cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A089410 Least common multiple of all cycle sizes (also the maximum cycle size) in range [A014137(n-1)..A014138(n-1)] of permutation A074679/A074680.

Original entry on oeis.org

1, 1, 2, 5, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78
Offset: 0

Views

Author

Antti Karttunen, Nov 29 2003

Keywords

Links

Crossrefs

Cf. A016825.

A089840 Signature permutations of non-recursive Catalan automorphisms (i.e., bijections of finite plane binary trees, with no unlimited recursion down to indefinite distances from the root), sorted according to the minimum number of opening nodes needed in their defining clauses.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 2, 2, 1, 0, 5, 7, 3, 2, 1, 0, 6, 8, 4, 3, 2, 1, 0, 7, 6, 6, 5, 3, 2, 1, 0, 8, 4, 5, 4, 5, 3, 2, 1, 0, 9, 5, 7, 6, 6, 6, 3, 2, 1, 0, 10, 17, 8, 7, 4, 5, 6, 3, 2, 1, 0, 11, 18, 9, 8, 7, 4, 4, 4, 3, 2, 1, 0, 12, 20, 10, 12, 8, 7, 5, 5, 4, 3, 2, 1, 0, 13, 21, 14, 13, 12, 8, 7, 6
Offset: 0

Views

Author

Antti Karttunen, Dec 05 2003; last revised Jan 06 2009

Keywords

Comments

Each row is a permutation of natural numbers and occurs only once. The table is closed with regards to the composition of its rows (see A089839) and it contains the inverse of each (their positions are shown in A089843). The permutations in table form an enumerable subgroup of the group of all size-preserving "Catalan bijections" (bijections among finite unlabeled rooted plane binary trees). The order of each element is shown at A089842.

References

  • A. Karttunen, paper in preparation, draft available by e-mail.

Links

Crossrefs

The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A069770, 2: A072796, 3: A089850, 4: A089851, 5: A089852, 6: A089853, 7: A089854, 8: A072797, 9: A089855, 10: A089856, 11: A089857, 12: A074679, 13: A089858, 14: A073269, 15: A089859, 16: A089860, 17: A074680, 18: A089861, 19: A073270, 20: A089862, 21: A089863.
Other rows: row 83: A154125, row 169: A129611, row 183: A154126, row 251: A129612, row 253: A123503, row 258: A123499, row 264: A123500, row 3608: A129607, row 3613: A129605, row 3617: A129606, row 3655: A154121, row 3656: A154123,row 3702: A082354, row 3747: A154122, row 3748: A154124, row 3886: A082353, row 4069: A082351, row 4207: A089865, row 4253: A082352, row 4299: A089866, row 65167: A129609, row 65352: A129610, row 65518: A123495, row 65796: A123496, row 79361: A123492, row 1653002: A123695, row 1653063: A123696, row 1654023: A073281, row 1654249: A123498, row 1654694: A089864, row 1654720: A129604,row 1655089: A123497, row 1783367: A123713, row 1786785: A123714.
Tables A122200, A122201, A122202, A122203, A122204, A122283, A122284, A122285, A122286, A122287, A122288, A122289, A122290, A130400-A130403 give various "recursive derivations" of these non-recursive automorphisms. See also A089831, A073200.
Index sequences to this table, giving various subgroups or other important constructions: A153826, A153827, A153829, A153830, A123694, A153834, A153832, A153833.

