A057163 -id:A057163 - cp's OEIS frontend

A122288 Signature permutations of KROF-transformations of Catalan automorphisms in table A122203.

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 2, 2, 1, 0, 5, 8, 3, 2, 1, 0, 6, 7, 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, 22, 8, 7, 4, 5, 6, 3, 2, 1, 0, 11, 21, 9, 8, 7, 4, 4, 4, 3, 2, 1, 0, 12, 20, 14, 13, 8, 7, 5, 5, 4, 3, 2, 1, 0, 13, 17, 11, 12, 13

Offset: 0

Antti Karttunen, Sep 01 2006, Jun 20 2007

Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th automorphism in the table A122203 with the recursion scheme "KROF", or equivalently row n is obtained as KROF(SPINE(n-th row of A089840)). See A122202 and A122203 for the description of KROF and SPINE. Moreover, each row of A122288 can be obtained as the "NEPEED" transform of the corresponding row in A122285. (See A122284 for the description of NEPEED). Each row occurs only once in this table. Inverses of these permutations can be found in table A122287. This table contains also all the rows of A122202 and A089840.

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

Index entries for signature-permutations of Catalan automorphisms

The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A069768, 2: A057164, 3: A130981, 4: A130983, 5: A130982, 6: A130984, 7: A130985, 8: A130987, 9: A130989, 10: A130991, 11: A130993, 12: A131009, 13: A130995, 14: A130997, 15: A130999, 16: A131001, 17: A057505, 18: A131003, 19: A131005, 20: A131007, 21: A057163. Other rows: 251: A122354, 3613: A057512, 65352: A074682.

A153141 Permutation of nonnegative integers: A059893-conjugate of A153151.

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 4, 5, 15, 14, 12, 13, 8, 9, 10, 11, 31, 30, 28, 29, 24, 25, 26, 27, 16, 17, 18, 19, 20, 21, 22, 23, 63, 62, 60, 61, 56, 57, 58, 59, 48, 49, 50, 51, 52, 53, 54, 55, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 127, 126, 124, 125, 120, 121

Offset: 0

Antti Karttunen, Dec 20 2008

This permutation is induced by a wreath recursion a = s(a,b), b = (b,b) (i.e., binary transducer, where s means that the bits at that state are toggled: 0 <-> 1) given on page 103 of the Bondarenko, Grigorchuk, et al. paper, starting from the active (swapping) state a and rewriting bits from the second most significant bit to the least significant end, continuing complementing as long as the first 1-bit is reached, which is the last bit to be complemented.
The automorphism group of infinite binary tree (isomorphic to an infinitely iterated wreath product of cyclic groups of two elements) embeds naturally into the group of "size-preserving Catalan bijections". Scheme-function psi gives an isomorphism that maps this kind of permutation to the corresponding Catalan automorphism/bijection (that acts on S-expressions). The following identities hold: *A069770 = psi(A063946) (just swap the left and right subtrees of the root), *A057163 = psi(A054429) (reflect the whole tree), *A069767 = psi(A153141), *A069768 = psi(A153142), *A122353 = psi(A006068), *A122354 = psi(A003188), *A122301 = psi(A154435), *A122302 = psi(A154436) and from *A154449 = psi(A154439) up to *A154458 = psi(A154448). See also comments at A153246 and A153830.
a(1) to a(2^n) is the sequence of row sequency numbers in a Hadamard-Walsh matrix of order 2^n, when constructed to give "dyadic" or Payley sequency ordering. - Ross Drewe, Mar 15 2014
In the Stern-Brocot enumeration system for positive rationals (A007305/A047679), this permutation converts the denominator into the numerator: A007305(n) = A047679(a(n)). - Yosu Yurramendi, Aug 01 2020

			18 = 10010 in binary and after complementing the second, third and fourth most significant bits at positions 3, 2 and 1, we get 1110, at which point we stop (because bit-1 was originally 1) and fix the rest, so we get 11100 (28 in binary), thus a(18)=28. This is the inverse of "binary adding machine". See pages 8, 9 and 103 in the Bondarenko, Grigorchuk, et al. paper.
19 = 10011 in binary. By complementing bits in (zero-based) positions 3, 2 and 1 we get 11101 in binary, which is 29 in decimal, thus a(19)=29.

