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This automorphism reflects the interpretations (pp)-(rr) of Stanley, obtained from the Dyck paths with the "rising slope mapping" illustrated on the example lines.

Deutsch shows in his 1998 paper that this automorphism maps the number of returns of Dyck path to the height of the last peak, i.e., that A057515(n) = A080237(A057503(n)) holds for all n, thus the two parameters have the same distribution.

From the recursive forms of A057161 and A057503 it is seen that both can be viewed as a convergent limits of a process where either the left or right side argument of A085201 in formula for A057501 is "iteratively recursivized", and on the other hand, both of these can then in turn be made to converge towards A057505, when the other side of the formula is also "recursivized" in the same way. - Antti Karttunen, Jun 06 2014

The original definition was: Number of rooted general plane trees which are symmetric and will stay symmetric after the underlying plane binary tree has been reflected, i.e., number of integers i in range [A014137(n-1)..A014138(n-1)] such that A057164(i) = i and A057164(A057163(i)) = A057163(i).

(Thus also) the number of fixed points in range [A014137(n-1)..A014138(n)] of permutation A071661 (= Donaghey's automorphism M "squared"), which is equal to condition A057164(i) = A069787(i) = i, i.e., the size of the intersection of fixed points of permutations A057164 and A069787 in the same range.

Additional comment from Antti Karttunen, Dec 13 2017: (Start)

However, David Callan's A123050 claims to give more correct version of that count from n=26 onward, so I probably made a little mistake when converting my insights into the formula given here. At that time I reckoned that if the conjecture given in A080070 were true, then it would imply that the formula given here were exact, otherwise it would give only a lower bound.

It would be nice to know what an empirical program would give as the count of fixed points of A071661 for n in range [A014137(25)..A014138(26)] = [6619846420553 .. 24987199492704], with total A000108(26) = 18367353072151 points to check.

(End)

Meeussen's observation about the orbits of a composition of two involutions F and R states that if the orbit size of the composition (acting on a particular element of the set) is odd, then it contains an element fixed by the other involution if and only if it contains also an element fixed by the other, on the (almost) opposite side of the cycle. Here those two involutions are A057163 and A057164, their composition is Donaghey's "Map M" A057505 and as the trees A080293/A080295 are symmetric as binary trees and the cycle sizes A080292 are odd, it follows that these are symmetric as general trees.

This Catalan bijection reflects the interpretations (pp)-(rr) of Stanley, obtained with the "descending slope mapping" from the Dyck paths encoded by A014486.

Like A069771, A069772, A125976 and A126315/A126316, this automorphism keeps symmetric Dyck paths symmetric, but not necessarily same.

Deutsch and Elizalde show in their paper that this automorphism converts certain properties concerning "tunnels" of Dyck path to another set of properties concerning the number of hills, even and odd rises, as well as the number of returns (A057515), thus proving the equidistribution of the said parameters.

This automorphism is implemented with function "tau" (Scheme code given below) that takes as its arguments an S-expression and a Catalan automorphism that permutes only the top level of the list (i.e., the top-level branches of a general tree, or the whole arches of a Dyck path) and thus when the permuting automorphism is applied to a list (parenthesization) of length 2n it induces some permutation of [1..2n].

This automorphism is induced in that manner by the automorphism *A127287 and likewise, *A127289 is induced by *A127285, *A057164 by *A057508, *A057501 by *A057509 and *A057502 by *A057510.

Note that so far these examples seem to satisfy the homomorphism condition, e.g., as *A127287 = *A127285 o *A057508 so is *A127291 = *A127289 o *A057164. and likewise, as *A057510 = *A057508 o *A057509 o *A057508, so is *A057502 = *A057164 o *A057501 o *A057164.

However, it remains open what are the exact criteria of the "picking automorphism" and the corresponding permutation that this method would induce a bijection. For example, if we give *A127288 (the inverse of *A127287) to function "tau" it will not induce *A127292 and actually not a bijection at all.

Instead, we have to compute the inverse of this automorphism with another, more specific algorithm that implements Deutsch's and Elizalde's description and is given in A127300.

This automorphism reflects non-crossing handshakes (the interpretation n of Stanley's exercise 19) over the diagonal that goes through corner at "1 o'clock".

This Catalan bijection rotates the interpretations (pp)-(rr) of Stanley, using the "descending slope" mapping illustrated in A086431.