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

0, 2, 12, 120, 1320, 17856, 273056, 4772624, 92626944, 1986317024, 46556867456, 1184827221584, 32524270418432, 958020105786536

Offset: 0

Author

Peter Kagey, Nov 22 2019

Keywords

• nonn
• more

Comments

a(n) <= ((n+1)! - 2*A131502(n))*A130783(n).

Every term is even because the n-th difference of a permutation and its reversal are the same up to sign.

Examples

			For n = 2, the second differences of the (2+1)! = 6 permutations of {0,1,2} are:
[0,1,2] ->  [1, 1] ->  0,
[0,2,1] ->  [2,-1] -> -3,
[1,0,2] -> [-1, 2] ->  3,
[1,2,0] ->  [1,-2] -> -3,
[2,0,1] -> [-2, 1] ->  3, and
[2,1,0] -> [-1,-1] ->  0.
The sum of the absolute values of these second differences is 0 + 3 + 3 + 3 + 3 + 0 = 12.
		

Crossrefs

Cf. A130783, A131502, A327845.

Programs

Mathematica

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 *)

Python

from math import comb
from itertools import permutations
def A329851(n):
    c = [-comb(n,i) if i&1 else comb(n,i) for i in range(n+1)]
    return sum(abs(sum(c[i]*p[i] for i in range(n+1))) for p in permutations(range(n+1)) if p[0]Chai Wah Wu, Jun 04 2024

Extensions

a(10) from Alois P. Heinz, Nov 22 2019

a(11)-a(13) from Giovanni Resta, Nov 23 2019