Results 1 to 10 of about 1,269,170 (176)
A Unified Generalization of the Catalan, Fuss, and Fuss—Catalan Numbers [PDF]
Mathematical and Computational Applications, 2019 In the paper, the authors introduce a unified generalization of the Catalan numbers, the Fuss numbers, the Fuss−Catalan numbers, and the Catalan−Qi function, and discover some properties of the unified generalization, including a product ...
Xiao-Ting Shi, Feng Qi, Pietro Cerone
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Three Identities of the Catalan-Qi Numbers
Mathematics, 2016 In the paper, the authors find three new identities of the Catalan-Qi numbers and provide alternative proofs of two identities of the Catalan numbers. The three identities of the Catalan-Qi numbers generalize three identities of the Catalan numbers.
Feng Qi, Mansour Mahmoud
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Some Properties of the Fuss–Catalan Numbers
Mathematics, 2018 In the paper, the authors express the Fuss⁻Catalan numbers as several forms in terms of the Catalan⁻Qi function, find some analytic properties, including the monotonicity, logarithmic convexity, complete monotonicity, and minimality, of the ...
Feng Qi, Pietro Cerone
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Some properties of the Catalan-Qi function related to the Catalan numbers. [PDF]
Springerplus, 2016 In the paper, the authors find some properties of the Catalan numbers, the Catalan function, and the Catalan-Qi function which is a generalization of the Catalan numbers. Concretely speaking, the authors present a new expression, asymptotic expansions, integral representations, logarithmic convexity, complete monotonicity, minimality, logarithmically ...
Qi F, Mahmoud M, Shi XT, Liu FF.
europepmc +4 more sources
Integral Representations of the Catalan Numbers and Their Applications
Mathematics, 2017 In the paper, the authors survey integral representations of the Catalan numbers and the Catalan–Qi function, discuss equivalent relations between these integral representations, supply alternative and new proofs of several integral representations ...
Bai-Ni Guo, Feng Qi
exaly +3 more sources
The Relationship of Multiset, Stirling Number, Bell Number, and Catalan Number
Science and Technology Indonesia, 2023 Catalan numbers is not as famous as Fibonacci numbers, however this number has own its beauty and arts. Catalan numbers was discovered by Ming Antu in 1730, however, this numbers is credited to Eugene Catalan when he was studying parentheses in 1838 ...
Wamiliana, Attiya Yuliana, Fitriani
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Eulerian-Catalan Numbers [PDF]
The Electronic Journal of Combinatorics, 2011 We show that the Eulerian-Catalan numbers enumerate Dyck permutations. We provide two proofs for this fact, the first using the geometry of alcoved polytopes and the second a direct combinatorial proof via an Eulerian-Catalan analogue of the Chung-Feller theorem.
Hoda Bidkhori, Seth Sullivant
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Mathematics, 2023 In the paper, the authors briefly survey several generalizations of the Catalan numbers in combinatorial number theory, analytically generalize the Catalan numbers, establish an integral representation of the analytic generalization of the Catalan ...
Jian Cao +4 more
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Noncommutative Catalan Numbers [PDF]
Annals of Combinatorics, 2019 The goal of this paper is to introduce and study noncommutative Catalan numbers $C_n$ which belong to the free Laurent polynomial algebra in $n$ generators. Our noncommutative numbers admit interesting (commutative and noncommutative) specializations, one of them related to Garsia-Haiman $(q,t)$-versions, another -- to solving noncommutative quadratic ...
Berenstein, A., Retakh, V.
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The Complexity of Trees, Universal Grammar and Economy Conditions
Biolinguistics, 2022 In this squib, I argue that the child faces a severe computational complexity problem in parsing even the simplest of trees: the number of possible trees consistent with UG grows exponentially as a function of the number of lexical items.
Chris Collins
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