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A114300 Number of non-intersecting cycle systems in a particular directed graph. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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 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.

REFERENCES

C. Hanusa (2005). A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows. PhD Thesis.  University of Washington, Seattle, USA.

LINKS

Table of n, a(n) for n=0..26.

FORMULA

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)

EXAMPLE

The number of non-intersecting cycle systems in the particular directed graph of order 4 is 40.

MAPLE

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);

MATHEMATICA

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 *)

CROSSREFS

See also A112831 and A112832.

Sequence in context: A118727 A183906 A042361 * A099207 A197918 A122566

Adjacent sequences:  A114297 A114298 A114299 * A114301 A114302 A114303

KEYWORD

easy,nonn

AUTHOR

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Nov 21 2005

STATUS

approved

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Last modified January 22 08:47 EST 2018. Contains 298042 sequences.