A symmetrical Eulerian identity
نویسندگان
چکیده
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.
منابع مشابه
A 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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