Higher Order Log-Concavity in Euler's Difference Table
William Y.C. Chen, Cindy C.Y. Gu, Kevin J. Ma and Larry X.W. Wang
Abstract: For 0 ≤ k ≤ n, let e^{k}_{n} be the entries in Euler's difference table and let d^{k}_{n} = e^{k}_{n}/k!. Dumont and Randrianarivony showed e^{k}_{n} equals the number of permutations on [n] whose fixed points are contained in {1, 2,..., k}. Rakotondrajao found a combinatorial interpretation of the number d^{k}_{n} in terms of k-fixed-points-permutations of [n]. We show that for any n ≥ 1, the sequence {d^{k}_{n}}_{0 ≤k≤n} is both 2-log-concave and reverse ultra logconcave. AMS Classification: 05A20, 05A10 Keywords: log-concavity, 2-log-concavity, reverse ultra log-concavity, Euler's difference table Download: pdf |