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

0,4

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.

Links

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

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

Example

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.

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 *)

Crossrefs

Cf. A059615.

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

Sequence in context: A090411 A154806 A133199 * A059615 A215855 A163031

Adjacent sequences: A059757 A059758 A059759 * A059761 A059762 A059763

Keyword

nonn

Author

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

Extensions

More terms from James A. Sellers, Feb 21 2001

Status

approved