A. Karttunen, Table of n, a(n) for n = 0..2047
Ievgen Bondarenko, Rostislav Grigorchuk, Rostyslav Kravchenko, Yevgen Muntyan, Volodymyr Nekrashevych, Dmytro Savchuk, and Zoran Sunic, Classification of groups generated by 3-state automata over a 2-letter alphabet, arXiv:0803.3555 [math.GR], 2008, pp. 8--9 & 103.
S. Wolfram and R. Lamy, Discussion on the NKS Forum
Index entries for sequences that are permutations of the natural numbers

Inverse: A153142. a(n) = A059893(A153151(A059893(n))) = A059894(A153152(A059894(n))) = A154440(A154445(n)) = A154442(A154443(n)). Corresponds to A069767 in the group of Catalan bijections. Cf. also A154435-A154436, A154439-A154448, A072376.
Differs from A006068 for the first time at n=14, where a(14)=10 while A006068(14)=11.
A240908-A240910 these give "natural" instead of "dyadic" sequency ordering values for Hadamard-Walsh matrices, orders 8,16,32. - Ross Drewe, Mar 15 2014

def ok(n): return n&(n - 1)==0
def a153151(n): return n if n<2 else 2*n - 1 if ok(n) else n - 1
def A(n): return (int(bin(n)[2:][::-1], 2) - 1)/2
def msb(n): return n if n<3 else msb(n/2)*2
def a059893(n): return A(n) + msb(n)
def a(n): return 0 if n==0 else a059893(a153151(a059893(n))) # Indranil Ghosh, Jun 09 2017

maxlevel <- 5 # by choice
a <- 1
for(m in 1:maxlevel){
a[2^m    ] <- 2^(m+1) - 1
a[2^m + 1] <- 2^(m+1) - 2
for (k in 1:(2^m-1)){
   a[2^(m+1) + 2*k    ] <- 2*a[2^m + k]
   a[2^(m+1) + 2*k + 1] <- 2*a[2^m + k] + 1}
}
a <- c(0,a)
# Yosu Yurramendi, Aug 01 2020

Conjecture: a(n) = f(a(f(a(A053645(n)))) + A053644(n)) for n > 0 where f(n) = A054429(n) for n > 0 with f(0) = 0. - Mikhail Kurkov, Oct 02 2023
From Mikhail Kurkov, Dec 22 2023: (Start)
a(n) < 2^k iff n < 2^k for k >= 0.
Conjectured formulas:
a(2^m + k) = f(2^m + f(k)) for m >= 0, 0 <= k < 2^m with a(0) = 0.
a(n) = f(A153142(f(n))) for n > 0 with a(0) = 0. (End)

A072797 Self-inverse permutation of natural numbers induced by a Catalan bijection acting on binary trees as encoded by A014486. See comments and examples for details.

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, 19, 21, 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, 53, 51, 52, 57, 56, 58, 59, 61, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 0

Antti Karttunen, Jun 12 2002

nonn

This bijection 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           A   C
    \ /             \ /
     x   C    -->    x   B               ()  A        ()   A
      \ /             \ /                 \ /    -->    \ /
       x               x                   x             x
  ((a . b) . c) --> ((a . c) . b)      (() . a) ---> (() . a)
See the example for an explanation of how to obtain a given integer sequence from this definition.
Notably for this permutation, A127301(a(n)) = A127301(n) does not always hold, even though for all n, A129593(a(n)) = A129593(n). - Antti Karttunen, Jan 14 2024

			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 the two subtrees in positions marked "B" and "C" in the diagram given in the comments are:
.
                          \/     \/       \/               \/
       \/     \/         \/       \/     \/       \/ \/     \/
      \/       \/       \/       \/       \/       \_/       \/
a(n)=  2        3        4        5        7        6        8
thus we obtain the first nine terms of this sequence: 0, 1, 2, 3, 4, 5, 7, 6, 8.

