منابع مشابه
Fuss - Catalan Numbers in Noncommutative Probability
We prove that if p, r ∈ R, p ≥ 1 and 0 ≤ r ≤ p then the Fuss-Catalan sequence ( mp+r m ) r mp+r is positive definite. We study the family of the corresponding probability measures μ(p, r) on R from the point of view of noncommutative probability. For example, we prove that if 0 ≤ 2r ≤ p and r + 1 ≤ p then μ(p, r) is ⊞-infinitely divisible. As a by-product, we show that the sequence m m m! is po...
متن کاملCatalan numbers
Catalan numbers appear in many places throughout mathematics, and are a fairly frequent subject for articles in this magazine. Indeed, a very nice introduction to their properties was given in this magazine by Vun and Belcher [3], and a historical account of their discovery in Chinese mathematics (before Catalan!) has also appeared in a recent article by Larcombe [1]. The article by Vun and Bel...
متن کاملCoincidences of Catalan and Q-catalan Numbers
Let Cn and Cn(q) be the nth Catalan number and the nth q-Catalan number, respectively. In this paper, we show that the Diophantine equation Cn = Cm(q) has only finitely many integer solutions (m,n, q) with m > 1, n > 1, q > 1. Moreover, they are all effectively computable. – To Professor Carl Pomerance on his 65th birthday
متن کاملParametric Catalan Numbers and Catalan Triangles
Here presented a generalization of Catalan numbers and Catalan triangles associated with two parameters based on the sequence characterization of Bell-type Riordan arrays. Among the generalized Catalan numbers, a class of large generalized Catalan numbers and a class of small generalized Catalan numbers are defined, which can be considered as an extension of large Schröder numbers and small Sch...
متن کاملMultivariate Fuss-Catalan numbers
are integers that appear in many combinatorial problems. These numbers first arose in the work of Catalan as the number of triangulations of a polygon by mean of nonintersecting diagonals. Stanley [13, 14] maintains a dynamic list of exercises related to Catalan numbers, including (at this date) 127 combinatorial interpretations. Closely related to Catalan numbers are ballot numbers. Their name...
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ژورنال
عنوان ژورنال: Annals of Combinatorics
سال: 2019
ISSN: 0218-0006,0219-3094
DOI: 10.1007/s00026-019-00473-4