A384133 Triangle read by rows: T(n,k) is the number of linear intervals of height k in the Tamari lattice Tam_n (0 <= k < n).

1, 2, 1, 5, 5, 2, 14, 21, 12, 2, 42, 84, 56, 14, 2, 132, 330, 240, 72, 16, 2, 429, 1287, 990, 330, 90, 18, 2, 1430, 5005, 4004, 1430, 440, 110, 20, 2, 4862, 19448, 16016, 6006, 2002, 572, 132, 22, 2, 16796, 75582, 63648, 24752, 8736, 2730, 728, 156, 24, 2

Offset: 1

Author

Ludovic Schwob, May 20 2025

Keywords

• nonn
• tabl

Comments

An interval is linear of height k if it is isomorphic to the total order on k+1 elements.

Examples

			Triangle begins:
   1;
   2,    1;
   5,    5,    2;
  14,   21,   12,    2;
  42,   84,   56,   14,    2;
 132,  330,  240,   72,   16,    2;
 ...
		

Links

• Clément Chenevière, Linear Intervals in the Tamari and the Dyck Lattices and in the alt-Tamari Posets, arXiv:2209.00418 [math.CO], 2022.

Crossrefs

Cf. A000108, A000260, A344136.

Programs

Mathematica

T[n_,k_]:=If[k==0, Binomial[2*n,n]/(n+1), If[k==1, Binomial[2*n-1,n-2], 2*Binomial[2*n-k,n-k-1]]]; Table[T[n,k],{n,10},{k,0,n-1}]//Flatten (* Stefano Spezia, May 26 2025 *)

Formula

Row sums give A344136.

T(n,0) = C(2*n,n)/(n+1), T(n,1) = C(2*n-1,n-2) and T(n,k) = 2*C(2*n-k,n-k-1) if k>1.