A057123 -id:A057123 - cp's OEIS frontend

A083927 Inverse function of N -> N injection A057123.

0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 5, 0

Offset: 0

Antti Karttunen, May 13 2003

nonn

a(0)=0 because A057123(0)=0, but a(n) = 0 also for those n which do not occur as values of A057123. All positive natural numbers occur here once.

If g(n) = A083927(f(A057123(n))) then we can say that Catalan bijection g embeds into Catalan bijection f in scale n:2n, using the obvious binary tree -> general tree embedding. E.g. we have: A057163 = A083927(A057164(A057123(n))), A057117 = A083927(A072088(A057123(n))), A057118 = A083927(A072089(A057123(n))), A069770 = A083927(A072796(A057123(n))), A072797 = A083927(A072797(A057123(n))).

Antti Karttunen, Gatomorphisms

a(A057123(n)) = n for all n. Cf. A083925-A083926, A083928-A083929, A083935.

A057122 The binary encoding (as a rooted planar tree) of each rooted planar binary tree. See A057123 for illustration.

Original entry on oeis.org

0, 10, 180, 210, 2920, 2980, 3380, 3490, 3730, 46800, 46920, 47720, 47940, 48420, 54120, 54180, 55860, 56130, 56610, 59700, 59810, 60690, 62610, 748960, 749200, 750800, 751240, 752200, 763600, 763720, 767080, 767620, 768580, 774760, 774980

Offset: 0

Antti Karttunen, Aug 11 2000

nonn

bintree_depth_first2tree given in A057119.

Cf. A057119, A057123, A057124. Subset of A057517.

Maple
```
map(bintree_depth_first2tree, A014486);
```

A057163 Signature-permutation of a Catalan automorphism: Reflect a rooted plane binary tree; Deutsch's 1998 involution on Dyck paths.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Aug 18 2000

nonn

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.

			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.

JungHwan Min, Table of n, a(n) for n = 0..10000
Emeric Deutsch, An involution on Dyck paths and its consequences, Discrete Math., 204 (1999), no. 1-3, 163-166.
Indranil Ghosh, Python program for computing this sequence, translated from Maple code.
Antti Karttunen, C program which computes this sequence.
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020.
Index entries for signature-permutations induced by Catalan automorphisms

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.

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))))));

A014486Q[0] = True; A014486Q[n_] := Catch[Fold[If[# < 0, Throw[False], If[#2 == 0, # - 1, # + 1]] &, 0, IntegerDigits[n, 2]] == 0]; tree[n_] := Block[{func, num = Append[IntegerDigits[n, 2], 0]}, func := If[num[[1]] == 0, num = Drop[num, 1]; 0, num = Drop[num, 1]; 1[func, func]]; func]; A057163L[n_] := Function[x, FirstPosition[x, FromDigits[Most@Cases[tree[#] /. 1 -> Reverse@*1, 0 | 1, All, Heads -> True], 2]][[1]] - 1 & /@ x][Select[Range[0, 2^n], A014486Q]]; A057163L[11] (* JungHwan Min, Dec 11 2016 *)

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

Equivalence with Deutsch's 1998 involution realized Dec 15 2006 and entry edited accordingly by Antti Karttunen, Jan 16 2007

A069770 Signature permutation of the first non-identity, nonrecursive Catalan automorphism in table A089840: swap the top branches of a binary tree. An involution of nonnegative integers.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Apr 16 2002

nonn

This is the simplest possible Catalan automorphism after the identity bijection (A001477). It effects the following transformation on the unlabeled rooted plane binary trees (letters A and B refer to arbitrary subtrees located on those vectices):
   A   B           B   A
    \ /     -->     \ /
     x               x
  (a . b)  -----> (b . a)
Applying this permutation recursively to the right hand side branch of the binary trees produces permutations A069767 and A069768 (that occur at the same index 1 in tables A122203 and A122204), and applying this recursively to the both branches of binary trees (as in pre- or postorder traversal) produces A057163 (which occurs at the same index 1 in tables A122201 and A122202) that reflects the whole binary tree.
For this permutation, A127302(a(n)) = A127302(n) for all n, [or equally, A153835(a(n)) = A153835(n)], and likewise for all such recursive derivations as mentioned above.

