login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A166860 Number of saturated chains in the poset of Dyck paths ordered by inclusion. 1
1, 1, 3, 16, 191, 9586, 3621062, 13539455808, 596242050871827, 358279069210950329112, 3339667708892016201497713938, 540966002417189385158099747634890008, 1685909333511453301447626182908204645875878754, 110859993072493750180447848516163015805399916591746521402 (list; graph; refs; listen; history; text; internal format)
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

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));

# 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-Fran├žois 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 19 20:41 EST 2020. Contains 331066 sequences. (Running on oeis4.)