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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]). 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,2
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
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
EXAMPLE
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
===================================
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 *)
CROSSREFS
Cf. A000108 (r=1), A000888 (n=2r+1).
Sequence in context: A225046 A079053 A224272 * A251554 A002518 A093727
KEYWORD
nonn,tabf
AUTHOR
Jonathan Vos Post, May 01 2011
EXTENSIONS
Rows 11-13 added by Andrey Zabolotskiy, Jan 19 2022
STATUS
approved

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Last modified March 18 22:09 EDT 2024. Contains 370951 sequences. (Running on oeis4.)