A057507 -id:A057507 - 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))).

A057506 Signature-permutation of a Catalan Automorphism: (inverse of) "Donaghey's map M", acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

0, 1, 3, 2, 8, 6, 7, 5, 4, 22, 19, 20, 15, 14, 21, 16, 18, 13, 11, 17, 12, 10, 9, 64, 60, 61, 52, 51, 62, 53, 55, 41, 39, 54, 40, 38, 37, 63, 56, 57, 43, 42, 59, 47, 50, 36, 33, 48, 34, 29, 28, 58, 44, 49, 35, 30, 46, 32, 27, 25, 45, 31, 26, 24, 23, 196, 191, 192, 178, 177

Offset: 0

Antti Karttunen, Sep 03 2000

This is inverse of A057505, which is a signature permutation of Catalan automorphism (bijection) known as "Donaghey's map M". See A057505 for more comments, links and references.

Antti Karttunen, Table of n, a(n) for n = 0..23713
A. Karttunen (et al) at OEIS Wiki, Maple implementations of CatalanRankGlobal, CatalanSequences, car, cdr, binexp2pars and pars2binexp functions
A. Karttunen at OEIS WIki, Scheme implementations of CatalanRankSexp and CatalanUnrankSexp
Indranil Ghosh, Python program for computing this sequence (after the functions mentioned in the OEIS wiki)
Index entries for signature-permutations of Catalan automorphisms

Inverse: A057505.
Cf. A057161, A057162, A057163, A057164, A057501, A057502, A057503, A057504 (for similar signature permutations of simple Catalan automorphisms).
Cf. A057507 (cycle counts).
The 2nd, 3rd, 4th, 5th and 6th "powers" of this permutation: A071662, A071664, A071666, A071668, A071670.
Row 12 of table A122287.
Cf. also A014486, A080300, A080069, A080070.

map(CatalanRankGlobal,map(DonagheysA057506,CatalanSequences(196))); # Where CatalanSequences(n) gives the terms A014486(0..n).
DonagheysA057506 := n -> pars2binexp(deepreverse(DonagheysA057505(deepreverse(binexp2pars(n)))));
DonagheysA057505 := h -> `if`((0 = nops(h)), h, [op(DonagheysA057505(car(h))), DonagheysA057505(cdr(h))]);
# The following corresponds to automorphism A057164:
deepreverse := proc(a) if 0 = nops(a) or list <> whattype(a) then (a) else [op(deepreverse(cdr(a))), deepreverse(a[1])]; fi; end;
# The rest of required Maple-functions: see the given OEIS Wiki page.

(define (A057506 n) (CatalanRankSexp (*A057506 (CatalanUnrankSexp n))))
(define (*A057506 bt) (let loop ((lt bt) (nt (list))) (cond ((not (pair? lt)) nt) (else (loop (cdr lt) (cons nt (*A057506 (car lt))))))))
;; Functions CatalanRankSexp and CatalanUnrankSexp can be found at OEIS Wiki page.

a(n) = A057163(A057164(n)).

a(n) = A057163(A057505(A057163(n))) = A057164(A057505(A057164(n))).

Entry revised by Antti Karttunen, May 30 2017

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.

A057513 Number of separate orbits to which permutations given in A057511/A057512 (induced by deep rotation of general parenthesizations/plane trees) partition each A000108(n) objects encoded by A014486 between (A014138(n-1)+1)-th and (A014138(n))-th terms.

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 56, 153, 451, 1357, 4212, 13308, 42898, 140276, 465324, 1561955, 5300285, 18156813, 62732842, 218405402, 765657940

Offset: 0

Antti Karttunen Sep 03 2000

It is much faster to compute this sequence empirically with the given C-program than to calculate the terms with the formula in its present form.

A. Karttunen, Gatomorphisms (with the complete Scheme source)
Index entries for sequences related to rooted trees
A. Karttunen, C-program for computing empirically the initial terms of this sequence

CountCycles given in A057502, for other procedures, follow A057511 and A057501.
Similarly generated sequences: A001683, A002995, A003239, A038775, A057507. Cf. also A000081.
Occurs for first time in A073201 as row 12. Cf. A057546 and also A000081.

A057513 := proc(n) local i; `if`((0=n),1,(1/A003418(n-1))*add(A079216bi(n,i),i=1..A003418(n-1))); end;
# Or empirically:
DeepRotatePermutationCycleCounts := 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,DeepRotateL(CatalanUnrank(n,r)))]; od; a := [op(a),CountCycles(b)]; od; RETURN(a); end;

a(0)=1, a(n) = (1/A003418(n-1))*Sum_{i=1..A003418(n-1)} A079216(n, i) [Needs improvement.] - Antti Karttunen, Jan 03 2003

A073201 Array of cycle count sequences for the table A073200.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 7, 4, 1, 1, 1, 22, 11, 3, 1, 1, 1, 66, 31, 7, 2, 1, 1, 1, 217, 96, 22, 4, 3, 1, 1, 1, 715, 305, 66, 11, 7, 2, 1, 1, 1, 2438, 1007, 217, 30, 22, 4, 2, 2, 1, 1, 8398, 3389, 715, 93, 66, 11, 3, 5, 1, 1, 1, 29414, 11636, 2438, 292, 217, 30, 6, 14, 2, 2, 1, 1

Offset: 0

Antti Karttunen, Jun 25 2002

Each row of this table gives the counts of separate orbits/cycles to which the Catalan bijection given in the corresponding row of A073200 partitions each A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.

Note that for involutions (self-inverse Catalan bijections) this is always (A000108(n)+Affffff(n))/2, where Affffff is the corresponding "fix-count sequence" from the table A073202.

