The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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.
Sequence in context: A026832 A225044 A325246 * A089408 A350287 A208888
KEYWORD
nonn,look,tabf
AUTHOR
Alois P. Heinz, Aug 02 2011
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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