A059615 a(n) is the number of non-parallel lines determined by a pair of vertices (extreme points) in the polytope of real n X n doubly stochastic matrices. The vertices are the n! permutation matrices.

0, 1, 15, 240, 6040, 217365, 10651011, 681667840, 55215038880, 5521504648185, 668102052847735, 96206695728917136, 16258931576714668920, 3186750589054271109325, 717018882536990087693835

Offset: 1

Author

Noam Katz (noamkj(AT)hotmail.com), Feb 18 2001

Keywords

• nonn

Examples

			a(3) = 15 because there are 3! = 6 vertices and C(6,2) lines and in this case all are nonparallel so a(3) = C(6,2) = 15.
		

References

• M. Marcus, Hermitian Forms and Eigenvalues, in Survey of Numerical Analysis, J. Todd, ed. McGraw-Hill, New York, 1962.

Crossrefs

Cf. A000166, A059760.

Programs

Maple

Digits := 200: with(combinat): d := n->n!*sum((-1)^j/j!,j=0..n): a059615 := n->1/2*sum( binomial(n,k)^2 * (n-k)!*d(n-k), k=0..n-2): for n from 1 to 30 do printf(`%d,`,round(evalf(a059615(n)))) od:

PARI

a(n) = (1/2)*sum(k=0, n-2, ((n!/k!)^2 * sum(m=0, n-k, (-1)^m/m!))); \\ Michel Marcus, Mar 14 2018

Formula

a(n) = (1/2)*Sum_{k=0...n-2} binomial(n,k)^2 * (n-k)! * d(n-k) for n >= 2, where d(n) is the number of derangements of n elements: permutations of n elements with no fixed points - sequence A000166. Using the formula: d(n) = n!*Sum_{k=0..n} (-1)^k/k!, a(n) = (1/2)*Sum_{k=0..n-2} ((n!/k!)^2 * Sum_{m=0..n-k} (-1)^m/m!).

Extensions

More terms from James Sellers, Feb 19 2001

Offset corrected by Michel Marcus, Mar 14 2018