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 , 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, on page 79 writes

The composite maps $M_{G}=L^{-1}\circ R$ and $M_{G}^{-1}=R^{-1}\circ L$ 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)