of them leads to a presentation using the γ function. 相似文献
11.
Matja? Konvalinka 《Journal of Combinatorial Theory, Series A》2007,114(6):1089-1100
We define a q generalization of weighted Catalan numbers studied by Postnikov and Sagan, and prove a result on the divisibility by p of such numbers when p is a prime and q its power. 相似文献
12.
中国数学家明安图在其《割圜密率捷法》中最先应用了Catalan数,取得优秀的研究成果.本文简介明安图的计数成就和Catalan数,综述国内外对明安图应用该数的研究.特别地,近两年来英国的Larcombe发表了5篇文章,对明安图的成果——包含Catalan数的sin(2pa)展开式,加以推广,并给出明安图确定Catalan 数的第二种方法的严格代数证明. 相似文献
13.
14.
J.M. Gutiérrez M.A. Hernández P.J. Miana N. Romero 《Journal of Mathematical Analysis and Applications》2008,341(1):52-61
In this paper we prove new identities in the Catalan triangle whose (n,p) entry is defined by
15.
Wenchang Chu 《Discrete Mathematics》2018,341(11):3159-3164
Three summation formulae on the -extended Catalan numbers are established by means of hypergeometric series approach with one of them being provided a combinatorial proof through lattice path countings. 相似文献
16.
Stefan Stanimirovi Predrag Stanimirovi Marko Miladinovi Aleksandar Ili 《Applied mathematics and computation》2009,215(2):796-805
We introduce the notion of the Catalan matrix whose non-zero elements are expressions which contain the Catalan numbers arranged into a lower triangular Toeplitz matrix. Inverse of the Catalan matrix is derived. Correlations between the matrix and the generalized Pascal matrix are considered. Some combinatorial identities involving Catalan numbers, binomial coefficients and the generalized hypergeometric function are derived using these correlations. Moreover, an additional explicit representation of the Catalan number, as well as an explicit representation of the sum of the first m Catalan numbers are given. 相似文献
17.
Toufik Mansour 《Discrete Mathematics》2009,309(12):4079-4088
We first establish the result that the Narayana polynomials can be represented as the integrals of the Legendre polynomials. Then we represent the Catalan numbers in terms of the Narayana polynomials by three different identities. We give three different proofs for these identities, namely, two algebraic proofs and one combinatorial proof. Some applications are also given which lead to many known and new identities. 相似文献
18.
Moubariz Z. Garaev 《Journal of Combinatorial Theory, Series A》2006,113(5):851-865
We estimate character sums with Catalan numbers and middle binomial coefficients modulo a prime p. We use this bound to show that the first at most p13/2(logp)6 elements of each sequence already fall in all residue classes modulo every sufficiently large p, which improves the previously known result requiring pO(p) elements. We also study, using a different technique, similar questions for sequences satisfying polynomial recurrence relations like the Apéry numbers. We show that such sequences form a finite additive basis modulo p for every sufficiently large prime p. 相似文献
19.
《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
20.
《Discrete Mathematics》2023,346(3):113247
A 3-dimensional Catalan word is a word on three letters so that the subword on any two letters is a Dyck path. For a given Dyck path D, a recently defined statistic counts the number of Catalan words with the property that any subword on two letters is exactly D. In this paper, we enumerate Dyck paths with this statistic equal to certain values, including all primes. The formulas obtained are in terms of Motzkin numbers and Motzkin ballot numbers. 相似文献