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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).

%I A384133 #11 May 30 2025 01:07:12
%S A384133 1,2,1,5,5,2,14,21,12,2,42,84,56,14,2,132,330,240,72,16,2,429,1287,
%T A384133 990,330,90,18,2,1430,5005,4004,1430,440,110,20,2,4862,19448,16016,
%U A384133 6006,2002,572,132,22,2,16796,75582,63648,24752,8736,2730,728,156,24,2
%N 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).
%C A384133 An interval is linear of height k if it is isomorphic to the total order on k+1 elements.
%H A384133 Clément Chenevière, <a href="https://arxiv.org/abs/2209.00418">Linear Intervals in the Tamari and the Dyck Lattices and in the alt-Tamari Posets</a>, arXiv:2209.00418 [math.CO], 2022.
%F A384133 Row sums give A344136.
%F A384133 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.
%e A384133 Triangle begins:
%e A384133    1;
%e A384133    2,    1;
%e A384133    5,    5,    2;
%e A384133   14,   21,   12,    2;
%e A384133   42,   84,   56,   14,    2;
%e A384133  132,  330,  240,   72,   16,    2;
%e A384133  ...
%t A384133 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 *)
%Y A384133 Cf. A000108, A000260, A344136.
%K A384133 nonn,tabl
%O A384133 1,2
%A A384133 _Ludovic Schwob_, May 20 2025