A362968 Number of integral points in 2 * permutohedron of order n.

1, 3, 19, 201, 3081, 62683, 1598955, 49180113, 1773405649, 73410669171, 3432267261699, 178922825114905, 10291053760222041, 647436905815864011, 44229766376059342171, 3260749830852693615777, 258039101519624535653025

Offset

1,2

Comments

Every vectorial sum of two permutations represents an integral point in 2*permutohedron, however the converse does not hold. Hence, a(n) >= A175176(n) for all n, where the equality holds only for n <= 5.

Number of points up to their components order is given by A007747.

Links

Table of n, a(n) for n=1..17.

C. Bebeacua, T. Mansour, A. Postnikov, and S. Severini. On the X-rays of permutations, arXiv:math/0506334 [math.CO], 2005.

Wikipedia, Permutohedron.

Formula

a(n) = Sum_{k=0..n-1} A138464(n,k) * 2^k, which is the value of the Ehrhart polynomial of permutohedron at t = 2.

E.g.f.: exp(-W(-2*x)/2 - W(-2*x)^2/4), where W() is the Lambert function.

Maple

w := LambertW(-2*x): egf := exp(-w * (2 + w) / 4): ser := series(egf, x, 20):
seq(n! * coeff(ser, x, n), n = 1..17); # Peter Luschny, Jun 19 2023

Prog

(PARI) a362968(n) = my(x=y+O(y^(n+1))); n! * polcoef( exp(-lambertw(-2*x)/2 - lambertw(-2*x)^2/4), n );

Crossrefs

Cf. A007747, A138464, A175176, A362967.

Sequence in context: A195895 A127502 A175176 * A233336 A208897 A027546

Adjacent sequences: A362965 A362966 A362967 * A362969 A362970 A362971

Keyword

nonn

Author

Max Alekseyev, Jun 17 2023

Status

approved