OFFSET
0,3
COMMENTS
This is also the number of interval-posets with connected Hasse diagram.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1031
Jose Bastidas, Félix Gélinas, Vincent Pilaud, Germain Poullot, Andrew Sack, and Eleni Tzanaki, Interval hypergraphic polytopes (or deformed associahedra), Tamari interval posets, and weeping willows, arXiv:2606.18376 [math.CO], 2026. See p. 22.
Frédéric Chapoton, Sur le nombre d'intervalles dans les treillis de Tamari, arXiv:math/0602368 [math.CO], 2006.
Frédéric Chapoton, Sur le nombre d'intervalles dans les treillis de Tamari, Séminaire Lotharingien de combinatoire, Vol. 55 (2006).
Alois Panholzer, Parking function varieties for combinatorial tree models, arXiv:2007.14676 [math.CO], 2020.
FORMULA
The generating series can be obtained by inverting the generating series of A000260.
EXAMPLE
Among the 3 interval-posets of size 2 :
1 --> 2 ; 1 <-- 2 ; 1 2,
only the third (which is an antichain) is not a connected poset.
MAPLE
h:= proc(n) h(n):= 2*(4*n+1)!/((n+1)!*(3*n+2)!) end:
a:= proc(n) a(n):= `if`(n=0, 0, h(n)-add(a(n-i)*h(i), i=1..n-1)) end:
seq(a(n), n=0..25); # Alois P. Heinz, Feb 28 2019
MATHEMATICA
terms = 25;
G[_] = 0; Do[G[x_] = 1 + x G[x]^4 + O[x]^terms, terms];
F[x_] = 1 - 1/((2 - G[x]) G[x]^2);
CoefficientList[F[x], x] (* Jean-François Alcover, Feb 14 2019 *)
PROG
(SageMath)
F = PowerSeriesRing(ZZ, 't')([1] + [(2 * binomial(4 * n + 1, n - 1)) // (n * (n + 1)) for n in range(1, 20)])
1 - F.inverse()
(Julia)
using Nemo
s(n) = div(Nemo.binom(4*n + 2, n + 1), (2*n + 1) * (3*n + 2))
R, z = PowerSeriesRing(ZZ, 25, "z")
F = sum(s(n)z^n for n in 0:25)
G = 1 - inv(F)
println([coeff(G, n) for n in 0:24]) # Peter Luschny, Feb 26 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
F. Chapoton, Feb 26 2018
STATUS
approved