A069787 -id:A069787 - cp's OEIS frontend

A243490 Fixed points of A069787: Numbers n such that A069787(n) = n.

0, 1, 2, 3, 4, 5, 8, 9, 10, 13, 16, 20, 22, 23, 24, 27, 30, 34, 36, 54, 55, 56, 64, 65, 66, 69, 72, 76, 78, 96, 97, 98, 106, 126, 136, 157, 158, 162, 165, 183, 186, 193, 196, 197, 198, 201, 204, 208, 210, 228, 229, 230, 238, 258, 268, 289, 290, 294, 297, 315

Offset: 0

Antti Karttunen, Jun 07 2014

nonn

Although in principle a list, the indexing of this sequence starts from zero, as 0 is always fixed by all Catalan bijections (permutations induced by bijective operations performed on A014486), so it is a trivial case, which can be skipped by considering only values from a(n>=1) onward.

Sequence gives also the positions of all zeros in A243492.

Antti Karttunen, Table of n, a(n) for n = 0..7059

Complement: A243489.
Fixed points of A069787, positions of zeros in A243492.
Cf. also A001405, A036256, A061856, A126312, A079442, A243493.

A243491 Matula-Goebel signature computed for trees rearranged by Catalan automorphism *A069787: a(n) = A127301(A069787(n)).

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 7, 6, 5, 16, 12, 14, 12, 10, 19, 13, 14, 12, 9, 17, 13, 10, 11, 32, 24, 28, 24, 20, 38, 26, 28, 24, 18, 34, 26, 20, 22, 53, 37, 43, 37, 29, 38, 26, 28, 24, 18, 21, 21, 18, 15, 67, 41, 43, 37, 23, 34, 26, 20, 15, 59, 41, 29, 22, 31, 64, 48, 56, 48, 40

Offset: 0

Antti Karttunen, Jun 07 2014

nonn

See the comments at A243492.

Antti Karttunen, Table of n, a(n) for n = 0..6917
A. Karttunen, On the fixed points of A071661 (Notes in OEIS Wiki)

A243492 gives the differences from A127301. Cf. also A243490, A243493.

(define (A243491 n) (A127301 (A069787 n)))

a(n) = A127301(A069787(n)).

A243493 Value of Matula-Goebel signature at the fixed points of A069787: a(n) = A127301(A243490(n)).

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 5, 16, 12, 10, 14, 13, 11, 32, 24, 20, 28, 26, 22, 37, 23, 34, 31, 64, 48, 40, 56, 52, 44, 74, 46, 68, 62, 76, 39, 89, 61, 47, 86, 101, 118, 109, 127, 128, 96, 80, 112, 104, 88, 148, 92, 136, 124, 152, 78, 178, 122, 94, 172, 202, 236, 218, 254

Offset: 0

Antti Karttunen, Jun 07 2014

The first duplicate value occurs at n=101, as a(101) = a(129) = 362. The corresponding A014486-indices are A243490(101) = 924 and A243490(129) = 1640, respectively.

Antti Karttunen, Table of n, a(n) for n = 0..525

A243494 gives the same terms sorted into ascending order with duplicates removed.

Cf. A243496, A209636, A209637, A069787, A127301, A243490, A243491, A000079, A007097, A036256.

a(n) = A127301(A243490(n)).
a(n) = A243491(A243490(n)).
For all n >= 0, a(A036256(n)-1) = A007097(n) and a(A036256(n)) = A000079(n+1).

A072798 Self-inverse permutation of nonnegative integers obtained by contracting the permutation A069787 induced by the telescoping Catalan bijection DeepRev1CarSide.

Original entry on oeis.org

0, 1, 3, 2, 4, 8, 9, 7, 5, 6, 12, 11, 10, 13, 22, 23, 26, 25, 27, 21, 24, 19, 14, 15, 20, 17, 16, 18, 35, 36, 34, 31, 32, 33, 30, 28, 29, 40, 39, 38, 37, 41, 64, 65, 68, 67, 69, 77, 78, 76, 73, 74, 82, 81, 80, 83, 63, 66, 75, 72, 79, 61, 70, 56, 42, 43, 57, 45, 44, 46, 62, 71

Offset: 0

Antti Karttunen, Jun 12 2002

nonn

A. Karttunen, Gatomorphisms (with the complete Scheme source)
Index entries for sequences that are permutations of the natural numbers

a(n) = A072799(n+1)-1.

A243489 Complement of A243490: Numbers n such that A069787(n) <> n.

Original entry on oeis.org

6, 7, 11, 12, 14, 15, 17, 18, 19, 21, 25, 26, 28, 29, 31, 32, 33, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 57, 58, 59, 60, 61, 62, 63, 67, 68, 70, 71, 73, 74, 75, 77, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 99, 100

Offset: 1

Antti Karttunen, Jun 07 2014

nonn

Numbers that are not fixed by permutation A069787.

Antti Karttunen, Table of n, a(n) for n = 1..6644

Complement: A243490. Cf. A069787.

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

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))).

A079438 a(0) = a(1) = 1, a(n) = 2*(floor((n+1)/3) + (if n >= 14) (floor((n-10)/4) + floor((n-14)/8))).

