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 A193691 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. 5
 1, 1, 1, 2, 1, 2, 2, 4, 5, 1, 2, 2, 4, 5, 2, 4, 4, 8, 10, 5, 10, 13, 14, 1, 2, 2, 4, 5, 2, 4, 4, 8, 10, 5, 10, 13, 14, 2, 4, 4, 8, 10, 4, 8, 8, 16, 20, 10, 20, 26, 28, 5, 10, 10, 20, 25, 13, 26, 34, 37, 14, 28, 37, 41, 42, 1, 2, 2, 4, 5, 2, 4, 4, 8, 10, 5, 10, 13, 14, 2, 4, 4, 8, 10, 4 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Alois P. Heinz, Rows n = 0..9, flattened EXAMPLE Dyck paths of semilength n=3 listed in lexicographic order: . /\ . /\ /\ /\/\ / \ . /\/\/\ /\/ \ / \/\ / \ / \ . 101010 101100 110010 110100 111000 . k = (1) (2) (3) (4) (5) . We have (1) <= (1); (1),(2) <= (2); (1),(3) <= (3); (1),(2),(3),(4) <= (4); and (1),(2),(3),(4),(5) <= (5), thus row 3 = [1, 2, 2, 4, 5]. Triangle begins: 1; 1; 1, 2; 1, 2, 2, 4, 5; 1, 2, 2, 4, 5, 2, 4, 4, 8, 10, 5, 10, 13, 14; 1, 2, 2, 4, 5, 2, 4, 4, 8, 10, 5, 10, 13, 14, 2, 4, 4, 8, ... MAPLE d:= proc(n, l) local m; m:= nops(l); `if`(n=m, [l], [seq(d(n, [l[], j])[], j=`if`(m=0, 1, max(m+1, l[-1]))..n)]) end: le:= proc(x, y) local i; for i to nops(x) do if x[i]>y[i] then return false fi od; true end: T:= proc(n) option remember; local l; l:= d(n, []); seq(add(`if`(le(l[i], l[j]), 1, 0), i=1..j), j=1..nops(l)) end: seq(T(n), n=0..6); MATHEMATICA 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[[i]], l[[j]]], 1, 0], {i, 1, j}], {j, 1, Length[l]}]]; Table[T[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, Feb 01 2017, after Alois P. Heinz *) CROSSREFS Row sums give A005700. Lengths and last elements of rows give A000108. Cf. A193692, A193693, A193694. Sequence in context: A026832 A225044 A325246 * A089408 A350287 A208888 Adjacent sequences: A193688 A193689 A193690 * A193692 A193693 A193694 KEYWORD nonn,look,tabf AUTHOR Alois P. Heinz, Aug 02 2011 STATUS approved

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Last modified May 21 19:35 EDT 2024. Contains 372738 sequences. (Running on oeis4.)