

A181519


Number of torsion pairs in the cluster category of type A_n up to AuslanderReiten translation.


1



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

Table of n, a(n) for n=3..25.
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(phi(d)/d*log(1/(1P(y^d))),d=1..infinity) 2/3*(P(y)^3+2*P(y^3))1/2*(3*P(y)^2+P(y^2))2*P(y)+y where P(y) is the G.f. for the number of torsion pairs in the cluster category of type A_n, A181517.


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.


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 (martin.rubey(AT)math.unihannover.de), Oct 26 2010


STATUS

approved



