OFFSET
1,2
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.
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.
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
EXAMPLE
All 3 intervals in the lattice of cardinality 2 are linear. Among 13 intervals in the pentagon, only one is not linear.
MATHEMATICA
Array[(3/2) Binomial[2 #, #]*(#^2 - # + 2)/(# + 2)/(# + 1) &, 24] (* Michael De Vlieger, Sep 09 2022 *)
PROG
(Sage)
[3/2*binomial(2*n, n)*(n**2-n+2)/(n+2)/(n+1) for n in range(1, 30)]
CROSSREFS
KEYWORD
nonn
AUTHOR
F. Chapoton, May 10 2021
STATUS
approved