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User talk:Antti Karttunen/Catalan bijections
However, I guess that we cannot a priori exclude the possibility that also any arbitrarily constructed size-preserving bijection would not have a simple description in the terms of symmetries of some yet unknown combinatorial interpretation of Catalan numbers.
For example, *A057163 is best understood as the reflection of plane binary trees and *A057164 as the reflection of plane general trees, but of their compositions *A057505 and *A057506, described by Donaghey in [1], it is hard to say, what exactly is the structural property which determines how they partition their domain into different equivalence classes. (But I cannot deny that possibility either).
Some reason for my confusing terminology, apart from my dilettantism, is that Donaghey in his Automorphisms on Catalan Trees and Bracketings[2], on page 79 writes
The composite maps and are inverse automorphisms mapping g.b. 1-1 onto itself.
etc. However, for the most of Catalan bijections appearing in peer-reviewed journals, the term bijection is used. -- Antti Karttunen 20:47, 22 June 2011 (UTC)
- I apologize for the current, sorry state of this and related pages (like Combinatorial interpretations of Catalan numbers). I cannot work on them before the October 2011 at earliest, because I got another deadline coming. Thanks for your patience! -- Antti Karttunen 20:23, 1 August 2011 (UTC)