Editing Combinatorial class (section)

==Counting sequences and isomorphism==
The ''counting sequence'' of a combinatorial class is the sequence of the numbers of elements of size ''i'' for ''i''&nbsp;=&nbsp;0, 1, 2, ...; it may also be described as a [[generating function]] that has these numbers as its coefficients. The counting sequences of combinatorial classes are the main subject of study of [[enumerative combinatorics]]. Two combinatorial classes are said to be isomorphic if they have the same numbers of objects of each size, or equivalently, if their counting sequences are the same.<ref name="ac">{{citation|title=Analytic Combinatorics|first1=Philippe|last1=Flajolet|author1-link=Philippe Flajolet|first2=Robert|last2=Sedgewick|author2-link=Robert Sedgewick (computer scientist)|publisher=Cambridge University Press|year=2009|isbn=9781139477161|at=Definition I.3, p.19|title-link= Analytic Combinatorics}}.</ref> Frequently, once two combinatorial classes are known to be isomorphic, a [[bijective proof]] of this equivalence is sought; such a proof may be interpreted as showing that the objects in the two isomorphic classes are [[cryptomorphism|cryptomorphic]] to each other.

For instance, the [[Polygon triangulation|triangulation]]s of [[regular polygon]]s (with size given by the number of sides of the polygon, and a fixed choice of polygon to triangulate for each size) and the set of [[Unrooted binary tree|unrooted binary]] [[Tree_(graph_theory)#Plane_tree|plane tree]]s (up to [[graph isomorphism]], with a fixed ordering of the leaves, and with size given by the number of leaves) are both counted by the [[Catalan number]]s, so they form isomorphic combinatorial classes. A bijective isomorphism in this case is given by [[Dual graph|planar graph duality]]: a triangulation can be transformed bijectively into a tree with a leaf for each polygon edge, an internal node for each triangle, and an edge for each two (polygon edges?) or triangles that are adjacent to each other.<ref>{{citation|title=Triangulations: Structures for Algorithms and Applications|volume=25|series=Algorithms and Computation in Mathematics|first1=Jesús A.|last1=De Loera|author1-link=Jesús A. De Loera|first2=Jörg|last2=Rambau|first3=Francisco|last3=Santos|publisher=Springer|year=2010|isbn=9783642129711|url=https://books.google.com/books?id=SxY1Xrr12DwC&pg=PA4|at=Theorem 1.1.3, pp. 4–6}}.</ref>