A079438 -id:A079438 - cp's OEIS frontend

A057505 Signature-permutation of a Catalan Automorphism: Donaghey's map M acting on the parenthesizations encoded by A014486.

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

Offset: 0

Antti Karttunen, Sep 03 2000

nonn

This is equivalent to map M given by Donaghey on page 81 of his paper "Automorphisms on ..." and also equivalent to the transformation procedure depicted in the picture (23) of Donaghey-Shapiro paper.

This can be also considered as a "more recursive" variant of A057501 or A057503 or A057161.

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).

Antti Karttunen, Table of n, a(n) for n = 0..2055
Robert Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 75-90.
Robert Donaghey and Louis W. Shapiro, Motzkin numbers, J. Combin. Theory, Series A, Vol. 23, No. 3 (1977), pp. 291-301.
Indranil Ghosh, Python program for computing this sequence (after the functions mentioned in the OEIS wiki).
Antti Karttunen, Illustration of symmetric general trees whose A057163-reflection is also symmetric (ones that occur in 2-cycles of A057505/A057506)
Antti Karttunen, On the fixed points of A071661 (Notes in OEIS Wiki regarding the 2-cycles of this automorphism)
D. E. Knuth, Pre-Fascicle 4a: Generating All Trees, Exercise 17, 7.2.1.6.
Index entries for signature-permutations of Catalan automorphisms

Inverse: A057506.
The 2nd, 3rd, 4th, 5th and 6th "power": A071661, A071663, A071665, A071667, A071669.
Other related permutations: A057501, A057503, A057161.
Cycle counts: A057507. Maximum cycle lengths: A057545. LCM's of all cycles: A060114. See A057501 for other Maple procedures.
Row 17 of table A122288.
Cf. A080981 (the "primitive elements" of this automorphism), A079438, A079440, A079442, A079444, A080967, A080968, A080972, A080272, A080292, A083929, A080973, A081164, A123050, A125977, A126312.

map(CatalanRankGlobal,map(DonagheysM, A014486)); or map(CatalanRankGlobal,map(DeepRotateTriangularization, A014486));
DonagheysM := n -> pars2binexp(DonagheysMP(binexp2pars(n)));
DonagheysMP := h -> `if`((0 = nops(h)),h,[op(DonagheysMP(car(h))),DonagheysMP(cdr(h))]);
DeepRotateTriangularization := proc(nn) local n,s,z,w; n := binrev(nn); z := 0; w := 0; while(1 = (n mod 2)) do s := DeepRotateTriangularization(BinTreeRightBranch(n))*2; z := z + (2^w)*s; w := w + binwidth(s); z := z + (2^w); w := w + 1; n := floor(n/2); od; RETURN(z); end;

a(0) = 0, and for n>=1, a(n) = A085201(a(A072771(n)), A057548(a(A072772(n)))). [This recurrence 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) = A057164(A057163(n)).
a(n) = A057163(A057506(A057163(n))).

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)).

A079442 Number of fixed points in range [A014137(n-1)..A014138(n)] of permutation A071663.

Original entry on oeis.org

1, 1, 0, 3, 0, 9, 0, 21, 0, 45, 0, 99, 0, 195, 0, 399, 0, 801, 0

Offset: 0

Antti Karttunen, Jan 27 2003

R. Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 75-90.
Antti Karttunen, C-program for counting the initial terms of this sequence
Antti Karttunen, Illustration of initial terms for trees of sizes n=3..11

Cf. A057505, A071663, A079438, A079441, A079443, A079444.

Occurs in A073202 as row 176609070820803.

For all n >= 0, a(2n+3)/3 = A079444(n).

Name corrected by Antti Karttunen, Dec 13 2017

A123050 Conjectured number of ordered trees on n edges for which the conjugate and transpose commute.

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, 30, 30, 30, 32, 38, 38, 40, 40, 46, 48, 48, 48, 58, 58, 58, 60, 68, 68, 70, 70, 80, 82, 82, 82, 94, 94, 94, 96, 108, 108, 110, 110, 122, 124, 124, 124, 140, 140, 140, 142, 156, 156, 158

Offset: 0

David Callan, Sep 25 2006

nonn

The conjugate of an ordered tree is given by flipping it over, while its transpose is given by flipping over the corresponding binary tree. A list of ordered trees for which the conjugate and transpose commute, counted by this sequence, is given in Exercise 17, Sec. 7.2.1.6 of the Knuth reference. (Knuth deletes the root from an ordered tree and works with the resulting forest instead.) This list is complete provided a certain set of ordered trees contains no self-conjugate members other than the "obvious" ones.

The set in question consists of all trees generated by repeatedly applying the following two productions to the one-edge tree: (i) T -> plant(T) (i.e. add an edge to the root to obtain a new root) and (ii) T -> add left root edge to the transpose of the conjugate of T. Computational evidence suggests that this proviso does indeed hold.

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).

