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A059760
a(n) is the number of edges (one-dimensional faces) in the convex polytope of real n X n doubly stochastic matrices.
13
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
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
KEYWORD
nonn
AUTHOR
Noam Katz (noamkj(AT)hotmail.com), Feb 20 2001
EXTENSIONS
More terms from James A. Sellers, Feb 21 2001
STATUS
approved