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A181519 Number of torsion pairs in the cluster category of type A_n up to Auslander-Reiten translation. 1

%I

%S 1,3,5,19,62,301,1413,7304,38294,208052,1149018,6466761,36899604,

%T 213245389,1245624985,7345962126,43688266206,261791220038,

%U 1579363550250,9586582997562,58513327318992,358957495385684,2212294939905234

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

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

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

%H Thorsten Holm, Peter Jorgensen, Martin Rubey, <a href="http://arxiv.org/abs/1010.1184">Ptolemy diagrams and torsion pairs in the cluster category of Dynkin type A_n</a>, arXiv:1010.1184v1 [math.RT], 2010

%F G.f.: 2 sum(phi(d)/d*log(1/(1-P(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.

%e 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.

%Y Cf. A181517

%K nonn

%O 3,2

%A Martin Rubey (martin.rubey(AT)math.uni-hannover.de), Oct 26 2010

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Last modified January 26 16:19 EST 2021. Contains 340442 sequences. (Running on oeis4.)