A166860 Number of saturated chains in the poset of Dyck paths ordered by inclusion.

1, 1, 3, 16, 191, 9586, 3621062, 13539455808, 596242050871827, 358279069210950329112, 3339667708892016201497713938, 540966002417189385158099747634890008, 1685909333511453301447626182908204645875878754, 110859993072493750180447848516163015805399916591746521402

Offset

0,3

Comments

Breakdown by length of chain:

n: chains

0: 1;

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.

References

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

Links

Table of n, a(n) for n=0..13.

J. Woodcock, Properties of the poset of Dyck paths ordered by inclusion

Formula

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) matrix.

Example

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.

Maple

# E.g., 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));
# second Maple program:
d:= proc(x, y, l) option remember;
      `if`(x<=1, [[y, l[]]], [seq(d(x-1, i, [y, l[]])[], i=x-1..y)])
    end:
a:= proc(n) option remember; local g;
      g:= proc(l) option remember;
            1 +add(`if`(l[i]>i and (i=1 or l[i-1]<l[i]),
                g(subsop(i=l[i]-1, l)), 0), i=1..n-1)
          end;
      add(g(j), j=d(n, n, []))
    end:
seq(a(n), n=0..10);  # Alois P. Heinz, Jul 27 2011

Mathematica

d[x_, y_, l_List] := d[x, y, l] = If[x <= 1, {Join[{y}, l]}, Flatten[Table[d[x-1, i, Join[{y}, l]], {i, x-1, y}], 1]]; a[n_] := a[n] = Module[{g}, g[l_List] := g[l] = 1 + Sum[If[l[[i]] > i && (i == 1 || l[[i-1]] < l[[i]]), g[ReplacePart[l, i -> l[[i]] - 1]], 0], {i, 1, n-1}]; Sum[g[j], {j, d[n, n, {}]}]]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Jul 06 2015, after Alois P. Heinz *)

Crossrefs

Cf. A143672, A000108, A005118.

Sequence in context: A045990 A007006 A174137 * A196562 A317073 A272658

Adjacent sequences: A166857 A166858 A166859 * A166861 A166862 A166863

Keyword

nonn

Author

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

Extensions

a(9)-a(13) from Alois P. Heinz, Jul 27 2011

Status

approved