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A114300
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Number of non-intersecting cycle systems in a particular directed graph.
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1
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1, 2, 5, 17, 40, 101, 260, 677, 1768, 4625, 12104, 31685, 82948, 217157, 568520, 1488401, 3896680, 10201637, 26708228, 69923045, 183060904, 479259665, 1254718088, 3284894597, 8599965700, 22515002501, 58945041800
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OFFSET
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0,2
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COMMENTS
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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 whenever j=i+2. For n+1<=i<j<=2n, create a directed edge from vertex j to vertex i whenever j=i+2. Create a directed edge from vertex 1 to vertex 2, from vertex n-1 to vertex n, from vertex n+2 to vertex n+1 and from vertex 2n to vertex 2n-1. 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 matrix.
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REFERENCES
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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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LINKS
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FORMULA
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G.f.: A(x) = (1-x-x^2+5*x^3-6*x^4-6*x^5+3*x^6)/(1-3*x+3*x^3-x^4)
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EXAMPLE
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The number of non-intersecting cycle systems in the particular directed graph of order 4 is 40.
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MAPLE
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A114300 := n->coeff(series((1-x-x^2+5*x^3-6*x^4-6*x^5+3*x^6)/(1-3*x+3*x^3-x^4), x=0, n+1), x, n);
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MATHEMATICA
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CoefficientList[Series[(3x^6-6x^5-6x^4+5x^3-x^2-x+1)/ (-x^4+3x^3-3x+1), {x, 0, 30}], x] (* Harvey P. Dale, Apr 19 2011 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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Christopher Hanusa (chanusa(AT)math.binghamton.edu), Nov 21 2005
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STATUS
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approved
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