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A112831
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Number of non-intersecting cycle systems in a particular directed graph.
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2
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1, 2, 5, 17, 74, 365, 1889, 9938, 52565, 278513, 1476506, 7828925, 41513921, 220137122, 1167334565, 6190107857, 32824743914, 174062236685, 923012961569, 4894530600818, 25954597551605, 137631407453873, 729828474212666
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Define a graph with 2n vertices. Vertices 1 through n will be on the top half, vertices n+1 through 2n will be on the bottom half. For 1 <= i < j <=n, create a directed edge from vertex i to vertex j. For n+1<=i<j<=2n, create a directed edge from vertex j to vertex i. Lastly, create a directed edge from i to n+i and vice versa for 1 <= i <= n. (A graph of this general type is called a hamburger.) The value a(n) gives the number of vertex-disjoint unions of directed cycles in this graph. Also calculable as the determinant of an n X n Toeplitz matrix.
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REFERENCES
| C. Hanusa (2005). A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows. PhD Thesis. University of Washington, Seattle, USA.
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FORMULA
| G.f.: A(x) = (1-5*x)/(9*x^2-7*x+1)
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EXAMPLE
| The number of non-intersecting cycle systems in the particular directed graph of order 4 is 74.
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MAPLE
| h:=n->transpose(ToeplitzMatrix([seq(-1, i=1..n-3), -1, -1, 2, seq(2^(i-2), i=2..n)])); B:=[1, 2, 5, seq(det(h(i)), i=3..25)];
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CROSSREFS
| See also A112832.
Sequence in context: A002135 A007868 A136726 * A081046 A000774 A118100
Adjacent sequences: A112828 A112829 A112830 * A112832 A112833 A112834
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KEYWORD
| easy,nonn
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AUTHOR
| Christopher Hanusa (chanusa(AT)math.binghamton.edu), Sep 21 2005
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