D. E. Knuth, Pre-Fascicle 4a: Generating All Trees, Exercise 17, 7.2.1.6.

This sequence updates the lower bound conjectured in A079438.

a[0]=a[1]=1; a[n_]/;n>=2 := 2(Floor[(n+1)/3] + Sum[Max[0,Floor[(n-(8k+2))/4]],{k,1,(n-2)/8}]); Table[a[n],{n,0,90}]

a(0)=a(1)=1, a(n) = 2(Floor[(n+1)/3] + Sum[Max[0,Floor[(n-(8k+2))/4]],{k,1,(n-2)/8}]) for n>=2. GF: 1 + x + 2x^2/((1-x)(1-x^3)) + 2x^14/((1-x)*(1-x^4)*(1-x^8))

A243492 Difference A243491(n) - A127301(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 2, -2, 0, 7, 4, 0, -7, -4, 7, 0, -7, 0, 0, 0, 4, -4, 0, 14, 8, 0, -14, -8, 14, 0, -14, 0, 29, 19, 25, 16, 14, 10, 5, -10, -29, -19, -5, -16, -25, -14, 47, 26, 17, 0, 0, 0, -17, -47, -26, 37, 12, -12, -37, 0, 0, 0, 8, -8, 0, 28, 16, 0, -28, -16, 28, 0, -28, 0

Offset: 0

Antti Karttunen, Jun 07 2014

A243490 gives the positions of zeros, which are also the fixed points of A069787. They correspond to the dots shown on the y=0 line of the arcsinh-version of scatter plot.

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

Cf. A069787, A243490, A057505, A079438, A123050, A080070, A153239, A153240, A153241, A154477.

(define (A243492 n) (- (A243491 n) (A127301 n)))

a(n) = A243491(n) - A127301(n) = A127301(A069787(n)) - A127301(n).

A079440 Number of transpositions (2-cycles) in range [A014137(n-1)..A014138(n-1)] of permutation A057505 (= Donaghey's automorphism M).

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 6, 6, 6, 7, 8, 8, 9, 9, 11, 12, 12, 12, 14, 14, 14, 15, 17, 17, 18, 18, 19, 20, 20, 20, 23, 23, 23, 24, 25, 25, 26, 26, 28, 29, 29, 29, 31, 31, 31, 32, 34, 34, 35, 35, 36, 37, 37, 37, 40, 40, 40, 41, 42, 42, 43, 43, 45, 46, 46, 46, 48, 48, 48

Offset: 0

Antti Karttunen, Jan 27 2003

nonn

Robert Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 75-90.
Antti Karttunen, Illustration of initial terms for trees of sizes n=2..18

From n>= 2 onward a(n) = A079438(n)/2 (with the same reservation). Cf. A079444.

A079440 := n -> floor((n+1)/3) + `if`((n>=14),floor((n-10)/4)+floor((n-14)/8),0);

A079437 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A071661.

Original entry on oeis.org

1, 1, 2, 3, 6, 16, 36, 83, 190, 448, 1056, 2514, 5872, 13806, 32424, 76609, 181434, 432062, 1032716

Offset: 0

Antti Karttunen, Jan 27 2003

That is, number of orbits to which "Catalan bijections" A071661/A071662 partition each A000108(n) Catalan tree structures encoded in A014486[A014137(n-1)..A014138(n-1)].

Antti Karttunen, C-program for counting the initial terms of this sequence

Occurs in A073201 as row 13373289.

Cf. A079438, A079439, A057507, A079441.

A079439 Longest cycle in range [A014137(n-1)..A014138(n-1)] of permutation A071661.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 12, 36, 72, 147, 294, 336, 1068, 5076, 5760, 14742, 58968, 135288, 328176

Offset: 0

Antti Karttunen, Jan 27 2003

Antti Karttunen, C-program for counting the initial terms of this sequence

Occurs in A073203 as row 13373289. Cf. A079437, A079438, A079443.

A328111 a(n) = A080069(n) OR A267357(n).

Original entry on oeis.org

1, 3, 15, 47, 191, 743, 2935, 12015, 47615, 190363, 737255, 3092431, 11777535, 48562151, 194672615, 778681963, 3117668351, 12677730147, 49850341191, 192901085003, 795560607711, 3243899871031, 12977889600367, 51055599708139, 204124618746111, 791262494980483, 3318011560984519, 12661179187462123, 52138250822737375, 212591566440951715, 836346216751952367, 3236342451194541807

Offset: 0

Antti Karttunen, Oct 05 2019

nonn

The pattern has a remarkably nice texture. A269174 gives the trajectory of 1-D Cellular Automaton rule 124 (which is a mirror image of rule 110), when started from a single alive cell. Trails of its evolution can be dimly discerned on the right hand side of given illustrations, while the left hand side shows the evolution of (left hand side of) iterated Dyck-path system A080069 unblemished.

Antti Karttunen, Table of n, a(n) for n = 0..1023
Antti Karttunen, Terms up to a(255) drawn as binary strings, with 1 bit = 3x3 pixels resolution
Antti Karttunen, Terms up to a(1023) drawn as binary strings, with 1 bit = 1 pixel resolution

Cf. A003986, A267357, A269174.
Cf. A080069, A080070, and also A079438 and A123050.
Cf. also A328103.

a(n) = A080069(n) OR A267357(n), where OR is bitwise-OR, A003986.

cp's OEIS Frontend

A057505 Signature-permutation of a Catalan Automorphism: Donaghey's map M acting on the parenthesizations encoded by A014486.

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A079442 Number of fixed points in range [A014137(n-1)..A014138(n)] of permutation A071663.

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Extensions

A123050 Conjectured number of ordered trees on n edges for which the conjugate and transpose commute.

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Mathematica

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A243492 Difference A243491(n) - A127301(n).

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A079440 Number of transpositions (2-cycles) in range [A014137(n-1)..A014138(n-1)] of permutation A057505 (= Donaghey's automorphism M).

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Maple

A079437 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A071661.

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A079439 Longest cycle in range [A014137(n-1)..A014138(n-1)] of permutation A071661.

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A328111 a(n) = A080069(n) OR A267357(n).

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