Row 8 of A089840.
Cf. A014486, A057163, A072796, A127301, A129593.
Counts for the fixed points and for the number of distinct cycles (in each subrange limited by A014137) are given by A073190 and A073191.

a(n) = A057163(A072796(A057163(n))).

Further comments added by Antti Karttunen, Jun 04 2011 and Mar 30 2024

A057502 Permutation of natural numbers: rotations of non-crossing handshakes encoded by A014486 (to opposite direction of A057501).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 8, 4, 5, 17, 16, 18, 14, 15, 20, 19, 21, 9, 10, 22, 11, 12, 13, 45, 44, 46, 42, 43, 48, 47, 49, 37, 38, 50, 39, 40, 41, 54, 53, 55, 51, 52, 57, 56, 58, 23, 24, 59, 25, 26, 27, 61, 60, 62, 28, 29, 63, 30, 31, 32, 64, 33, 34, 35, 36, 129, 128, 130, 126, 127

Offset: 0

Antti Karttunen, Sep 03 2000

nonn

In A057501 and A057502, the cycles between (A014138(n-1)+1)-th and (A014138(n))-th term partition A000108(n) objects encoded by the corresponding terms of A014486 into A002995(n+1) equivalence classes of planar trees, thus the latter sequence can be produced also with Maple procedure RotHandshakesPermutationCycleCounts given below.

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 A057501 and the car/cdr-flipped conjugate of A069774, i.e. A057502(n) = A057163(A069774(A057163(n))). Cf. also A057507, A057510, A057513, A069771, A069772.

map(CatalanRankGlobal,map(RotateHandshakesR, A014486));
RotateHandshakesR := n -> pars2binexp(deepreverse(RotateHandshakesP(deepreverse(binexp2pars(n)))));
deepreverse := proc(a) if 0 = nops(a) or list <> whattype(a) then (a) else [op(deepreverse(cdr(a))), deepreverse(a[1])]; fi; end;
with(group); CountCycles := b -> (nops(convert(b,'disjcyc')) + (nops(b)-convert(map(nops,convert(b,'disjcyc')),`+`)));
RotHandshakesPermutationCycleCounts := 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,RotateHandshakes(CatalanUnrank(n,r)))]; od; a := [op(a),CountCycles(b)]; od; RETURN(a); end;
# For other procedures, follow A057501.

A057508 Self-inverse permutation of natural numbers induced by the function 'reverse' (present in programming languages like Lisp, Scheme, Prolog and Haskell) when it acts on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen Sep 03 2000

nonn

A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
Index entries for the sequences induced by list functions of Lisp
Index entries for signature-permutations induced by Catalan automorphisms

The car/cdr-flipped conjugate of A069769, i.e. A057508(n) = A057163(A069769(A057163(n))). Cf. also A057164, A057509, A057510, A033538.

Similar function for Maple lists can be implemented as: reverse := proc(a) if 0 = nops(a) then (a) else [op(reverse(cdr(a))),a[1]]; fi; end;

A080070 Decimal encoding of parenthesizations produced by simple iteration starting from empty parentheses and where each successive parenthesization is obtained from the previous by reflecting it as a general tree/parenthesization, then adding an extra stem below the root and then reflecting the underlying binary tree.

Original entry on oeis.org

0, 10, 1010, 101100, 10110010, 1011100100, 101100110100, 10111001001100, 1011100110100010, 101110011010011000, 10110011101001100010, 1011110010011011000100, 101100111011010001100100

Offset: 0

Antti Karttunen, Jan 27 2003

Corresponding Lisp/Scheme S-expressions are (), (()), (()()), (()(())), (()(())()), (()((())())), (()(())(()())), ...

Conjecture: only the terms in positions 0,1,2 and 4 are symmetric, i.e., A057164(A080068(n)) = A080068(n) (equivalently: A036044(A080069(n)) = A080069(n)) only when n is one of {0,1,2,4}. If this is true, then the formula given in A079438 is exact. I (AK) have checked this up to n=404631 with no other occurrence of a symmetric (general) tree.

