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

0,3

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.

Links

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

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

Example

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

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 *)

Crossrefs

Column k=0 of A062866 or of A062867.

Cf. A179567, A196687, A196688, A320337.

Sequence in context: A263741 A263742 A263743 * A054936 A006443 A152103

Adjacent sequences: A062865 A062866 A062867 * A062869 A062870 A062871

Keyword

nice,nonn

Author

Olivier Gérard, Jun 26 2001

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

Status

approved