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

0,4

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

Links

Table of n, a(n) for n=0..22.

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.

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A356429 A228568 A007170 * A171455 A136722 A085615

Adjacent sequences: A355485 A355486 A355487 * A355489 A355490 A355491

Keyword

nonn

Author

F. Chapoton, Jul 04 2022

Status

approved