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A193692 Triangle T(n,k), n>=0, 1<=k<=C(n), read by rows: T(n,k) = number of elements >= k-th path in the poset of Dyck paths of semilength n ordered by inclusion.

%I A193692 #21 Feb 01 2017 12:28:45
%S A193692 1,1,2,1,5,3,3,2,1,14,9,10,7,4,9,6,7,5,3,4,3,2,1,42,28,32,23,14,32,22,
%T A193692 26,19,12,17,13,9,5,28,19,22,16,10,23,16,19,14,9,13,10,7,4,14,10,12,9,
%U A193692 6,9,7,5,3,5,4,3,2,1,132,90,104,76,48,107,75,89,66,43,62,48,34,20,104
%N A193692 Triangle T(n,k), n>=0, 1<=k<=C(n), read by rows: T(n,k) = number of elements >= k-th path in the poset of Dyck paths of semilength n ordered by inclusion.
%H A193692 Alois P. Heinz, <a href="/A193692/b193692.txt">Rows n = 0..9, flattened</a>
%e A193692 Dyck paths of semilength n=3 listed in lexicographic order:
%e A193692 .                                               /\
%e A193692 .                  /\      /\        /\/\      /  \
%e A193692 .     /\/\/\    /\/  \    /  \/\    /    \    /    \
%e A193692 .     101010    101100    110010    110100    111000
%e A193692 . k = (1)       (2)       (3)       (4)       (5)
%e A193692 .
%e A193692 We have (1),(2),(3),(4),(5) >= (1); (2),(4),(5) >= (2); (3),(4),(5) >= (3);
%e A193692 (4),(5) >= (4); and (5) >= (5), thus row 3 = [5, 3, 3, 2, 1].
%e A193692 Triangle begins:
%e A193692 1;
%e A193692 1;
%e A193692 2,   1;
%e A193692 5,   3,  3,  2,  1;
%e A193692 14,  9, 10,  7,  4,  9,  6,  7,  5,  3,  4,  3, 2, 1;
%e A193692 42, 28, 32, 23, 14, 32, 22, 26, 19, 12, 17, 13, 9, 5, 28, 19, 22, 16, ...
%p A193692 d:= proc(n, l) local m; m:= nops(l);
%p A193692       `if`(n=m, [l], [seq(d(n, [l[], j])[],
%p A193692                      j=`if`(m=0, 1, max(m+1, l[-1]))..n)])
%p A193692     end:
%p A193692 le:= proc(x, y) local i;
%p A193692        for i to nops(x) do if x[i]>y[i] then return false fi od; true
%p A193692      end:
%p A193692 T:= proc(n) option remember; local l;
%p A193692       l:= d(n, []);
%p A193692       seq(add(`if`(le(l[j], l[i]), 1, 0), i=j..nops(l)), j=1..nops(l))
%p A193692     end:
%p A193692 seq(T(n), n=0..6);
%t A193692 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}]]]; le[x_, y_] := Module[{i}, For[i = 1, i <= Length[x], i++, If[x[[i]] > y[[i]] , Return[False]]]; True]; T[n_] := T[n] = Module[{l}, l = d[n, {}]; Table[Sum[If[le[l[[j]], l[[i]]], 1, 0], {i, j, Length[l]}], {j, 1, Length[l]}]]; Table[T[n], {n, 0, 6}] // Flatten (* _Jean-François Alcover_, Feb 01 2017, after _Alois P. Heinz_ *)
%Y A193692 Row sums give A005700.
%Y A193692 Lengths and first elements of rows give A000108.
%Y A193692 Cf. A193691, A193693, A193694.
%K A193692 nonn,look,tabf
%O A193692 0,3
%A A193692 _Alois P. Heinz_, Aug 02 2011