Catalan numbers

Catalan numbers

[′kat·əl·ən ‚nəm·bərz]
(mathematics)
The numbers, cn, which count the ways to insert parentheses in a string of n terms so that their product may be unambiguously carried out by multiplying two quantities at a time.
References in periodicals archive ?
An immediate consequence is: if G is a set of irreducible permutations such that the wreath product G [??] I is enumerated by the Catalan numbers ([16], A00018) then G is enumerated by the Motzkin numbers ([16], A001006).
The sequence of Catalan numbers begins (1, 2, 5, 14, 42, ....
This text has been developed out of a minicourse given by Grimaldi (mathematics, Rose-Hulman Institute of Technology) at national mathematics meetings on examples, properties, and applications of the sequences of the Fibonacci and Catalan numbers. The material is presented in a manner that is intended to allow a broad range of mathematicians to understand at least significant portions of the material even if they are unfamiliar with the subject area and so should be considered an introduction to the two important number sequences.
[4] Liu Jianjun, Ming Antu and Catalan Numbers, Journal of Mathematical Research and Exposition, 22(2002), 589-594.
For the root system of type [A.sub.n-1], these reduce to objects counted by the classical Catalan numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], namely the set of noncrossing partitions of [n] = {1, 2, ..., n}, the set of nonnesting partitions of [n] and the set of triangulations of a convex (n + 2)-gon, respectively.
Loehr, Multivariate analogues of Catalan numbers, parking functions, and their extensions.
For any positive integer n, the classical Catalan numbers [b.sub.n] are defined as follows:
This provides a link between parking functions and various combinatorial objects counted by Catalan numbers. In a series of papers Garsia, Haglund, Haiman, et al.
It is well known that Catalan numbers [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] enumerate many combinatorial objects, such as binary trees and parallelogram polyominoes.
Our results show that all such polynomials have nonnegative coefficients, conjectured by Kazhdan and Lusztig (1979), and give a combinatorial interpretation of them in terms of Catalan numbers and the Coxeter graph of the group.
is the well known Carlitz q-analogue of Catalan numbers.
Both of these generalizations can be motivated from Garsia's and Haiman's [GH] observation that the Catalan numbers play a deep role in representation theory.