A122203 Signature permutations of SPINE-transformations of non-recursive Catalan automorphisms in table A089840.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 2, 2, 1, 0, 5, 7, 3, 2, 1, 0, 6, 8, 4, 3, 2, 1, 0, 7, 6, 6, 5, 3, 2, 1, 0, 8, 5, 5, 4, 5, 3, 2, 1, 0, 9, 4, 7, 6, 6, 6, 3, 2, 1, 0, 10, 17, 8, 7, 4, 5, 6, 3, 2, 1, 0, 11, 18, 9, 8, 7, 4, 4, 4, 3, 2, 1, 0, 12, 20, 11, 12, 8, 7, 5, 5, 4, 3, 2, 1, 0, 13, 21, 14, 13, 12
Offset: 0

Views

Author

Antti Karttunen, Sep 01 2006, Jun 06 2007

Keywords

Comments

Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th nonrecursive automorphism in the table A089840 with the recursion scheme "SPINE". In this recursion scheme the given automorphism is first applied at the root of binary tree, before the algorithm recurses down to the new right-hand side branch. The associated Scheme-procedures SPINE and !SPINE can be used to obtain such a transformed automorphism from any constructively or destructively implemented automorphism. Each row occurs only once in this table. Inverses of these permutations can be found in table A122204.
The recursion scheme SPINE has a well-defined inverse, that is, it acts as a bijective mapping on the set of all Catalan automorphisms. Specifically, if g = SPINE(f), then (f s) = (cond ((pair? s) (let ((t (g s))) (cons (car t) (g^{-1} (cdr t))))) (else s)) that is, to obtain an automorphism f which gives g when subjected to recursion scheme SPINE, we compose g with its own inverse applied to the cdr-branch of a S-expression. This implies that for any non-recursive automorphism f in the table A089840, SPINE^{-1}(f) is also in A089840, which in turn implies that the rows of table A089840 form a (proper) subset of the rows of this table.

References

  • A. Karttunen, paper in preparation, draft available by e-mail.

Links

Crossrefs

Cf. The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A069767, 2: A057509, 3: A130341, 4: A130343, 5: A130345, 6: A130347, 7: A122282, 8: A082339, 9: A130349, 10: A130351, 11: A130353, 12: A074685, 13: A130355, 14: A130357, 15: A130359, 16: A130361, 17: A057501, 18: A130363, 19: A130365, 20: A130367, 21: A069770. Other rows: row 251: A089863, row 253: A123717, row 3608: A129608, row 3613: A072796, row 65352: A074680, row 79361: A123715.
See also tables A089840, A122200, A122201-A122204, A122283-A122284, A122285-A122288, A122289-A122290.

Programs

  • Scheme
    (define (SPINE foo) (letrec ((bar (lambda (s) (let ((t (foo s))) (if (pair? t) (cons (car t) (bar (cdr t))) t))))) bar))
(define (!SPINE foo!) (letrec ((bar! (lambda (s) (cond ((pair? s) (foo! s) (bar! (cdr s)))) s))) bar!))

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

Original entry on oeis.org

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

Views

Author

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

Keywords

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

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.

A074679 Signature permutation of a Catalan automorphism: Rotate binary tree left if possible, otherwise swap its sides.

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 8, 4, 5, 14, 15, 16, 17, 18, 19, 20, 21, 9, 10, 22, 11, 12, 13, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 23, 24, 59, 25, 26, 27, 60, 61, 62, 28, 29, 63, 30, 31, 32, 64, 33, 34, 35, 36, 107, 108, 109, 110, 111
Offset: 0

Views

Author

Antti Karttunen, Sep 11 2002

Keywords

Comments

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node.)
...B...C.......A...B
....\./.........\./
.A...x....-->....x...C.................A..().........()..A..
..\./.............\./...................\./....-->....\./...
...x...............x.....................x.............x....
(a . (b . c)) -> ((a . b) . c) ____ (a . ()) --> (() . a)
That is, we rotate the binary tree left, in case it is possible and otherwise (if the right hand side of a tree is a terminal node) swap the left and right subtree (so that the terminal node ends to the left hand side), i.e., apply the automorphism *A069770. Look at the example in A069770 to see how this will produce the given sequence of integers.
This is the first multiclause nonrecursive automorphism in table A089840 and the first one whose order is not finite, i.e., the maximum size of cycles in this permutation is not bounded (see A089842). The cycle counts in range [A014137(n-1)..A014138(n)] of this permutation is given by A001683(n+1), which is otherwise the same sequence as for Catalan automorphisms *A057161/*A057162, but shifted once right. For an explanation, please see the notes in OEIS Wiki.