			This demonstrates how to get the fourth term 10110010 from the 3rd term 101100. The corresponding binary and general trees plus parenthesizations are shown. The first operation reflects the general tree, the second adds a new stem under the root and the third reflects the underlying binary tree, which induces changes on the corresponding general tree:
..............................................
.....\/................\/\/..........\/\/.....
......\/......\/\/......\/............\/......
.....\/........\/........\/..........\/.......
......(A057164).(A057548)..(A057163)..........
........................o.....................
........................|.....................
........o.....o.........o...o.........o.......
........|.....|..........\./..........|.......
....o...o.....o...o.......o.........o.o.o.....
.....\./.......\./........|..........\|/......
......*.........*.........*...........*.......
..[()(())]..[(())()]..[((())())]..[()(())()]..
...101100....110010....11100100....10110010...

A. Karttunen, Illustration of initial terms
A. Karttunen, Python program for computing this sequence.
A. Karttunen, Terms a(1)-a(256) plotted as a Wolframesque triangle.
A. Karttunen, Terms a(1)-a(512) plotted as a Wolframesque triangle.

Compare to similar Wolframesque plots given in A122229, A122232, A122235, A122239, A122242, A122245. See also A079438, A080067, A080071, A057119.

a(n) = A007088(A080069(n)) = A063171(A080068(n)).

A122287 Signature permutations of FORK-transformations of Catalan automorphisms in table A122204.

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, 14, 13, 8, 7, 5, 5, 4, 3, 2, 1, 0, 13, 21, 11, 12, 13

Offset: 0

Antti Karttunen, Sep 01 2006, Jun 20 2007

Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th automorphism in the table A122204 with the recursion scheme "FORK", or equivalently row n is obtained as FORK(ENIPS(n-th row of A089840)). See A122201 and A122204 for the description of FORK and ENIPS. Moreover, each row of A122287 can be obtained as the "DEEPEN" transform of the corresponding row in A122286. (See A122283 for the description of DEEPEN). Each row occurs only once in this table. Inverses of these permutations can be found in table A122288. This table contains also all the rows of A122201 and A089840.

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

Index entries for signature-permutations of Catalan automorphisms

The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A069767, 2: A057164, 3: A130981, 4: A130983, 5: A130982, 6: A130984, 7: A130986, 8: A130988, 9: A130994, 10: A130992, 11: A130990, 12: A057506, 13: A131004, 14: A131006, 15: A057163, 16: A131008, 17: A131010, 18: A130996, 19: A130998, 20: A131002, 21: A131000. Other rows: 169: A122353, 3617: A057511, 65167: A074681.

A089851 Permutation of natural numbers induced by Catalan automorphism *A089851 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 4, 7, 8, 12, 13, 15, 16, 19, 11, 14, 9, 17, 18, 10, 20, 21, 22, 31, 32, 34, 35, 36, 40, 41, 43, 44, 47, 52, 53, 56, 60, 30, 33, 39, 42, 51, 28, 37, 23, 45, 46, 24, 48, 49, 50, 29, 38, 25, 54, 55, 26, 57, 58, 59, 27, 61, 62, 63, 64, 87, 88, 90, 91, 92, 96, 97, 99

Offset: 0

Antti Karttunen, Nov 29 2003

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...........C...A
....\./.............\./
.A...x....-->....B...x.................A..().........A...()..
..\./.............\./...................\./....-->....\./...
...x...............x.....................x.............x....
(a . (b . c)) -> (b . (c . a)) ____ (a . ()) ---> (a . ())
In terms of S-expressions, this rotates car, cadr and cddr of an S-exp
if its length > 1, otherwise keeps it intact.
Note that the first clause corresponds to generator C of Thompson's groups T and V.
(Cf. also A072796, A074679 and A154121 for other related generators).
See "Catalan Automorphisms" OEIS-Wiki page for a detailed explanation how to obtain a given integer sequence from this definition.

