0,3

Deutsch shows in his 1999 paper that this automorphism maps the number of doublerises of Dyck paths to number of valleys and height of the first peak to the number of returns, i.e. that A126306(n) = A127284(a(n)) and A126307(n) = A057515(a(n)) hold for all n.

The A000108(n-2) n-gon triangularizations can be reflected over n axes of symmetry, which all can be generated by appropriate compositions of the permutations A057161/A057162 and A057163.

Composition with A057164 gives signature permutation for Donaghey's Map M (A057505/A057506). Embeds into itself in scale n:2n+1 as a(n) = A083928(a(A080298(n))). A127302(a(n)) = A127302(n) and A057123(A057163(n)) = A057164(A057123(n)) hold for all n.

E. Deutsch, An involution on Dyck paths and its consequences, Discrete Math., 204 (1999), no. 1-3, 163-166.

A. Karttunen, C-program which computes this sequence.

Index entries for signature-permutations induced by Catalan automorphisms

a(n) = A083927(A057164(A057123(n)))

This involution (self-inverse permutation) of natural numbers is induced when we reflect the rooted plane binary trees encoded by A014486. E.g. we have A014486(5) = 44 (101100 in binary), A014486(7) = 52 (110100 in binary) and these encode the following rooted plane binary trees, which are reflections of each other:

....0...0.............0...0...

.....\./...............\./....

......1...0.........0...1.....

.......\./...........\./......

....0...1.............1...0...

.....\./...............\./....

......1.................1.....

thus a(5)=7 and a(7)=5.

a(n) = A080300(ReflectBinTree(A014486(n)))

ReflectBinTree := n -> ReflectBinTree2(n)/2; ReflectBinTree2 := n -> (`if`((0 = n), n, ReflectBinTreeAux(A030101(n))));

ReflectBinTreeAux := proc(n) local a, b; a := ReflectBinTree2(BinTreeLeftBranch(n)); b := ReflectBinTree2(BinTreeRightBranch(n)); RETURN((2^(A070939(b)+A070939(a))) + (b * (2^(A070939(a)))) + a); end;

NextSubBinTree := proc(nn) local n, z, c; n := nn; c := 0; z := 0; while(c < 1) do z := 2*z + (n mod 2); c := c + (-1)^n; n := floor(n/2); od; RETURN(z); end;

BinTreeLeftBranch := n -> NextSubBinTree(floor(n/2));

BinTreeRightBranch := n -> NextSubBinTree(floor(n/(2^(1+A070939(BinTreeLeftBranch(n))))));

(Scheme implementations of this automorphism that acts on S-expressions, i.e. list-structures:)

(CONSTRUCTIVE IMPLEMENTATION:) (define (*A057163 s) (cond ((not (pair? s)) s) (else (cons (*A057163 (cdr s)) (*A057163 (car s))))))

(DESTRUCTIVE IMPLEMENTATION:) (define (*A057163! s) (cond ((pair? s) (*A069770! s) (*A057163! (car s)) (*A057163! (cdr s)))) s)

This automorphism conjugates between the car/cdr-flipped variants of other automorphisms, e.g. A057162(n) = a(A057161(a(n))), A069768(n) = a(A069767(a(n))), A069769(n) = a(A057508(a(n))), A069773(n) = a(A057501(a(n))), A069774(n) = a(A057502(a(n))), A069775(n) = a(A057509(a(n))), A069776(n) = a(A057510(a(n))), A069787(n) = a(A057164(a(n))).

Row 1 of tables A122201 and A122202, that is, obtained with FORK (and KROF) transformation from even simpler automorphism *A069770. Cf. A122351.

Sequence in context: A130360 A082348 A122339 this_sequence A130918 A021308 A060921

Adjacent sequences: A057160 A057161 A057162 this_sequence A057164 A057165 A057166

nonn

Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Aug 18 2000. Equivalence with Deutsch's 1998 involution realized Dec 15 2006 and entry edited accordingly by AK, Jan 16 2007.