OFFSET
0,3
COMMENTS
a(n) is also the dimension of the span of chromatic quasi-symmetric invariants of generalized permutahedra.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..435
Vladimir Grujić, Counting faces of graphical zonotopes, Ars Math. Contemp., 13(1), 2017; arXiv preprint, arXiv:1604.06931 [math.CO], 2017.
Raul Penaguiao, The kernel of chromatic quasisymmetric functions on graphs and hypergraphic polytopes, Journal of Combinatorial Theory, Series A, 175, 105258, 2020.
FORMULA
Asymptotic growth: a(n) = n! * b^(-n) * c, where b is the unique positive root of exp(2*x) = 1 + e^x + x*e^x, and c is 0.722487... .
From Andrew Howroyd, Sep 27 2024: (Start)
a(n) = Sum_{k=0..n} Sum_{j=0..floor(k/2)} binomial(n,k)*A358623(k,j)*j!*(j+1)^(n-k).
E.g.f.: exp(x)/(1 - exp(x)*(exp(x)-x-1)). (End)
In the notation above, c = 1/(b*(2*exp(b) - b - 2)). - Vaclav Kotesovec, Nov 21 2024
EXAMPLE
a(2) = 2 because the ordered set partitions 1|2 and 2|1 are counted only once.
a(3) = 8, all ordered set partitions with length 3 (e.g. 1|2|3) are counted only once.
a(4) = 40 counts 1|34|2 separately to 2|34|1, but treats 1|2|34 as the same as 2|1|34 since only adjacent singletons can commute.
MAPLE
b:= proc(n, p) option remember; `if`(n=0, 1/p!, add(
b(n-j, 0)*binomial(n, j)/p!, j=2..n)+b(n-1, p+1)*n)
end:
a:= n-> b(n, 0):
seq(a(n), n=0..21); # Alois P. Heinz, Nov 19 2024
PROG
B(n, k) = {sum(i=0, k, (-1)^i * binomial(n, i) * stirling(n-i, k-i, 2) ); }
a(n)={sum(k=0, n, binomial(n, k)*sum(j=0, k\2, B(k, j)*j!*(j+1)^(n-k)))} \\ Andrew Howroyd, Sep 27 2024
(PARI) seq(n)=my(g=exp(x + O(x*x^n))); Vec(serlaplace(g/(1 - g*(g-x-1)))) \\ Andrew Howroyd, Sep 27 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Raul Penaguiao, Sep 27 2024
EXTENSIONS
a(10) onwards from Andrew Howroyd, Sep 27 2024
STATUS
approved