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A193629 Triangle T(n,k), n>=0, 0<=k<=C(n,2), read by rows: T(n,k) = number of k-length chains in the poset of Dyck paths of semilength n ordered by inclusion. 3
1, 1, 2, 1, 5, 9, 7, 2, 14, 70, 176, 249, 202, 88, 16, 42, 552, 3573, 13609, 33260, 54430, 60517, 45248, 21824, 6144, 768, 132, 4587, 72490, 653521, 3785264, 15104787, 43358146, 91942710, 146186256, 175196202, 157630704, 104922224, 50152960, 16290560, 3221504 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Rows n = 0..11, flattened

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

EXAMPLE

Poset of Dyck paths of semilength n=3:

.

.      A       A:/\      B:

.      |        /  \      /\/\

.      B       /    \    /    \

.     / \

.    C   D     C:        D:        E:

.     \ /         /\      /\

.      E       /\/  \    /  \/\    /\/\/\

.

Chains of length k=0: A, B, C, D, E (5); k=1: A-B, A-C, A-D, A-E, B-C, B-D, B-E, C-E, D-E (9); k=2: A-B-C, A-B-D, A-B-E, A-C-E, A-D-E, B-C-E, B-D-E (7), k=3: A-B-C-E, A-B-D-E (2) => [5, 9, 7, 2].

Triangle begins:

:   1;

:   1;

:   2,    1;

:   5,    9,     7,      2;

:  14,   70,   176,    249,     202,       88,       16;

:  42,  552,  3573,  13609,   33260,    54430,    60517,    45248, ...

: 132, 4587, 72490, 653521, 3785264, 15104787, 43358146, 91942710, ...

MAPLE

d:= proc(x, y, l) option remember;

      `if`(x<=1, [[l[], y]], [seq(d(x-1, i, [l[], y])[], i=x-1..y)])

    end:

le:= proc(l1, l2) local i;

       for i to nops(l1) do if l1[i]>l2[i] then return false fi od; true

     end:

T:= proc(n) option remember; local h, l, m, g, r;

      l:= d(n, n, []); m:= nops(l);

      g:= proc(t) option remember; local r, d;

            r:= [1];

            for d to t-1 do if le(l[d], l[t]) then

              r:= zip((x, y)->x+y, r, [0, g(d)[]], 0)

            fi od; r

          end;

      r:= [];

      for h to m do

        r:= zip((x, y)->x+y, r, g(h), 0)

      od; r[]

    end:

seq(T(n), n=0..7);

CROSSREFS

Row sums give: A143672-A057427. Column k=0 gives: A000108. Last elements of rows give: A005118. Row lengths give: A000124(n-1). Cf. A193536.

Sequence in context: A176665 A199050 A306539 * A021467 A011132 A178627

Adjacent sequences:  A193626 A193627 A193628 * A193630 A193631 A193632

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Aug 01 2011

STATUS

approved

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Last modified October 27 11:55 EDT 2021. Contains 348276 sequences. (Running on oeis4.)