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Search: id:A059615
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| A059615 |
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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. |
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2
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| 0, 1, 15, 240, 6040, 217365, 10651011, 681667840, 55215038880, 5521504648185, 668102052847735, 96206695728917136, 16258931576714668920, 3186750589054271109325, 717018882536990087693835
(list; graph; listen)
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OFFSET
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0,3
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REFERENCES
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M. Marcus, Hermitian Forms and Eigenvalues, in Survey of Numerical Analysis, J.Todd, ed. McGraw-Hill, New York, 1962.
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FORMULA
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a(n) = 1/2*sum k=0...n-2 C(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((-1)^k/k!, k=0..n), a(n) = 1/2*Sum k=0...n-2 ((n!)^2/(k!)^2 *Sum((-1)^m/m!, m=0..n-k))
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EXAMPLE
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a(3) = 15 because there are 3! = 6 vertices and C(6,2) lines and in this case all are non parallel so a(3) = C(6,2) = 15
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MAPLE
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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:
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CROSSREFS
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Cf. A000166.
Sequence in context: A154806 A133199 A059760 this_sequence A163031 A065920 A093147
Adjacent sequences: A059612 A059613 A059614 this_sequence A059616 A059617 A059618
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KEYWORD
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nonn
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AUTHOR
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Noam Katz (noamkj(AT)hotmail.com), Feb 18 2001
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EXTENSIONS
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More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 19 2001
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