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A181519
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Number of torsion pairs in the cluster category of type A_n up to Auslander-Reiten translation.
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2
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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
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3,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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]
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EXAMPLE
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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.
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PROG
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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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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