منابع مشابه
A symmetrical Eulerian identity
We give three proofs for the following symmetrical identity involving binomial coefficients ( n m ) and Eulerian numbers 〈 n m 〉 : ∑ k ( a + b k ) 〈 k a − 1 〉 = ∑ k ( a + b k ) 〈 k b − 1 〉 for any positive integers a and b (where we take 〈 0 0 〉 = 0). We also show how this fits into a family of similar (but more complicated) identities for Eulerian numbers.
متن کاملA SYMMETRICAL q - EULERIAN IDENTITY
We find a q-analog of the following symmetrical identity involving binomial coefficients ( n m ) and Eulerian numbers An,m, due to Chung, Graham and Knuth [J. Comb., 1 (2010), 29–38]: ∑ k≥0 ( a + b k ) Ak,a−1 = ∑ k≥0 ( a + b k ) Ak,b−1. We give two proofs, using generating function and bijections, respectively.
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Symmetrical, extracellular fibrils, which are related to the "special fibrils" of the dermis described by Palade and Farquhar, have been found along the outer surface of the basement membrane covering the notochord in the tail of Rana catesbeiana (bullfrog) tadpoles. The fibrils are approximately 7,500 A long and occur singly or in clusters. The single fibrils are characterized by a symmetrical...
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ژورنال
عنوان ژورنال: Journal of Combinatorics
سال: 2010
ISSN: 2156-3527,2150-959X
DOI: 10.4310/joc.2010.v1.n1.a2