Links

Crossrefs

This automorphism has several variants, where the first clause is same (rotate binary tree to the left, if possible), but something else is done (than just swapping sides), in case the right hand side is empty: A082335, A082349, A123499, A123695. The following automorphisms can be derived recursively from this one: A057502, A074681, A074683, A074685, A074687, A074690, A089865, A120706, A122321, A122332. See also somewhat similar ones: A069773, A071660, A071656, A071658, A072091, A072095, A072093.
Inverse: A074680.
Row 12 of A089840.
Occurs also in A073200 as row 557243 because a(n) = A073283(A073280(A072796(n))). a(n) = A083927(A123498(A057123(n))).
Number of cycles: LEFT(A001683). Number of fixed points: LEFT(A019590). Max. cycle size & LCM of all cycle sizes: A089410 (in range [A014137(n-1)..A014138(n)] of this permutation).

Extensions

Description clarified Oct 10 2006

A074684 Permutation of natural numbers induced by Catalan Automorphism *A074684 acting on the parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Sep 11 2002

Keywords

Comments

This bijection maps between the "standard" ordering of binary trees as encoded by A014486 and "variant A quaternary encoding" as explained in the sequence A085184.
This is a rare example of a simply defined Catalan Automorphism where the cycle count sequence (A089411) is not monotone. (See A127296 for a much more complex example.)

Links

Crossrefs

Row 17 of A122201. Inverse of A074683. a(n) = A057163(A074681(A057163(n))).
The number of cycles, maximum cycle sizes and LCM's of all cycle sizes in subpermutations limited by A014137 and A014138 are given by A089411, A086586 and A089412.

A057161 Signature-permutation of a Catalan Automorphism: rotate one step counterclockwise the triangulations of polygons encoded by A014486.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Aug 18 2000; entry revised Jun 06 2014

Keywords

Comments

This is a permutation of natural numbers induced when Euler's triangulation of convex polygons, encoded by the sequence A014486 in a straightforward way (via binary trees, cf. the illustration of the rotation of a triangulated pentagon, given in the Links section) are rotated counterclockwise.
The number of cycles in range [A014137(n-1)..A014138(n)] of this permutation is given by A001683(n+2), otherwise the same sequence as for Catalan bijections *A074679/*A074680, but shifted once left (for an explanation, see the related notes in OEIS Wiki).
E.g., in range [A014137(0)..A014138(1)] = [1,1] there is one cycle (as a(1)=1), in range [A014137(1)..A014138(2)] = [2,3] there is one cycle (as a(2)=3 and a(3)=2), in range [A014137(2)..A014138(3)] = [4,8] there is also one cycle (as a(4) = 7, a(7) = 6, a(6) = 5, a(5) = 8 and a(8) = 4), and in range [A014137(3)..A014138(4)] = [9,22] there are A001683(4+2) = 4 cycles.
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 by the same method, when the other side of the formula is also "recursivized".

Links

Crossrefs

Inverse: A057162.
Also, a "SPINE"-transform of A069774, and thus occurs as row 12 of A130403.
Other related permutations: A057163, A057164, A057501, A057504, A057505.
Cf. A001683 (cycle counts), A057544 (max cycle lengths).

Programs

  • Maple
    a(n) = CatalanRankGlobal(RotateTriangularization(A014486[n]))
CatalanRankGlobal given in A057117 and the other Maple procedures in A038776.
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+binwidth(BinTreeLeftBranch(n))))));
RotateTriangularization := proc(nn) local n,s,z,w; n := binrev(nn); z := 0; w := 0; while(1 = (n mod 2)) do s := BinTreeRightBranch(n); z := z + (2^w)*s; w := w + binwidth(s); z := z + (2^w); w := w + 1; n := floor(n/2); od; RETURN(z); end;