A. Karttunen, Catalan Automorphisms
A. Karttunen, C-program for computing this sequence
Index entries for signature-permutations induced by Catalan automorphisms

Inverse of A089853. a(n) = A089850(A072796(n)) = A057163(A089857(A057163(n))). Row 4 of A089840.

Number of cycles: A089847. Number of fixed-points: A089848 (in each range limited by A014137 and A014138).

The new mail-address, further comments and constructive implementation of Scheme-function (*A089851) added by Antti Karttunen, Jun 04 2011

A057509 Permutation of natural numbers: rotations of the bottom branches of the rooted plane trees encoded by A014486.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 11, 14, 16, 19, 10, 15, 12, 17, 18, 13, 20, 21, 22, 23, 25, 28, 30, 33, 37, 39, 42, 44, 47, 51, 53, 56, 60, 24, 29, 38, 43, 52, 26, 40, 31, 45, 46, 32, 48, 49, 50, 27, 41, 34, 54, 55, 35, 57, 58, 59, 36, 61, 62, 63, 64, 65, 67, 70, 72, 75, 79, 81

Offset: 0

Antti Karttunen, Sep 03 2000

nonn

The number of objects (rooted planar trees, mountain ranges, parenthesizations) fixed by this permutation can be computed with procedure fixedcount, which gives A034731.

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 A057510 and the car/cdr-flipped conjugate of A069775 and also composition of A069770 & A057501, i.e. A057509(n) = A057163(A069775(A057163(n))) = A057501(A069770(n)).

Cycle counts given by A003239. Cf. also A057511.

map(CatalanRankGlobal,map(RotateBottomBranchesL, A014486));
RotateBottomBranchesL := n -> pars2binexp(rotateL(binexp2pars(n)));
rotateL := proc(a) if 0 = nops(a) then (a) else [op(cdr(a)), a[1]]; fi; end;
fixedcount := proc(n) local d,z; z := 0; for d in divisors(n) do z := z+C(d-1); od; RETURN(z); end;

A074683 Permutation of natural numbers induced by the Catalan Automorphism *A074683 acting on parenthesizations as encoded and ordered by A014486/A063171.

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 8, 5, 4, 17, 16, 18, 15, 14, 20, 19, 22, 12, 11, 21, 13, 10, 9, 45, 44, 46, 43, 42, 48, 47, 50, 40, 39, 49, 41, 38, 37, 54, 53, 55, 52, 51, 61, 60, 63, 31, 30, 62, 32, 29, 28, 57, 56, 64, 34, 33, 59, 36, 26, 25, 58, 35, 27, 24, 23, 129, 128, 130, 127, 126

Offset: 0

Antti Karttunen, Sep 11 2002

nonn

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 Catalan Automorphism (with simple definition) where the cycle count sequence (A089411) is not monotone. (See A127296 for more complex example.)

Row 12 of A122202. Inverse of A074684. a(n) = A057163(A074682(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.

cp's OEIS Frontend

A122288 Signature permutations of KROF-transformations of Catalan automorphisms in table A122203.

Views

Author

Keywords

Comments

References

Links

Crossrefs

A153141 Permutation of nonnegative integers: A059893-conjugate of A153151.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

R

Formula

A072797 Self-inverse permutation of natural numbers induced by a Catalan bijection acting on binary trees as encoded by A014486. See comments and examples for details.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A057502 Permutation of natural numbers: rotations of non-crossing handshakes encoded by A014486 (to opposite direction of A057501).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

A057508 Self-inverse permutation of natural numbers induced by the function 'reverse' (present in programming languages like Lisp, Scheme, Prolog and Haskell) when it acts on symbolless S-expressions encoded by A014486/A063171.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A122287 Signature permutations of FORK-transformations of Catalan automorphisms in table A122204.

Views

Author

Keywords

Comments

References

Links

Crossrefs

A089851 Permutation of natural numbers induced by Catalan automorphism *A089851 acting on the binary trees/parenthesizations encoded by A014486/A063171.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A057509 Permutation of natural numbers: rotations of the bottom branches of the rooted plane trees encoded by A014486.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs