A355488 Expansion of g.f. f/(1+2*f) where f is the g.f. of nonempty permutations.

0, 1, 0, 2, 8, 48, 328, 2560, 22368, 216224, 2291456, 26430336, 329805952, 4429255168, 63730438656, 978479250944, 15972310317056, 276292865550336, 5049672714569728, 97245533647568896, 1968395389124714496, 41783552069858877440, 928204423021249003520

Offset: 0

Author

F. Chapoton, Jul 04 2022

Keywords

• nonn

Comments

This is to factorials what Fine numbers are to Catalan numbers. There is no known combinatorial interpretation.

The same construction, applied to the central binomials, leads to A126984, apart from signs and the first term. - Peter Luschny, Jul 22 2022

a(n) is the number of permutations of [n] whose number of components is odd minus the number of those permutations with an even number of components. - Peter Luschny, Sep 10 2022

Examples

			Consider the permutations of [3]: [2,3,1], [3,1,2] and [3,2,1] have 1 component,
[1,3,2] and [2,1,3] have 2 components, and [1,2,3] has three components. Thus 3 - 2 + 1 = 2 = a(3). - _Peter Luschny_, Sep 10 2022
		

Links

• David Callan, Counting Stabilized-Interval-Free Permutations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.1.8.
• FindStat - Combinatorial Statistic Finder, The number of connected components of a permutation.

Crossrefs

Cf. A000108, A003319, A000957, A000142, A052186, A126984, A059438, A122827.

Programs

Maple

a:= n-> (f-> coeff(series(f/(1+2*f), x, n+1), x, n))(add(j!*x^j, j=1..n)):
seq(a(n), n=0..23);  # Alois P. Heinz, Jul 20 2022

Mathematica

nmax=22; f[x_]:=Sum[i! x^i,{i,nmax}]; CoefficientList[Series[f[x]/(1+2f[x]),{x,0,nmax}],x] (* Stefano Spezia, Jul 04 2022 *)

SageMath

A = QQ[['t']]
f = A([0] + [factorial(n) for n in range(1,30)]).O(30)
print(list(f/(1+2*f)))

SageMath

# Uses function A059438_triangle.
def A355488_list(size):
    triangle = A059438_triangle(size)
    return [0] + [sum((-1)^k*t for (k,t) in enumerate(row)) for row in triangle]
print(A355488_list(20))  # Peter Luschny, Sep 10 2022

Formula

G.f.: f/(1+2*f) where f is (the g.f. of A000142) - 1.

a(n) = -Sum_{k=1..n} (-1)^k * A059438(n, k) for n >= 1. - Peter Luschny, Sep 10 2022