A057543 -id:A057543 - cp's OEIS frontend

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

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

nonn

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

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

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.

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.

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.

A057544 Maximum cycle length (orbit size) in the rotation permutation of n+2 side polygon triangularizations.

Original entry on oeis.org

1, 1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68

Offset: 0

Antti Karttunen, Sep 07 2000

I.e., in permutations A057161 and A057162 (also A057503 and A057504), the longest cycle among all cycles between the (A014138(n-2)+1)-th and (A014138(n-1))-th terms.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A057542, A057543, A057545.

a(n)=if(n>2,n+2,1) \\ Charles R Greathouse IV, Jun 20 2024

a(0)=1, a(1)=1, a(2)=2, a(n)=n+2.
From Chai Wah Wu, Jul 28 2022: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 4.
G.f.: (-2*x^4 + 2*x^3 + x^2 - x + 1)/(x - 1)^2. (End)

More terms from Sean A. Irvine, Jun 13 2022

A086427 Permutation of natural numbers induced by the Catalan bijection gma086427 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 8, 6, 7, 5, 4, 22, 19, 20, 15, 14, 21, 16, 18, 13, 12, 17, 10, 11, 9, 64, 60, 61, 52, 51, 62, 53, 55, 41, 40, 54, 38, 39, 37, 63, 56, 57, 43, 42, 59, 47, 50, 36, 34, 49, 35, 32, 31, 58, 44, 46, 27, 26, 48, 29, 33, 30, 45, 24, 25, 28, 23, 196, 191, 192, 178, 177

Offset: 0

Antti Karttunen, Jun 23 2003

nonn

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

A. Karttunen, Gatomorphisms (With the complete Scheme source)
Index entries for signature-permutations induced by Catalan automorphisms

Inverse: A086428. a(n) = A086431(A086428(A086431(n))) = A057164(A085173(A057164(n))) = A086425(A057501(A086426(n))). Occurs in A073200. Cf. also A086429 (whole step rotate).

Number of cycles: A002995. Number of fixed points: A019590. Max. cycle size: A057543. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A057542 Maximum cycle length in each permutation between A038776(1) and A038776(A000108(n)).

Original entry on oeis.org

1, 1, 1, 3, 4, 16, 87, 202, 607, 1441, 4708, 41888, 44741, 339108, 1617551

Offset: 0

Antti Karttunen, Sep 07 2000

Sean A. Irvine, Java program (github)

Cycle lengths of permutation A038776 given in A038774.
Cf. A057543, A057544, A057545.
LCM's of all cycles: A060113.

map(lmax,Bf2DfBinTreePermutationCycleLengths(some_value)); (e.g. 10)
bf2df := s -> (btbf2df(binrev(s),0,1)/2); # btbf2df and binrev given in A038776
Bf2DfBinTreePermutationCycleLengths := proc(upto_n) local u,n,a,r,b; a := []; for n from 0 to upto_n do b := []; u := (binomial(2*n,n)/(n+1)); for r from 0 to u-1 do b := [op(b),1+CatalanRank(n,bf2df(CatalanUnrank(n,r)))]; od; a := [op(a),CycleLengths1(b)]; od; RETURN(a); end;
CycleLengths1 := b -> [[(nops(b)-convert(map(nops,convert(b,'disjcyc')),`+`)),`*`,1],op(map(nops,convert(b,'disjcyc')))];
last_term := proc(l) local n: n := nops(l); if(0 = n) then ([]) else (op(n,l)): fi: end:
lmax := proc(a) local e,z; z := 0; for e in a do if whattype(e) = list then e := last_term(e); fi; if e > z then z := e; fi; od; RETURN(z); end;

a(11)-a(14) from Sean A. Irvine, Jun 13 2022

A073203 Array of maximum cycle length sequences for the table A073200.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 3, 2, 1, 1, 2, 2, 2, 6, 2, 2, 1, 1, 2, 2, 2, 8, 2, 3, 2, 1, 1, 2, 2, 2, 10, 2, 6, 4, 1, 1, 1, 2, 2, 2, 12, 2, 8, 8, 1, 2, 1, 1, 2, 2, 2, 14, 2, 10, 16, 1, 4, 1, 1, 1, 2, 2, 2, 16, 2, 12, 32, 1, 8, 2, 2, 1, 1

Offset: 0

Antti Karttunen, Jun 25 2002

Each row of this table gives the longest cycle/orbit produced by the Catalan bijection (given in the corresponding row of A073200) when it acts on A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.

