A062868 Number of permutations of degree n with barycenter 0.

1, 1, 2, 4, 14, 46, 282, 1394, 12658, 83122, 985730, 8012962, 116597538, 1127575970, 19410377378, 217492266658, 4320408974978, 55023200887938, 1238467679662722, 17665859065690754, 444247724347355554, 7015393325151055906, 194912434760367113570, 3375509056735963889634

Offset: 0

Author

Olivier Gérard, Jun 26 2001

Keywords

• nice
• nonn

Comments

The barycenter or signcenter of a permutation is the sum of the signs of the difference between initial and final positions of the objects.

Examples

			(4,1,3,5,2) has difference (3,-1,0,1,-3) and signs (1,-1,0,1,-1) with total 0.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450 (first 151 terms from Maxwell Jiang)

Crossrefs

Column k=0 of A062866 or of A062867.

Cf. A179567, A196687, A196688, A320337.

Programs

Maple

b:= proc(s, t) option remember; (n-> `if`(abs(t)>n, 0, `if`(n=0, 1,
      add(b(s minus {j}, t+signum(n-j)), j=s))))(nops(s))
    end:
a:= n-> b({$1..n}, 0):
seq(a(n), n=0..14);  # Alois P. Heinz, Jul 31 2018

Mathematica

E1[n_ /; n >= 0, 0] = 1;
E1[n_, k_] /; k < 0 || k > n = 0;
E1[n_, k_] := E1[n, k] = (n-k) E1[n-1, k-1] + (k+1) E1[n-1, k];
b[n_] := Sum[(-1)^(n-k) E1[n+k, n] Binomial[2n, n-k], {k, 0, n}];
a[n_] := Sum[Binomial[n, n-2k] b[k], {k, 0, n/2}];
a /@ Range[0, 150] (* Jean-François Alcover, Oct 29 2020, after Peter Luschny in A320337 *)

Formula

a(n) = Sum_{k=0..floor(n/2)} binomial(n, n-2*k)*A320337(k). - Maxwell Jiang, Dec 19 2018 (added by editors)

a(n) ~ sqrt(3) * (1 + exp(-2)*(-1)^n) * n^n / exp(n). - Vaclav Kotesovec, Oct 29 2020

Extensions

One more term from Vladeta Jovovic, Jun 28 2001

a(11)-a(14) from Hugo Pfoertner, Sep 23 2004

a(15)-a(18) from R. H. Hardin, Jul 18 2010

a(19)-a(22) from Kyle G Hess, Jul 30 2018

a(0)=1 prepended by Alois P. Heinz, Jul 30 2018

Terms a(23) and beyond from Maxwell Jiang, Dec 19 2018