Counting the number of elements in the mutation classes of \tilde{A}_n-quivers

In this article we prove explicit formulae counting the elements in the mutation classes of quivers of type Ãn. In particular, we obtain the number of non-isomorphic clustertilted algebras of type Ãn. Furthermore, we give an alternative proof for the number of quivers of Dynkin type Dn which was first determined by Buan and Torkildsen in [4].

[1]  Gilbert Labelle,et al.  Combinatorial species and tree-like structures , 1997, Encyclopedia of mathematics and its applications.

[2]  Janine Bastian,et al.  Mutation classes of $\tilde{A}_n$-quivers and derived equivalence classification of cluster tilted algebras of type $\tilde{A}_n$ , 2009, 0901.1515.

[3]  Aslak Bakke Buan,et al.  Cluster-tilted algebras of type $A_n$ , 2007 .

[4]  Philippe Flajolet,et al.  Analytic Combinatorics , 2009 .

[5]  Aslak Bakke Buan,et al.  The Number of Elements in the Mutation Class of a Quiver of Type Dn , 2008, Electron. J. Comb..

[6]  Ralf Schiffler,et al.  Quivers with relations arising from clusters $(A_n$ case) , 2004 .

[7]  Aslak Bakke Buan,et al.  Derived equivalence classification for cluster-tilted algebras of type An , 2007 .

[8]  Idun Reiten,et al.  CLUSTER MUTATION VIA QUIVER REPRESENTATIONS , 2004 .

[9]  S. Fomin,et al.  Cluster algebras I: Foundations , 2001, math/0104151.

[10]  Hermund Andr'e Torkildsen.  COUNTING CLUSTER-TILTED ALGEBRAS OF TYPE An , 2008, 0801.3762.

[11]  Ibrahim Assem,et al.  Gentle algebras arising from surface triangulations , 2009, 0903.3347.

[12]  Sergey Fomin,et al.  Cluster algebras and triangulated surfaces. Part I: Cluster complexes , 2006 .

[13]  Dagfinn F. Vatne,et al.  The Mutation Class of D n Quivers , 2008, 0810.4789.

[14]  F. Jotzo,et al.  Double counting and the Paris Agreement rulebook , 2019, Science.

[15]  Martin Rubey,et al.  Extended rate, more GFUN , 2007, J. Symb. Comput..