A. Karttunen, Gatomorphisms (With the complete source and explanation)

Only the first known occurrence(s) given (marked with ? if not yet proved/unclear): rows 0, 2, 4, etc.: A007595, Row 1: A073191, Rows 6 (& 8): A073431, Row 7: A000108, Rows 12, 14, 20, ...: A057513, Rows 16, 18, ...: A003239, Row 57, ..., 164: A007123, Row 168: A073193, Row 261: A002995, Row 2614: A057507, Row 2618 (?), row 17517: A001683.

A057545 Maximum cycle size in range [A014137(n-1)..A014138(n-1)] of permutation A057505/A057506.

Original entry on oeis.org

1, 1, 2, 3, 6, 6, 24, 72, 144, 147, 588, 672, 2136, 10152, 11520, 29484, 117936, 270576, 656352, 2062368, 4040160

Offset: 0

Antti Karttunen, Sep 07 2000

For the convenience of the range notation above, we define A014137(-1) and A014138(-1) as zero.
Equal to the degree of the polynomials M_n(x) Donaghey gives on the page 81 of his paper.
Factored terms: 1, 1, 2, 3, 2*3, 2*3, 2^3 * 3, 2^3 * 3^2, 2^4 * 3^2, 3 * 7^2, 2^2 * 3 * 7^2, 2^5 * 3 * 7, 2^3 * 3 * 89, 2^3 * 3^3 * 47, 2^8 * 3^2 * 5, 2^2 * 3^4 * 7 * 13, 2^4 * 3^4 * 7 * 13, 2^4 * 3^2 * 1879, 2^5 * 3^2 * 43 * 53, 2^5 * 3^3 * 7 * 11 * 31, 2^5 * 3 * 5 * 19 * 443

Robert Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 75-90.

Cf. A057507, A060114, A080967, A081165.

Occurs for first time in A073203 as row 2614.

A060114 Least common multiple of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutation A057505/A057506.

Original entry on oeis.org

1, 1, 2, 6, 6, 30, 120, 720, 15120, 1164240, 15135120, 283931716867999200, 14510088480716327580681600, 3280681990411073806237542217555200, 936436634805345771521186435213604447980767985241556128000

Offset: 0

Antti Karttunen, Mar 01 2001

nonn

For the convenience of the range notation above, we define A014137(-1) and A014138(-1) as zero.

This sequence grows so fast that it seems hopeless to count A057507 with Burnside's (orbit-counting) lemma.

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

Cf. A057505, A057507, A057545, A081165, A081166, A078491.

Occurs for first time in A073204 as row 2614.

A081150 Number of even cycles in range [A014137(n-1)..A014138(n-1)] of permutation A057505/A057506.

Original entry on oeis.org

0, 0, 1, 1, 3, 6, 18, 37, 95, 212, 528, 1226, 2936, 6822, 16212, 38081, 90717, 215414, 516358

Offset: 0

Wouter Meeussen and Antti Karttunen, Mar 10 2003

For the convenience of the range notation above, we define A014137(-1) and A014138(-1) as zero.

Antti Karttunen, Counted with the help of this C program.

Bisections: A081151, A081152.
Cf. A081157, A081160, A081162, A057507, A081148.
Cf. A014137, A014138, A057505, A057506.

a(n) = A081157(n) + A081160(n) + A081162(n) = A057507(n) - A081148(n).

A081148 Number of odd cycles in range [A014137(n-1)..A014138(n-1)] of permutation A057505/A057506.

Original entry on oeis.org

1, 1, 0, 1, 0, 4, 0, 9, 0, 24, 0, 62, 0, 162, 0, 447, 0, 1234, 0

Offset: 0

Wouter Meeussen and Antti Karttunen, Mar 10 2003

For the convenience of the range notation above, we define A014137(-1) and A014138(-1) as zero.

Antti Karttunen, Counted with the help of this C program.

Bisection: A081149.
Cf. A081153, A081155, A057507, A081150.
Cf. A014137, A014138, A057505, A057506.

a(n) = A081153(n) + A081155(n) = A057507(n) - A081150(n).

A081151 Number of even cycles in range [A014137(2n-1)..A014138(2n-1)] of permutation A057505/A057506.

Original entry on oeis.org

0, 1, 3, 18, 95, 528, 2936, 16212, 90717, 516358, 2979992

Offset: 0

Wouter Meeussen and Antti Karttunen, Mar 10 2003

Equal to the number of all cycles in the same range (after n>0).

Cf. A081152.
Cf. A081150, A081158, A001700, A057507.
Cf. A014137, A014138, A057505, A057506.

a(n) = A081150(2n) = A081158(n)+A001700(n-1) [where A001700(-1) is taken as 0 here] = A057507(2n) [After n>0].

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Formula

A057506 Signature-permutation of a Catalan Automorphism: (inverse of) "Donaghey's map M", acting on the parenthesizations encoded by A014486.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Scheme

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

A057513 Number of separate orbits to which permutations given in A057511/A057512 (induced by deep rotation of general parenthesizations/plane trees) partition each A000108(n) objects encoded by A014486 between (A014138(n-1)+1)-th and (A014138(n))-th terms.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula

A073201 Array of cycle count sequences for the table A073200.

Views

Author

Keywords

Comments

Links

Crossrefs

A057545 Maximum cycle size in range [A014137(n-1)..A014138(n-1)] of permutation A057505/A057506.

Views

Author

Keywords

Comments

Links

Crossrefs

A060114 Least common multiple of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutation A057505/A057506.

Views

Author

Keywords

Comments

Links

Crossrefs

A081150 Number of even cycles in range [A014137(n-1)..A014138(n-1)] of permutation A057505/A057506.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A081148 Number of odd cycles in range [A014137(n-1)..A014138(n-1)] of permutation A057505/A057506.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula