A189942 Table, read by rows, of the number of quivers of type Ã_(n-1) according to the parameter k (n >= 2, 1 <= k <= [n/2]).

1, 2, 5, 4, 14, 12, 42, 36, 22, 132, 108, 100, 429, 349, 315, 172, 1430, 1144, 1028, 980, 4862, 3868, 3432, 3240, 1651, 16796, 13260, 11700, 10920, 10584, 58786, 46210, 40520, 37556, 36036, 18028, 208012, 162792, 142120, 130900, 124740, 121968

Offset: 2

Author

Jonathan Vos Post, May 01 2011

Keywords

• nonn
• tabf

Comments

Table 1, p. 15 of Bastian.

There is a bijection with dissections of an annulus [Hermund André Torkildsen]. - N. J. A. Sloane, Jan 31 2013

Examples

			The table begins
===================================
n  |  r=1 | r=2 | r=3 | r=4 | r=5 |
===================================
n=2     1
n=3     2
n=4     5      4
n=5    14     12
n=6    42     36    22
n=7   132    108   100
n=8   429    349   315   172
n=9  1430   1144  1028   980
n=10 4862   3868  3432  3240  1651
===================================
		

References

• Francois Bergeron, Gilbert Labelle and Pierre Leroux, Combinatorial species and tree-like structures, Encyclopedia of Mathematics and its Applications, vol. 67, Cambridge University Press, Cambridge, 1998, Translated from the 1994 French original by Margaret Readdy, With a foreword by Gian-Carlo Rota.

Links

• Ibrahim Assem, Thomas Brustle, Gabrielle Charbonneau-Jodoin and Pierre-Guy Plamondon, Gentle algebras arising from surface triangulations, Algebra & Number Theory 4 (2010), no. 2, 201-229; arXiv:0903.3347 [math.RT], 2009.
• Janine Bastian, Thomas Prellberg, Martin Rubey and Christian Stump, Counting the number of elements in the mutation classes of Ã_n-quivers, arXiv:0906.0487 [math.CO], 2009-2011.
• Janine Bastian, Mutation classes of Ã_n-quivers and derived equivalence classification of cluster tilted algebras of type Ã_n, Algebra Number Theory 5 (2011), no. 5, 567-594; arXiv:0901.1515 [math.RT], 2009-2012.
• Hermund André Torkildsen, A geometric realization of the m-cluster category of type Ã, arXiv 1208.2138 [math.RT], 2012. - From _N. J. A. Sloane_, Jan 31 2013

Crossrefs

Cf. A000108 (r=1), A000888 (n=2r+1).

Programs

Mathematica

a[r_, r_] := 1/2 (Binomial[2 r, r]/2 + Sum[EulerPhi[k]/(4 r) Binomial[2 r/k, r/k]^2, {k, Divisors@r}]);
a[r_, s_] := 1/2 Sum[EulerPhi[k]/(r + s) Binomial[2 r/k, r/k] Binomial[2 s/k, s/k], {k, Intersection[Divisors@r, Divisors@s]}];
Table[a[r, n - r], {n, 2, 10}, {r, n/2}] // TableForm
(* Andrey Zabolotskiy, Jan 19 2022 *)

Extensions

Rows 11-13 added by Andrey Zabolotskiy, Jan 19 2022