%I #20 Oct 08 2022 10:26:39
%S 1,1,2,2,5,4,4,5,5,14,10,11,10,8,10,9,10,12,12,8,12,14,14,42,29,33,26,
%T 18,33,25,29,26,21,21,22,21,18,29,22,25,23,19,26,23,26,29,28,22,29,32,
%U 31,18,19,21,28,30,21,32,38,39,18,31,39,42,42,132,91,105,79,52,108,78,92,73,52
%N Triangle T(n,k), n>=0, 1<=k<=C(n), read by rows: T(n,k) = number of elements comparable to the k-th path in the poset of Dyck paths of semilength n ordered by inclusion.
%H Alois P. Heinz, <a href="/A193693/b193693.txt">Rows n = 0..9, flattened</a>
%e Dyck paths of semilength n=3 listed in lexicographic order:
%e . /\
%e . /\ /\ /\/\ / \
%e . /\/\/\ /\/ \ / \/\ / \ / \
%e . 101010 101100 110010 110100 111000
%e . k = (1) (2) (3) (4) (5)
%e .
%e Only paths (2) and (3) are incomparable, thus row 3 = [5, 4, 4, 5, 5].
%e Triangle begins:
%e 1;
%e 1;
%e 2, 2;
%e 5, 4, 4, 5, 5;
%e 14, 10, 11, 10, 8, 10, 9, 10, 12, 12, 8, 12, 14, 14
%e 42, 29, 33, 26, 18, 33, 25, 29, 26, 21, 21, 22, 21, 18, 29, 22, 25, 23, ...
%p d:= proc(n, l) local m; m:= nops(l);
%p `if`(n=m, [l], [seq(d(n, [l[], j])[],
%p j=`if`(m=0, 1, max(m+1, l[-1]))..n)])
%p end:
%p le:= proc(x, y) local i;
%p for i to nops(x) do if x[i]>y[i] then return false fi od; true
%p end:
%p T:= proc(n) option remember; local l;
%p l:= d(n, []);
%p seq(add(`if`(le(l[i], l[j]) or le(l[j], l[i]), 1, 0),
%p i=1..nops(l)), j=1..nops(l))
%p end:
%p seq(T(n), n=0..6);
%t d[n_, l_] := d[n, l] = Module[{m}, m = Length[l]; If[n == m, {l}, Flatten[#, 1]&@Table[d[n, Append[l, j]], {j, If[m == 0, 1, Max[m + 1, Last[l]]], n}]]];
%t le[x_, y_] := Module[{i}, For[i = 1, i <= Length[x], i++, If[x[[i]] > y[[i]], Return[False]]]; True];
%t T[n_] := T[n] = Module[{l}, l = d[n, {}]; Table[Sum[ If[le[l[[i]], l[[j]]] || le[l[[j]], l[[i]]], 1, 0], {i, 1, Length[l]}], {j, 1, Length[l]}]];
%t Table[T[n], {n, 0, 6}] // Flatten (* _Jean-François Alcover_, Oct 08 2022, after _Alois P. Heinz_ *)
%Y Lengths, first, last and second to last elements of rows give A000108.
%Y Cf. A193691, A193692, A193694.
%K nonn,look,tabf
%O 0,3
%A _Alois P. Heinz_, Aug 02 2011