A263191 Triangle read by rows: T(n>=0, 1<=k<=A000108(n)) is the number of Dyck paths of length 2n having k smaller elements in Tamari order.

1, 1, 1, 1, 1, 2, 1, 0, 1, 1, 3, 2, 2, 2, 0, 2, 0, 1, 0, 0, 0, 0, 1, 1, 4, 3, 5, 4, 2, 4, 0, 5, 2, 0, 2, 0, 3, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 4, 9, 6, 7, 6, 3, 10, 6, 4, 4, 0, 9, 5, 2, 0, 4, 4, 4, 0, 0, 4, 3, 1, 0, 2, 4, 0, 4, 0, 0, 0, 3, 0, 0, 2

Offset

0,6

Comments

Row sums give A000108.

Links

Alois P. Heinz, Rows n = 0..10, flattened

FindStat - Combinatorial Statistic Finder, The number of elements smaller than the given Dyck path in the Tamari Order.

Wikipedia, Tamari lattice.

Formula

Sum_{k=1..A000108(n)} k * T(n,k) = A000260(n). - Alois P. Heinz, Nov 15 2015

Example

Triangle begins:
1;
1;
1,1;
1,2,1,0,1;
1,3,2,2,2,0,2,0,1,0,0,0,0,1;
1,4,3,5,4,2,4,0,5,2,0,2,0,3,0,1,0,0,2,0,0,0,2,0,0,0,0,1,0,0,0,0,0,0,0, 0,0,0,0,0,0,1;
...

Crossrefs

Cf. A000108, A000260.

Sequence in context: A221857 A133607 A103631 * A192517 A309896 A083856

Adjacent sequences: A263188 A263189 A263190 * A263192 A263193 A263194

Keyword

nonn,tabf

Author

Christian Stump, Oct 19 2015

Extensions

Two terms (for rows 0 and 1) prepended by Alois P. Heinz, Nov 15 2015

Status

approved