cp's OEIS Frontend

A344136 Number of linear intervals in the Tamari lattices.

%I A344136 #29 Jan 17 2024 10:35:22
%S A344136 1,3,12,49,198,792,3146,12441,49062,193154,759696,2986458,11737820,
%T A344136 46134090,181350630,713046345,2804421510,11033453970,43424181240,
%U A344136 170965500030,673354218420,2652994345560,10456457024052,41227321016394
%N A344136 Number of linear intervals in the Tamari lattices.
%C A344136 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.
%C A344136 Apparently odd exactly when n is a power of 2.
%H A344136 Michael De Vlieger, <a href="/A344136/b344136.txt">Table of n, a(n) for n = 1..1663</a>
%H A344136 Clément Chenevière, <a href="https://arxiv.org/abs/2209.00418">Linear Intervals in the Tamari, Dyck and alt-Tamari Lattices</a>, arXiv:2209.00418 [math.CO], 2022.
%H A344136 Clément Chenevière, <a href="https://theses.hal.science/tel-04255439">Enumerative study of intervals in lattices of Tamari type</a>, Ph. D. thesis, Univ. Strasbourg (France), Ruhr-Univ. Bochum (Germany), HAL tel-04255439 [math.CO], 2024. See p. 151.
%F A344136 a(n) = (3/2)*binomial(2*n, n)*(n^2 - n + 2)/(n + 2)/(n + 1).
%F A344136 a(n) = binomial(2*n, n)/(n + 1) + binomial(2*n-1, n-2) + 2*binomial(2*n-1, n-3).
%F A344136 a(n) ~ (3/2) * 4^n * (1 - 33/(8*n)) / sqrt(n*Pi). - _Peter Luschny_, May 10 2021
%F A344136 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
%e A344136 All 3 intervals in the lattice of cardinality 2 are linear. Among 13 intervals in the pentagon, only one is not linear.
%t A344136 Array[(3/2) Binomial[2 #, #]*(#^2 - # + 2)/(# + 2)/(# + 1) &, 24] (* _Michael De Vlieger_, Sep 09 2022 *)
%o A344136 (Sage)
%o A344136 [3/2*binomial(2*n,n)*(n**2-n+2)/(n+2)/(n+1) for n in range(1,30)]
%Y A344136 Cf. A000260.
%K A344136 nonn
%O A344136 1,2
%A A344136 _F. Chapoton_, May 10 2021