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 ekn be the entries in Euler's difference table and let dkn = ekn/k!. Dumont and Randrianarivony showed ekn equals the number of permutations on [n] whose fixed points are contained in {1, 2,..., k}. Rakotondrajao found a combinatorial interpretation of the number dkn in terms of k-fixed-points-permutations of [n]. We show that for any n ≥ 1, the sequence {dkn}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

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