0,2

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.

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

G.f.: A(x) = (1-x-x^2+5*x^3-6*x^4-6*x^5+3*x)/(1-3*x+3*x^3-x^4)

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

A114300 := n->coeff(series((1-x-x^2+5*x^3-6*x^4-6*x^5+3*x)/(1-3*x+3*x^3-x^4), x=0, n+1), x, n);

See also A112831 and A112832.

Sequence in context: A064168 A118727 A042361 this_sequence A099207 A122566 A118500

Adjacent sequences: A114297 A114298 A114299 this_sequence A114301 A114302 A114303

easy,nonn

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