منابع مشابه
Catalan Triangle Numbers and Binomial Coefficients
The binomial coefficients and Catalan triangle numbers appear as weight multiplicities of the finite-dimensional simple Lie algebras and affine Kac–Moody algebras. We prove that any binomial coefficient can be written as weighted sums along rows of the Catalan triangle. The coefficients in the sums form a triangular array, which we call the alternating Jacobsthal triangle. We study various subs...
متن کاملPermutations, Pattern Avoidance, and the Catalan Triangle
In the study of various objects indexed by permutations, a natural notion of minimal excluded structure, now known as a permutation pattern, has emerged and found diverse applications. One of the earliest results from the study of permutation pattern avoidance in enumerative combinatorics is that the Catalan numbers cn count the permutations of size n that avoid any fixed pattern of size three....
متن کاملFactors of binomial sums from the Catalan triangle
By using the Newton interpolation formula, we generalize the recent identities on the Catalan triangle obtained by Miana and Romero as well as those of Chen and Chu. We further study divisibility properties of sums of products of binomial coefficients and an odd power of a natural number. For example, we prove that for all positive integers n1, . . . , nm, nm+1 = n1, and any nonnegative integer...
متن کاملSome New Binomial Sums Related to the Catalan Triangle
In this paper, we derive many new identities on the classical Catalan triangle C = (Cn,k)n>k>0, where Cn,k = k+1 n+1 ( 2n−k n ) are the well-known ballot numbers. The first three types are based on the determinant and the fourth is relied on the permanent of a square matrix. It not only produces many known and new identities involving Catalan numbers, but also provides a new viewpoint on combin...
متن کاملNew Sums Identities In Weighted Catalan Triangle With The Powers Of Generalized Fibonacci And Lucas Numbers
In this paper, we consider a generalized Catalan triangle de ned by km n 2n n k for positive integer m: Then we compute the weighted half binomial sums with the certain powers of generalized Fibonacci and Lucas numbers of the form n X k=0 2n n+ k km n X tk; where Xn either generalized Fibonacci or Lucas numbers, t and r are integers for 1 m 6: After we describe a general methodology to show how...
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 1976
ISSN: 0012-365X
DOI: 10.1016/0012-365x(76)90009-1