			To obtain the signature permutation, we apply these transformations to the binary trees as encoded and ordered by A014486 and for each n, a(n) will be the position of the tree to which the n-th tree is transformed to, as follows:
.
                   one tree of one internal
  empty tree         (non-leaf) node
      x                      \/
n=    0                      1
a(n)= 0                      1               (both are always fixed)
.
the next 7 trees, with 2-3 internal nodes, in range [A014137(1), A014137(2+1)-1] = [2,8] are:
.
                          \/     \/                 \/     \/
       \/     \/         \/       \/     \/ \/     \/       \/
      \/       \/       \/       \/       \_/       \/       \/
n=     2        3        4        5        6        7        8
.
and the new shapes after swapping their left and right hand subtrees are:
.
                        \/     \/                     \/     \/
     \/         \/     \/       \/       \/ \/       \/       \/
      \/       \/       \/       \/       \_/       \/       \/
a(n)=  3        2        7        8        6        4        5
thus we obtain the first nine terms of this sequence: 0, 1, 3, 2, 7, 8, 6, 4, 5.

Row 1 of A089840.
The number of cycles and the number of fixed points in each subrange limited by terms of A014137 are given by A007595 and A097331.
Other related sequences: A014486, A057163, A069767, A069768, A089864, A123492, A154125, A154126.
Cf. also A127302, A153835.

a(n) = A154125(A154126(n)) = A154126(A154125(n)).

a(n) = A083927(A072796(A057123(n))) = A083927(A057508(A057123(n))) = A083927(A057509(A057123(n))).

Entry revised by Antti Karttunen, Oct 11 2006 and Mar 30 2024

A057164 Self-inverse permutation of natural numbers induced by reflections of the rooted plane trees and mountain ranges encoded by A014486.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 14, 11, 16, 19, 10, 15, 12, 17, 20, 13, 18, 21, 22, 23, 37, 28, 42, 51, 25, 39, 30, 44, 53, 33, 47, 56, 60, 24, 38, 29, 43, 52, 26, 40, 31, 45, 54, 34, 48, 57, 61, 27, 41, 32, 46, 55, 35, 49, 58, 62, 36, 50, 59, 63, 64, 65, 107, 79, 121, 149, 70

Offset: 0

Antti Karttunen, Aug 18 2000

nonn

CatalanRankGlobal given in A057117 and the other Maple procedures in A056539.

Composition with A057163 gives Donaghey's Map M (A057505/A057506).

			a(10)=14 and a(14)=10, A014486[10] = 172 (10101100 in binary), A014486[14] = 202 (11001010 in binary) and these encode the following mountain ranges (and the corresponding rooted plane trees), which are reflections of each other:
...../\___________/\
/\/\/__\_________/__\/\/\
...
...../...........\
..\|/.............\|/

Rémy Sigrist, Table of n, a(n) for n = 0..6917 (first 197 terms from Indranil Ghosh)
Antti Karttunen, Gatomorphisms (includes the complete Scheme program for computing this sequence).
Antti Karttunen, C program which implements this Catalan bijection.
Indranil Ghosh, Python program for computing the sequence.
Rémy Sigrist, PARI program.
Index entries for signature-permutations induced by Catalan automorphisms

A057123(A057163(n)) = A057164(A057123(n)) for all n. Also the car/cdr-flipped conjugate of A069787, i.e., A057164(n) = A057163(A069787(A057163(n))). Fixed terms are given by A061856. Cf. also A057508, A069772.

Row 2 of tables A122287 and A122288.

a(n) = CatalanRankGlobal(runcounts2binexp(reverse(binexp2runcounts(A014486[n])))) # i.e., reverse and complement the totally balanced binary sequences

PARI
```
See Links section.
```

a(n) = A080300(A036044(A014486(n))) = A080300(A056539(A014486(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

Antti Karttunen, Sep 11 2002

nonn

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.

A. Karttunen, Table of n, a(n) for n = 0..2055
A. Karttunen, Introductory Survey of Catalan Automorphisms and Bijections, (an unfinished draft)
A. Karttunen, Notes on the orbits of this permutation, OEIS Wiki.
A. Karttunen, Prolog-program which illustrates the construction of this and similar nonrecursive bijections of oriented binary trees
Index entries for signature-permutations of Catalan automorphisms

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).

Description clarified Oct 10 2006

A074680 Signature permutation of the seventeenth nonrecursive Catalan automorphism in table A089840. (Rotate binary tree right if possible, otherwise swap its sides.)

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Sep 11 2002

nonn

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..-->.....A...x................()..B.......B..()
...\./............\./..................\./...-->...\./.
....x..............x....................x...........x..
((a . b) . c) -> (a . (b . c)) __ (() . b) --> (b . ())
That is, we rotate the binary tree right, in case it is possible and otherwise (if the left hand side of a tree is a terminal node) swap the right and left subtree (so that the terminal node ends to the right 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.
See also the comments at A074679.

