A181519 Number of torsion pairs in the cluster category of type A_n up to Auslander-Reiten translation.

1, 3, 5, 19, 62, 301, 1413, 7304, 38294, 208052, 1149018, 6466761, 36899604, 213245389, 1245624985, 7345962126, 43688266206, 261791220038, 1579363550250, 9586582997562, 58513327318992, 358957495385684, 2212294939905234

Offset

3,2

Comments

a(n) is also the number of Ptolemy diagrams on n vertices up to rotation.

a(n) is the sum over all polygon dissections up to rotation, where each region of size at least four has weight two.

Links

Andrew Howroyd, Table of n, a(n) for n = 3..500

Thorsten Holm, Peter Jorgensen, Martin Rubey, Ptolemy diagrams and torsion pairs in the cluster category of Dynkin type A_n, arXiv:1010.1184v1 [math.RT], 2010

Formula

G.f.: (2*Sum_{d>=1} phi(d)*log(1/(1-P(y^d)))/d ) - (1/3)*(P(y)^3+2*P(y^3)) - (1/2)*(3*P(y)^2+P(y^2)) - 2*P(y) + y*P(y) - y^2 where y*P(y) - y^2 is the g.f. of A181517. [corrected by Andrew Howroyd, May 09 2023]

Example

For n=5 there are 5 Ptolemy diagrams up to rotation: the pentagon with no diagonal, the pentagon with one diagonal, the pentagon with two noncrossing diagonals, the pentagon with three diagonals and the pentagon with all five diagonals.

Prog

(PARI) seq(n)={my(p=serreverse(x - x^2*(1 + x)/(1 - x) + O(x*x^n)), P(d)=subst(p + O(x^(n\d+1)), x, x^d)); Vec(2*sum(d=1, n, eulerphi(d)/d*log(1/(1-P(d)))) - (P(1)^3 + 2*P(3))/3 - (3*P(1)^2+P(2))/2 - (2 - x)*P(1) - x^2)} \\ Andrew Howroyd, May 09 2023

Crossrefs

Cf. A181517.

Sequence in context: A174132 A106918 A182357 * A265778 A257866 A251617

Adjacent sequences: A181516 A181517 A181518 * A181520 A181521 A181522

Keyword

nonn

Author

Martin Rubey, Oct 26 2010

Status

approved