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

Author

Christian Stump, Oct 19 2015

Keywords

• nonn
• tabf

Comments

Row sums give A000108.

Examples

			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;
...
		

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.

Crossrefs

Cf. A000108, A000260.

Formula

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

Extensions

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