1,2

Breakdown by length of chain:

n:chains

1: 1

2: 2,1

3: 5,5,4,2

4: 14,21,30,38,40,32,16

5: 42,84,168,322,578,952,1408,1808,1920,1536,768

Note that for each n, there are C_n chains of length 0 (A000108) and the number of maximal chains is A005118.

R. P. Stanley, Enumerative Combinatorics 1, Cambridge University Press, New York, 1997.

1) For D_i,D_j in D_n, the number of saturated chains = sum_{D_i<=D_j} (number of standard Young tableaux for D_j\D_i partition)

2) Define zeta(x,y)=1 if x=y or if y immediately covers x in the poset and delta is the identity function.

Then the number of saturated chains = sum of entries in the (2*delta - zeta)^(-1)(x,y) matrix.

For n=3, the Hasse diagram consists of 5 vertices corresponding to the 5 Dyck paths. With area as the rank function, we have one vertex of rank 0, two of rank 1, one of rank 2 and one of rank 3.

There are 16 saturated chains with 5 chains on one vertex, 5 chains on two vertices, 4 chains on three vertices and the 2 maximal chains on four vertices.

eg. for n=3, using John Stembridge's Symmetric Functions package:

withSF();

AA:=add(s[op(la)], la=subPar([2, 1])); tos(skew(AA, AA));

scalar(%, add(h1^r, r=0..4));

Cf. A143672, A000108, A005118

Sequence in context: A152554 A045990 A007006 this_sequence A113597 A000273 A071897

Adjacent sequences: A166857 A166858 A166859 this_sequence A166861 A166862 A166863

nonn

Jennifer Woodcock (jennifer.woodcock(AT)ugdsb.on.ca), Oct 21 2009