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
Keywords
Examples
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.
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
Crossrefs
Cf. A181517.
Programs
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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
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]
Comments