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. 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.

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) = (1-5*x)/(9*x^2-7*x+1)

The number of non-intersecting cycle systems in the particular directed graph of order 4 is 74.

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)];

See also A112832.

Sequence in context: A002135 A007868 A136726 this_sequence A000774 A081046 A118100

Adjacent sequences: A112828 A112829 A112830 this_sequence A112832 A112833 A112834

easy,nonn

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