A283828 Number of bounded regions in the Linial arrangement L_{n-1}.

0, 0, 1, 4, 26, 212, 2108, 24720, 334072, 5112544, 87396728, 1650607040, 34132685120, 767025716736, 18612106195456, 485013257865472, 13509071081429888, 400505695457942528, 12592502771190979712, 418524228123134068224, 14661145374751901317888

Offset: 1

Author

N. J. A. Sloane, Mar 19 2017

Keywords

• nonn

Comments

Except for the initial 0, these are the absolute values of A349719. - Ira M. Gessel, Nov 01 2023

Links

• Richard P. Stanley, Hyperplane arrangements, interval orders, and trees, Proc. Natl. Acad. Sci. USA 93 (1996), 2620-2625.
• Vasu Tewari, Gessel polynomials, rooks, and extended Linial arrangements, arXiv preprint arXiv:1604.06894 [math.CO], 2016.

Crossrefs

Cf. A007889, A349719.

Formula

From Ira M. Gessel, Nov 01 2023: (Start)

a(n) = (1/2^n) * Sum_{k=0..n} (k-1)^(n-1) * binomial(n,k) for n>=2.

In the following generating functions we take a(1)=1 rather than a(1)=0.

E.g.f.: 1 + (1/2)*x/LambertW(-(1/2)*x*exp(x/2)).

E.g.f.: 1-1/B(x), where B(x) is the e.g.f. of A007889. See Corollary 4.2 of Stanley's paper. (End)

a(n) ~ sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (exp(n) * 2^n * LambertW(exp(-1))^(n-1)). - Vaclav Kotesovec, Nov 13 2023

Extensions

More terms from Ira M. Gessel, Nov 01 2023