A. Karttunen, Gatomorphisms (With the complete source and explanation)

Cf. also A073201, A073202, A073204.
Few EIS-sequences which occur in this table. Only the first known occurrence(s) given:.
Rows 6 and 8: A011782, Row 7: A000012, Row 12, 14: A000793 (shifted right and prepended with 1), Row 261: A057543, Row 2614: A057545, Rows 2618, 17517: A057544.

A086428 Permutation of natural numbers induced by the Catalan bijection gma086428 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 5, 6, 4, 22, 20, 21, 18, 17, 13, 12, 15, 19, 16, 10, 11, 14, 9, 64, 61, 62, 55, 54, 63, 57, 59, 50, 49, 58, 46, 48, 45, 36, 34, 35, 32, 31, 41, 40, 52, 60, 53, 43, 56, 47, 44, 27, 26, 29, 33, 30, 38, 39, 51, 42, 24, 25, 28, 37, 23, 196, 192, 193, 181, 180

Offset: 0

Antti Karttunen, Jun 23 2003

nonn

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

A. Karttunen, Gatomorphisms (With the complete Scheme source)
Index entries for signature-permutations induced by Catalan automorphisms

Inverse: A086427. a(n) = A086431(A086427(A086431(n))) = A057164(A085174(A057164(n))) = A086425(A057502(A086426(n))). Occurs in A073200. Cf. also A086430 (whole step rotate).

Number of cycles: A002995. Number of fixed points: A019590. Max. cycle size: A057543. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A085173 Permutation of natural numbers induced by the Catalan bijection gma085173 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 5, 6, 4, 22, 21, 18, 20, 17, 13, 12, 15, 19, 14, 10, 16, 11, 9, 64, 63, 59, 62, 58, 50, 49, 55, 61, 54, 46, 57, 48, 45, 36, 35, 32, 34, 31, 41, 40, 52, 60, 51, 38, 56, 39, 37, 27, 26, 43, 47, 42, 29, 53, 33, 28, 24, 44, 30, 25, 23, 196, 195, 190, 194, 189

Offset: 0

Antti Karttunen, Jun 23 2003

nonn

This Catalan bijection rotates by "half step" the interpretations (pp)-(rr) of Stanley, using the "rising slope" mapping illustrated in A085161.

A. Karttunen, Gatomorphisms (With the complete Scheme source)
Index entries for signature-permutations induced by Catalan automorphisms

Inverse: A085174. a(n) = A085161(A085174(A085161(n))) = A085169(A057501(A085170(n))) = A074684(A057501(A074683(n))). Occurs in A073200. Cf. also A085159 (whole step rotate), A086427.

Number of cycles: A002995. Number of fixed points: A019590. Max. cycle size: A057543. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A085174 Permutation of natural numbers induced by the Catalan bijection gma085174 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 8, 6, 7, 5, 4, 22, 19, 21, 15, 14, 18, 16, 20, 13, 11, 17, 12, 10, 9, 64, 60, 63, 52, 51, 59, 56, 62, 41, 39, 58, 40, 38, 37, 50, 47, 49, 43, 42, 55, 53, 61, 36, 33, 54, 35, 29, 28, 46, 44, 57, 32, 30, 48, 34, 27, 25, 45, 31, 26, 24, 23, 196, 191, 195, 178, 177

Offset: 0

Antti Karttunen, Jun 23 2003

nonn

This Catalan bijection rotates by "half step" the interpretations (pp)-(rr) of Stanley, using the "rising slope" mapping illustrated in A085161.

A. Karttunen, Gatomorphisms (With the complete Scheme source)
Index entries for signature-permutations induced by Catalan automorphisms

Inverse: A085173. a(n) = A085161(A085173(A085161(n))) = A085169(A057502(A085170(n))) = A074684(A057502(A074683(n))). Occurs in A073200. Cf. also A085160 (whole step rotate), A086428.

Number of cycles: A002995. Number of fixed points: A019590. Max. cycle size: A057543. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

cp's OEIS Frontend

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

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A057544 Maximum cycle length (orbit size) in the rotation permutation of n+2 side polygon triangularizations.

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A086427 Permutation of natural numbers induced by the Catalan bijection gma086427 acting on symbolless S-expressions encoded by A014486/A063171.

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A057542 Maximum cycle length in each permutation between A038776(1) and A038776(A000108(n)).

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Maple

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A073203 Array of maximum cycle length sequences for the table A073200.

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A086428 Permutation of natural numbers induced by the Catalan bijection gma086428 acting on symbolless S-expressions encoded by A014486/A063171.

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A085173 Permutation of natural numbers induced by the Catalan bijection gma085173 acting on symbolless S-expressions encoded by A014486/A063171.

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A085174 Permutation of natural numbers induced by the Catalan bijection gma085174 acting on symbolless S-expressions encoded by A014486/A063171.

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