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

This automorphism has several variants, where the first clause is same (rotate binary tree to the right, if possible), but something else is done (than just swapping sides), in case the left hand side is empty: A082336, A082350, A123500, A123696. The following automorphisms can be derived recursively from this one: A057501, A074682, A074684, A074686, A074688, A074689, A089866, A120705, A122322, A122331. See also somewhat similar ones: A069774, A071659, A071655, A071657, A072090, A072094, A072092.
Inverse: A074679. Row 17 of A089840. Occurs also in A073200 as row 2156396687 as a(n) = A072796(A073280(A073282(n))). a(n) = A083927(A123497(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-1)] of this permutation).

Description clarified Oct 10 2006

A127302 Matula-Goebel signatures for plane binary trees encoded by A014486.

Original entry on oeis.org

1, 4, 14, 14, 86, 86, 49, 86, 86, 886, 886, 454, 886, 886, 301, 301, 301, 886, 886, 301, 454, 886, 886, 13766, 13766, 6418, 13766, 13766, 3986, 3986, 3986, 13766, 13766, 3986, 6418, 13766, 13766, 3101, 3101, 1589, 3101, 3101, 1849, 1849, 3101, 13766

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

This sequence maps A000108(n) oriented (plane) rooted binary trees encoded in range [A014137(n-1)..A014138(n-1)] of A014486 to A001190(n+1) non-oriented rooted binary trees, encoded by their Matula-Goebel numbers (when viewed as a subset of non-oriented rooted general trees). See also the comments at A127301.
If the signature-permutation of a Catalan automorphism SP satisfies the condition A127302(SP(n)) = A127302(n) for all n, then it preserves the non-oriented form of a binary tree. Examples of such automorphisms include A069770, A057163, A122351, A069767/A069768, A073286-A073289, A089854, A089859/A089863, A089864, A122282, A123492-A123494, A123715/A123716, A127377-A127380, A127387 and A127388.
A153835 divides natural numbers to same equivalence classes, i.e. a(i) = a(j) <=> A153835(i) = A153835(j) - Antti Karttunen, Jan 03 2013

			A001190(n+1) distinct values occur each range [A014137(n-1)..A014138(n-1)]. As an example, terms A014486(4..8) encode the following five plane binary trees:
........\/.....\/.................\/.....\/...
.......\/.......\/.....\/.\/.....\/.......\/..
......\/.......\/.......\_/.......\/.......\/.
n=.....4........5........6........7........8..
The trees in positions 4, 5, 7 and 8 all produce Matula-Goebel number A000040(1)*A000040(A000040(1)*A000040(A000040(1)*A000040(1))) = 2*A000040(2*A000040(2*2)) = 2*A000040(14) = 2*43 = 86, as they are just different planar representations of the one and same non-oriented tree. The tree in position 6 produces Matula-Goebel number A000040(A000040(1)*A000040(1)) * A000040(A000040(1)*A000040(1)) = A000040(2*2) * A000040(2*2) = 7*7 = 49. Thus a(4..8) = 86,86,49,86,86.

Index entries for sequences related to Matula-Goebel numbers

Cf A127301, A153835, A153829.

a(n) = A127301(A057123(n)).

Can be also computed directly as a fold, see the Scheme-program. - Antti Karttunen, Jan 03 2013

A123497 Signature permutation of a nonrecursive Catalan automorphism: row 1655089 of table A089840.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 11 2006

nonn

Inverse: A123498. Row 1655089 of A089840. Used to construct automorphism *A123501. A074680(n) = A083927(a(A057123(n))).

A123498 Signature permutation of a nonrecursive Catalan automorphism: row 1654249 of table A089840.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 11 2006

nonn

Inverse: A123497. Row 1654249 of A089840. Used to construct automorphism *A123502. A074679(n) = A083927(a(A057123(n))).

cp's OEIS Frontend

A083927 Inverse function of N -> N injection A057123.

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A057122 The binary encoding (as a rooted planar tree) of each rooted planar binary tree. See A057123 for illustration.

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A057163 Signature-permutation of a Catalan automorphism: Reflect a rooted plane binary tree; Deutsch's 1998 involution on Dyck paths.

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A069770 Signature permutation of the first non-identity, nonrecursive Catalan automorphism in table A089840: swap the top branches of a binary tree. An involution of nonnegative integers.

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A057164 Self-inverse permutation of natural numbers induced by reflections of the rooted plane trees and mountain ranges encoded by A014486.

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Maple

PARI

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

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A074680 Signature permutation of the seventeenth nonrecursive Catalan automorphism in table A089840. (Rotate binary tree right if possible, otherwise swap its sides.)

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A127302 Matula-Goebel signatures for plane binary trees encoded by A014486.

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A123497 Signature permutation of a nonrecursive Catalan automorphism: row 1655089 of table A089840.

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