A344136 Number of linear intervals in the Tamari lattices.

1, 3, 12, 49, 198, 792, 3146, 12441, 49062, 193154, 759696, 2986458, 11737820, 46134090, 181350630, 713046345, 2804421510, 11033453970, 43424181240, 170965500030, 673354218420, 2652994345560, 10456457024052, 41227321016394

Offset: 1

Author

F. Chapoton, May 10 2021

Keywords

• nonn

Comments

The description is conjectural. An interval is linear if it is isomorphic to a total order. The conjecture has been checked up to the term 49062 for n=9.

Apparently odd exactly when n is a power of 2.

Examples

			All 3 intervals in the lattice of cardinality 2 are linear. Among 13 intervals in the pentagon, only one is not linear.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..1663
• Clément Chenevière, Linear Intervals in the Tamari, Dyck and alt-Tamari Lattices, arXiv:2209.00418 [math.CO], 2022.
• Clément Chenevière, Enumerative study of intervals in lattices of Tamari type, Ph. D. thesis, Univ. Strasbourg (France), Ruhr-Univ. Bochum (Germany), HAL tel-04255439 [math.CO], 2024. See p. 151.

Crossrefs

Cf. A000260.

Programs

Mathematica

Array[(3/2) Binomial[2 #, #]*(#^2 - # + 2)/(# + 2)/(# + 1) &, 24] (* Michael De Vlieger, Sep 09 2022 *)

Sage

[3/2*binomial(2*n,n)*(n**2-n+2)/(n+2)/(n+1) for n in range(1,30)]

Formula

a(n) = (3/2)*binomial(2*n, n)*(n^2 - n + 2)/(n + 2)/(n + 1).

a(n) = binomial(2*n, n)/(n + 1) + binomial(2*n-1, n-2) + 2*binomial(2*n-1, n-3).

a(n) ~ (3/2) * 4^n * (1 - 33/(8*n)) / sqrt(n*Pi). - Peter Luschny, May 10 2021

a(n) = a(n-1)*2*(2*n - 1)*(n^2 - n + 2)/((n + 2)*(n^2 - 3*n + 4)) for n > 1. - Chai Wah Wu, May 13 2021