Original entry on oeis.org

1, 1, 2, 2, 2, 4, 4, 4, 6, 6, 6, 8, 8, 8, 12, 12, 12, 14, 16, 16, 18, 18, 22, 24, 24, 24, 28, 28, 28, 30, 34, 34, 36, 36, 38, 40, 40, 40, 46, 46, 46, 48, 50, 50, 52, 52, 56, 58, 58, 58, 62, 62, 62, 64, 68, 68, 70, 70, 72, 74, 74, 74, 80, 80, 80, 82, 84, 84, 86, 86, 90, 92, 92, 92

Offset: 0

Antti Karttunen, Jan 27 2003

nonn

The original definition was: Number of rooted general plane trees which are symmetric and will stay symmetric after the underlying plane binary tree has been reflected, i.e., number of integers i in range [A014137(n-1)..A014138(n-1)] such that A057164(i) = i and A057164(A057163(i)) = A057163(i).
(Thus also) the number of fixed points in range [A014137(n-1)..A014138(n)] of permutation A071661 (= Donaghey's automorphism M "squared"), which is equal to condition A057164(i) = A069787(i) = i, i.e., the size of the intersection of fixed points of permutations A057164 and A069787 in the same range.
Additional comment from Antti Karttunen, Dec 13 2017: (Start)
However, David Callan's A123050 claims to give more correct version of that count from n=26 onward, so I probably made a little mistake when converting my insights into the formula given here. At that time I reckoned that if the conjecture given in A080070 were true, then it would imply that the formula given here were exact, otherwise it would give only a lower bound.
It would be nice to know what an empirical program would give as the count of fixed points of A071661 for n in range [A014137(25)..A014138(26)] = [6619846420553 .. 24987199492704], with total A000108(26) = 18367353072151 points to check.
(End)

D. E. Knuth, The Art of Computer Programming, Volume 4, Fascicle 4: Generating All Trees--History of Combinatorial Generation, vi+120pp. ISBN 0-321-33570-8 Addison-Wesley Professional; 1ST edition (Feb 06, 2006).

G. C. Greubel, Table of n, a(n) for n = 0..10000
R. Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 75-90.
A. Karttunen, C-program for counting the initial terms of this sequence (empirically)
A. Karttunen, Illustration of initial terms for trees of sizes n=2..18
A. Karttunen, On the fixed points of A071661 (Notes in OEIS Wiki)
D. E. Knuth, Pre-Fascicle 4a: Generating All Trees, Exercise 17, 7.2.1.6.

Cf. A000108, A057163, A057164, A057505, A069787, A071661, A079437, A079439, A079442, A080070, A243490,  A243491, A243492.
From n>= 2 onward A079440(n) = a(n)/2.
Occurs in A073202 as row 13373289.
Differs from A123050 for the first time at n=26.

A079438 := n -> `if`((n<2),1,2*(floor((n+1)/3) + `if`((n>=14),floor((n-10)/4)+floor((n-14)/8),0)));

a[0]:= 1; a[1]:= 1; a[n_]:= a[n] = 2*Floor[(n+1)/3] +2*If[ n >= 14, (Floor[(n-10)/4] +Floor[(n-14)/8]), 0]; Table[a[n], {n, 0, 100}] (* G. C. Greubel, Jan 18 2019 *)

{a(n) = if(n==0, 1, if(n==1, 1, 2*floor((n+1)/3) + 2*if(n >= 14, floor( (n-10)/4) + floor((n-14)/8), 0)))}; \\ G. C. Greubel, Jan 18 2019

a(0) = a(1) = 1, a(n) = 2*(floor((n+1)/3) + (if n >= 14) (floor((n-10)/4) + floor((n-14)/8))).

Entry edited (the definition replaced by a formula, the old definition moved to the comments) - Antti Karttunen, Dec 13 2017

A069775 Permutation of natural numbers induced by the automorphism gma069775! acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Apr 16 2002

nonn

A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
Index entries for sequences that are permutations of the natural numbers

Inverse of A069776. a(n) = A057163(A057509(A057163(n))) = A069773(A069770(n)). Cf. also A069787, A072797.

Number of cycles: A003239. Number of fixed points: A034731. Max. cycle size: A028310. LCM of cycle sizes: A003418. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A069776 Permutation of natural numbers induced by the automorphism gma069776! acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Apr 16 2002

nonn

A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
Index entries for sequences that are permutations of the natural numbers

Inverse of A069775. a(n) = A057163(A057510(A057163(n))) = A069770(A069774(n)). Cf. also A069787, A072797.

Number of cycles: A003239. Number of fixed points: A034731. Max. cycle size: A028310. LCM of cycle sizes: A003418. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

cp's OEIS Frontend

A243490 Fixed points of A069787: Numbers n such that A069787(n) = n.

Views

Author

Keywords

Comments

Links

Crossrefs

A243491 Matula-Goebel signature computed for trees rearranged by Catalan automorphism *A069787: a(n) = A127301(A069787(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A243493 Value of Matula-Goebel signature at the fixed points of A069787: a(n) = A127301(A243490(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A072798 Self-inverse permutation of nonnegative integers obtained by contracting the permutation A069787 induced by the telescoping Catalan bijection DeepRev1CarSide.

Views

Author

Keywords

Links

Crossrefs

A243489 Complement of A243490: Numbers n such that A069787(n) <> n.

Views

Author

Keywords

Comments

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

A079438 a(0) = a(1) = 1, a(n) = 2*(floor((n+1)/3) + (if n >= 14) (floor((n-10)/4) + floor((n-14)/8))).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A069775 Permutation of natural numbers induced by the automorphism gma069775! acting on the parenthesizations encoded by A014486.

Views