Yet another triangle for the Genocchi numbers
A permutation \(\pi= a_1a_2\dots a_n\) is exceedance-alternating, if it satisfies the following conditions: \(a_i>i\), if \(i\) is odd and \(i< n\); and \(a_i\leq i\), if \(i\) is even. For \(2\leq k\leq n\), let \(E^k_n\) denote the number of exceedance-alternating permutations of \(S_n\) with \(a_1= k\). Let \(E^1_n\) denote the number of permutations of \(S_n\) with \(a_1= 1\), which satisfy the exceedence-alternating property for all \(i>1\). The paper relates the numbers \(E^k_n\) to the Seidel triangle and the Genocchi numbers, and to a recent refinement of Genocchi numbers by \textit{G. Kreweras} [Eur. J. Comb. 18, No. 1, 49-58 (1997; Zbl 0869.05002)] which counts exceedance-alternating permutations by their last letter.
scientific article; zbMATH DE number 1512034