cp's OEIS Frontend

A329851 Sum of absolute values of n-th differences over all permutations of {0, 1, ..., n}.

%I A329851 #23 Jun 05 2024 01:25:13
%S A329851 0,2,12,120,1320,17856,273056,4772624,92626944,1986317024,46556867456,
%T A329851 1184827221584,32524270418432,958020105786536
%N A329851 Sum of absolute values of n-th differences over all permutations of {0, 1, ..., n}.
%C A329851 a(n) <= ((n+1)! - 2*A131502(n))*A130783(n).
%C A329851 Every term is even because the n-th difference of a permutation and its reversal are the same up to sign.
%e A329851 For n = 2, the second differences of the (2+1)! = 6 permutations of {0,1,2} are:
%e A329851 [0,1,2] ->  [1, 1] ->  0,
%e A329851 [0,2,1] ->  [2,-1] -> -3,
%e A329851 [1,0,2] -> [-1, 2] ->  3,
%e A329851 [1,2,0] ->  [1,-2] -> -3,
%e A329851 [2,0,1] -> [-2, 1] ->  3, and
%e A329851 [2,1,0] -> [-1,-1] ->  0.
%e A329851 The sum of the absolute values of these second differences is 0 + 3 + 3 + 3 + 3 + 0 = 12.
%t A329851 a[n_] := Block[{x, k}, k = CoefficientList[(x - 1)^n, x]; Sum[Abs[k.p], {p, Permutations@ Range[0, n]}]]; Array[a, 10, 0] (* _Giovanni Resta_, Nov 23 2019 *)
%o A329851 (Python)
%o A329851 from math import comb
%o A329851 from itertools import permutations
%o A329851 def A329851(n):
%o A329851     c = [-comb(n,i) if i&1 else comb(n,i) for i in range(n+1)]
%o A329851     return sum(abs(sum(c[i]*p[i] for i in range(n+1))) for p in permutations(range(n+1)) if p[0]<p[-1])<<1 # _Chai Wah Wu_, Jun 04 2024
%Y A329851 Cf. A130783, A131502, A327845.
%K A329851 nonn,more
%O A329851 0,2
%A A329851 _Peter Kagey_, Nov 22 2019
%E A329851 a(10) from _Alois P. Heinz_, Nov 22 2019
%E A329851 a(11)-a(13) from _Giovanni Resta_, Nov 23 2019