A069776 -id:A069776 - cp's OEIS frontend

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

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

A057510 Permutation of natural numbers: rotations of the bottom branches of the rooted plane trees encoded by A014486. (to opposite direction of A057509).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 14, 10, 16, 19, 11, 15, 12, 17, 18, 13, 20, 21, 22, 23, 37, 24, 42, 51, 25, 38, 26, 44, 47, 27, 53, 56, 60, 28, 39, 29, 43, 52, 30, 40, 31, 45, 46, 32, 48, 49, 50, 33, 41, 34, 54, 55, 35, 57, 58, 59, 36, 61, 62, 63, 64, 65, 107, 66, 121, 149, 67

Offset: 0

Antti Karttunen, Sep 03 2000

nonn

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

Inverse of A057509 and the car/cdr-flipped conjugate of A069776 and also composition of A057502 & A069770, i.e. A057510(n) = A057163(A069776(A057163(n))) = A069770(A057502(n)).

Cycle counts given by A003239. Cf. also A057512, A057513.

# reverse given in A057508, for CountCycles, see A057502, for other procedures, follow A057501.
map(CatalanRankGlobal,map(RotateBottomBranchesR, A014486));
RotateBottomBranchesR := n -> pars2binexp(rotateR(binexp2pars(n)));
rotateR := a -> reverse(rotateL(reverse(a)));
RotBBPermutationCycleCounts := 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,RotateBottomBranchesL(CatalanUnrank(n,r)))]; od; a := [op(a),CountCycles(b)]; od; RETURN(a); end;
A003239 := RotBBPermutationCycleCounts(some_value); (e.g. 9. Cf. A057502, A057162)

A069774 Permutation of natural numbers induced by the automorphism RoblDownCar_et_SwapInv! acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

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

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 A069773, the car/cdr-flipped conjugate of A057502, i.e. A069774(n) = A057163(A057502(A057163(n))). Cf. also A069776.

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).

A130936 Signature permutation of a Catalan automorphism: row 2 of A130403.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 6, 8, 9, 10, 12, 11, 13, 17, 18, 16, 14, 15, 20, 21, 19, 22, 23, 24, 26, 25, 27, 31, 32, 30, 28, 29, 34, 35, 33, 36, 45, 46, 48, 49, 50, 44, 47, 42, 37, 38, 43, 40, 39, 41, 54, 55, 57, 58, 59, 53, 56, 51, 52, 61, 62, 63, 60, 64, 65, 66, 68, 67, 69, 73, 74

Offset: 0

Antti Karttunen, Jun 11 2007

nonn

The signature-permutation of the Catalan automorphism which is derived from *A069776 with recursion schema SPINE (see A122203 for the definition).

Inverse: A130935.

A122314 Row 8 of A122284.

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, 35, 34, 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, Sep 01 2006

nonn

The signature-permutation of the automorphism which is derived from the eighth non-recursive automorphism *A072797 with recursion schema NEPEED (see A122284 for the definition).

Index entries for signature-permutations of Catalan automorphisms

Inverse: A122313. A082326(n) = A083927(A122314(A057123(n))). Differs from A069776 for the first time at n=34, where a(n)=35, while A069776(n)=34.

cp's OEIS Frontend

A082338 Duplicate of A069776.

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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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A057510 Permutation of natural numbers: rotations of the bottom branches of the rooted plane trees encoded by A014486. (to opposite direction of A057509).

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A069774 Permutation of natural numbers induced by the automorphism RoblDownCar_et_SwapInv! acting on the parenthesizations encoded by A014486.

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

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A130936 Signature permutation of a Catalan automorphism: row 2 of A130403.

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A122314 Row 8 of A122284.

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