Formula

a(0) = 0, and for n>=1, a(n) = A085201(a(A072771(n)), A057548(A072772(n))). [This formula reflects the S-expression implementation given first in the Program section: 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 languages), and A057548 corresponds to unary form of function 'list'.]
As a composition of related permutations:
a(n) = A069767(A069769(n)).
a(n) = A057163(A057162(A057163(n))).
a(n) = A057164(A057504(A057164(n))). [For a proof, see pp. 53-54 in the "Introductory survey ..." draft]

A089855 Permutation of natural numbers induced by Catalan Automorphism *A089855 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 6, 9, 10, 11, 12, 13, 17, 18, 20, 21, 22, 16, 19, 14, 15, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 45, 46, 48, 49, 50, 54, 55, 57, 58, 59, 61, 62, 63, 64, 44, 47, 53, 56, 60, 42, 51, 37, 38, 43, 52, 39, 40, 41, 65, 66, 67, 68, 69, 70, 71, 72
Offset: 0

Views

Author

Antti Karttunen, Nov 29 2003

Keywords

Comments

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node).
.A...B...........B...C
..\./.............\./
...x...C....-->....x...A...............()..A.........()..A..
....\./.............\./.................\./....-->....\./...
.....x...............x...................x.............x....
((a . b) . c) -> ((b . c) . a) ___ (() . a) ---> (() . a)
In terms of S-expressions, this automorphism rotates caar, cdar and cdr of an S-exp if possible, i.e., if car-side is not ().
See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

Links

Crossrefs

Inverse of A089857. a(n) = A089860(A069770(n)) = A069770(A074680(n)) = A057163(A089853(A057163(n))). Row 9 of A089840.
Number of cycles: A089847. Number of fixed-points: A089848 (in each range limited by A014137 and A014138).

Extensions

Further comments and constructive implementation of Scheme-function (*A089855) added by Antti Karttunen, Jun 04 2011

A089860 Permutation of natural numbers induced by Catalan automorphism *A089860 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 8, 6, 7, 4, 5, 21, 22, 19, 14, 15, 20, 16, 17, 9, 10, 18, 11, 12, 13, 58, 59, 62, 63, 64, 56, 60, 51, 37, 38, 52, 39, 40, 41, 57, 61, 53, 42, 43, 54, 44, 45, 23, 24, 46, 25, 26, 27, 55, 47, 48, 28, 29, 49, 30, 31, 32, 50, 33, 34, 35, 36, 170, 171, 174, 175, 176, 184
Offset: 0

Views

Author

Antti Karttunen, Nov 29 2003

Keywords

Comments

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node).
.....B...C.......C...A
......\./.........\./
...A...x...-->... .x...B...............A..().........()..A..
....\./.............\./.................\./....-->....\./...
.....x...............x...................x.............x....
(a . (b . c)) --> ((c . a) . b) ___ (a . ()) --> (() . a)
See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

Links

Crossrefs

Row 16 of A089840. Inverse of A089862. a(n) = A089855(A069770(n)) = A069770(A089851(n)) = A069770(A074680(A069770(n))) = A057163(A089862(A057163(n))).
Number of cycles: A001683 (probably, not checked). Number of fixed points: A019590. Max. cycle size & LCM of all cycle sizes: A089410 (in each range limited by A014137 and A014138).

Extensions

A graphical description and constructive version of Scheme-implementation added by Antti Karttunen, Jun 04 2011

A120705 Permutation of natural numbers induced by the Catalan bijection gma120705 acting on the binary trees encoded by A014486/A063171.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 28 2006

Keywords

Comments

When the automorphisms A120705/A120705 act on the full Stern-Brocot tree (A007305/A047679), which is an infinite binary tree, they will move each fraction r to the position of 2*r (or r/2 respectively). See comments at A065249 and A065251. (Proof in preparation, to be published.)

Links

Crossrefs

Inverse of A120706. Cf. A074680.
Number of cycles: A120707. Number of fixed-points: A019590. Max. cycle size: A120708. LCM of cycle sizes: A120709.
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