A057501 Signature-permutation of a Catalan Automorphism: Rotate non-crossing chords (handshake) arrangements; rotate the root position of general trees as encoded by A014486.

0, 1, 3, 2, 7, 8, 5, 4, 6, 17, 18, 20, 21, 22, 12, 13, 10, 9, 11, 15, 14, 16, 19, 45, 46, 48, 49, 50, 54, 55, 57, 58, 59, 61, 62, 63, 64, 31, 32, 34, 35, 36, 26, 27, 24, 23, 25, 29, 28, 30, 33, 40, 41, 38, 37, 39, 43, 42, 44, 47, 52, 51, 53, 56, 60, 129, 130, 132, 133, 134

Offset: 0

Author

Antti Karttunen, Sep 03 2000; entry revised Jun 06 2014

Keywords

• nonn

Comments

This is a permutation of natural numbers induced when "noncrossing handshakes", i.e., Stanley's interpretation (n), "n nonintersecting chords joining 2n points on the circumference of a circle", are rotated.

The same permutation is induced when the root position of plane trees (Stanley's interpretation (e)) is successively changed around the vertices.

For a good illustration how the rotation of the root vertex works, please see the Figure 6, "Rotation of an ordered rooted tree" in Torsten Mütze's paper (on page 24 in 20 May 2014 revision).

For yet another application of this permutation, please see the attached notes for A085197.

By "recursivizing" either the left or right hand side argument of A085201 in the formula, one ends either with A057161 or A057503. By "recursivizing" the both sides, one ends with A057505. - Antti Karttunen, Jun 06 2014

Links

• Antti Karttunen, Table of n, a(n) for n = 0..2055
• A. Karttunen, Introductory Survey of Catalan Automorphisms and Bijections (an unfinished draft), pp. 56-57.
• A. Karttunen et al, Combinatorial interpretations of Catalan numbers, OEIS Wiki.
• Torsten Mütze, Proof of the middle levels conjecture, arXiv preprint arXiv:1404.4442 [math.CO], 2014 (p. 24).
• R. P. Stanley, Exercises on Catalan and Related Numbers (This sequence is concerned with rotations of the interpretations (e) and (n) in the Exercise 19)
• Index entries for signature-permutations of Catalan automorphisms

Crossrefs

Inverse: A057502.

Also, a "SPINE"-transform of A074680, and thus occurs as row 17 of A122203. (Also as row 65167 of A130403.)

Successive powers of this permutation, a^2(n) - a^6(n): A082315, A082317, A082319, A082321, A082323.

Other related permutations: A057161, A057163, A057503, A057505, A057508, A057509, A057511, A069770, A069771, A069772, A069773, A069888, A069889, A082313, A082314, A085173, A086427, A123501, A127291, A127292.

Cf. also A057548, A072771, A072772, A085201, A002995 (cycle counts), A057543 (max cycle lengths), A085197, A129599, A057517, A064638, A064640.

Programs

Maple

map(CatalanRankGlobal,map(RotateHandshakes, A014486));
RotateHandshakes := n -> pars2binexp(RotateHandshakesP(binexp2pars(n)));
RotateHandshakesP := h -> `if`((0 = nops(h)),h,[op(car(h)),cdr(h)]); # This does the trick! In Lisp: (defun RotateHandshakesP (h) (append (car h) (list (cdr h))))
car := proc(a) if 0 = nops(a) then ([]) else (op(1,a)): fi: end: # The name is from Lisp, takes the first element (head) of the list.
cdr := proc(a) if 0 = nops(a) then ([]) else (a[2..nops(a)]): fi: end: # As well. Takes the rest (the tail) of the list.
PeelNextBalSubSeq := proc(nn) local n,z,c; if(0 = nn) then RETURN(0); fi; n := nn; c := 0; z := 0; while(1 = 1) do z := 2*z + (n mod 2); c := c + (-1)^n; n := floor(n/2); if(c >= 0) then RETURN((z - 2^(floor_log_2(z)))/2); fi; od; end;
RestBalSubSeq := proc(nn) local n,z,c; n := nn; c := 0; while(1 = 1) do c := c + (-1)^n; n := floor(n/2); if(c >= 0) then break; fi; od; z := 0; c := -1; while(1 = 1) do z := 2*z + (n mod 2); c := c + (-1)^n; n := floor(n/2); if(c >= 0) then RETURN(z/2); fi; od; end;
pars2binexp := proc(p) local e,s,w,x; if(0 = nops(p)) then RETURN(0); fi; e := 0; for s in p do x := pars2binexp(s); w := floor_log_2(x); e := e * 2^(w+3) + 2^(w+2) + 2*x; od; RETURN(e); end;
binexp2pars := proc(n) option remember; `if`((0 = n),[],binexp2parsR(binrev(n))); end;
binexp2parsR := n -> [binexp2pars(PeelNextBalSubSeq(n)),op(binexp2pars(RestBalSubSeq(n)))];
# Procedure CatalanRankGlobal given in A057117, other missing ones in A038776.

Formula

a(0) = 0, and for n>=1, a(n) = A085201(A072771(n), A057548(A072772(n))). [This formula reflects directly the given non-destructive Lisp/Scheme function: A085201 is a 2-ary function corresponding to 'append', A072771 and A072772 correspond to 'car' and 'cdr' (known also as first/rest or head/tail in some dialects), and A057548 corresponds to unary form of function 'list'].

As a composition of related permutations:

a(n) = A057509(A069770(n)).

a(n) = A057163(A069773(A057163(n))).

Invariance-identities:

A129599(a(n)) = A129599(n) holds for all n.