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A376544 a(n) is the number of singleton commuting ordered set partitions.

%I A376544 #37 Dec 01 2024 20:40:40
%S A376544 1,1,2,8,40,242,1784,15374,151008,1669010,20503768,277049126,
%T A376544 4083693200,65211041690,1121435565384,20662801363790,406100030507200,
%U A376544 8480197575505442,187500501495191480,4376026842424336886,107506303414618515696,2773174380946415844266
%N A376544 a(n) is the number of singleton commuting ordered set partitions.
%C A376544 a(n) is also the dimension of the span of chromatic quasi-symmetric invariants of generalized permutahedra.
%H A376544 Alois P. Heinz, <a href="/A376544/b376544.txt">Table of n, a(n) for n = 0..435</a>
%H A376544 Vladimir Grujić, <a href="https://doi.org/10.26493/1855-3974.1132.fae">Counting faces of graphical zonotopes</a>, Ars Math. Contemp., 13(1), 2017; <a href="https://arxiv.org/abs/1604.06931">arXiv preprint</a>, arXiv:1604.06931 [math.CO], 2017.
%H A376544 Raul Penaguiao, <a href="https://doi.org/10.1016/j.jcta.2020.105258">The kernel of chromatic quasisymmetric functions on graphs and hypergraphic polytopes</a>, Journal of Combinatorial Theory, Series A, 175, 105258, 2020.
%F A376544 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... .
%F A376544 From _Andrew Howroyd_, Sep 27 2024: (Start)
%F A376544 a(n) = Sum_{k=0..n} Sum_{j=0..floor(k/2)} binomial(n,k)*A358623(k,j)*j!*(j+1)^(n-k).
%F A376544 E.g.f.: exp(x)/(1 - exp(x)*(exp(x)-x-1)). (End)
%F A376544 In the notation above, c = 1/(b*(2*exp(b) - b - 2)). - _Vaclav Kotesovec_, Nov 21 2024
%e A376544 a(2) = 2 because the ordered set partitions 1|2 and 2|1 are counted only once.
%e A376544 a(3) = 8, all ordered set partitions with length 3 (e.g. 1|2|3) are counted only once.
%e A376544 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.
%p A376544 b:= proc(n, p) option remember; `if`(n=0, 1/p!, add(
%p A376544       b(n-j, 0)*binomial(n, j)/p!, j=2..n)+b(n-1, p+1)*n)
%p A376544     end:
%p A376544 a:= n-> b(n, 0):
%p A376544 seq(a(n), n=0..21);  # _Alois P. Heinz_, Nov 19 2024
%o A376544 (PARI) \\ here B(n,k) is A008299 or A358623.
%o A376544 B(n, k) = {sum(i=0, k, (-1)^i * binomial(n, i) * stirling(n-i, k-i, 2) ); }
%o A376544 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
%o A376544 (PARI) seq(n)=my(g=exp(x + O(x*x^n))); Vec(serlaplace(g/(1 - g*(g-x-1)))) \\ _Andrew Howroyd_, Sep 27 2024
%Y A376544 Corresponds to a subset of elements counted in A000670.
%Y A376544 Cf. A005840, A113059, A307389, A358623.
%K A376544 nonn
%O A376544 0,3
%A A376544 _Raul Penaguiao_, Sep 27 2024
%E A376544 a(10) onwards from _Andrew Howroyd_, Sep 27 2024