%I #18 May 30 2017 09:57:14

%S 1,1,2,1,5,9,7,2,14,70,176,249,202,88,16,42,552,3573,13609,33260,

%T 54430,60517,45248,21824,6144,768,132,4587,72490,653521,3785264,

%U 15104787,43358146,91942710,146186256,175196202,157630704,104922224,50152960,16290560,3221504

%N 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.

%H Alois P. Heinz, <a href="/A193629/b193629.txt">Rows n = 0..11, flattened</a>

%H J. Woodcock, <a href="http://garsia.math.yorku.ca/~zabrocki/papers/DPfinal.pdf">Properties of the poset of Dyck paths ordered by inclusion</a>

%e Poset of Dyck paths of semilength n=3:

%e .

%e . A A:/\ B:

%e . | / \ /\/\

%e . B / \ / \

%e . / \

%e . C D C: D: E:

%e . \ / /\ /\

%e . E /\/ \ / \/\ /\/\/\

%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].

%e Triangle begins:

%e : 1;

%e : 1;

%e : 2, 1;

%e : 5, 9, 7, 2;

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

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

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

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

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

%p end:

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

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

%p end:

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

%p l:= d(n, n, []); m:= nops(l);

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

%p r:= [1];

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

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

%p fi od; r

%p end;

%p r:= [];

%p for h to m do

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

%p od; r[]

%p end:

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

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

%K nonn,tabf

%O 0,3

%A _Alois P. Heinz_, Aug 01 2011