A059760 a(n) is the number of edges (one-dimensional faces) in the convex polytope of real n X n doubly stochastic matrices.

0, 0, 1, 15, 240, 5040, 147240, 5959800, 323850240, 22800476160, 2017745251200, 219066851203200, 28615863103027200, 4425987756321331200, 799788468703877452800, 166940001463941433728000, 39857401887591969128448000, 10792266259145851457961984000

Offset: 0

Author

Noam Katz (noamkj(AT)hotmail.com), Feb 20 2001

Keywords

• nonn

Comments

The vertices are the n! permutation matrices. If A(p1) and A(p2) are two permutation matrices corresponding to permutations p1 and p2 the closed interval between these two matrices forms an edge of the polytope iff the permutation p1*(p2^-1) is a cycle, i.e. its cycle decomposition in the symmetric group S_n contains exactly one nontrivial cycle.

Examples

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A059615.

Note that b(n) = (Sum k=2...n C(n,k)*(k-1)!) gives sequence A006231.

Programs

Maple

with(combinat): for n from 1 to 30 do printf(`%d,`,1/2* n! * sum(binomial(n,k)*(k-1)!, k=2..n)) od:

Mathematica

a[n_] = If[n==0, 0, (n*n!/2)*(HypergeometricPFQ[{1, 1, 1-n}, {2}, -1]-1)]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 19 2017 *)

Formula

a(n) = 1/2* n! * Sum_{k=2...n} C(n,k)*(k-1)!.

a(n) ~ Pi * n^(2*n) / exp(2*n - 1). - Vaclav Kotesovec, Jun 09 2019

Extensions

More terms from James Sellers, Feb 21 2001