A061856 -id:A061856 - cp's OEIS frontend

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

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
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See Links section.
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a(n) = A080300(A036044(A014486(n))) = A080300(A056539(A014486(n))).

A061855 Symmetric totally balanced binary sequences: those terms of A014486 which are equal to their reversed complement.

Original entry on oeis.org

0, 2, 10, 12, 42, 52, 56, 170, 178, 204, 212, 232, 240, 682, 722, 738, 812, 852, 868, 920, 936, 976, 992, 2730, 2762, 2866, 2898, 2978, 3010, 3244, 3276, 3380, 3412, 3492, 3524, 3640, 3672, 3752, 3784, 3888, 3920, 4000, 4032, 10922, 11082, 11146

Offset: 0

Antti Karttunen, May 11 2001

nonn

These encode symmetric (palindromic) structures in many of the Catalan families, e.g. mountain ranges, parenthesizations, unlabeled rooted plane trees.

			E.g. the 45th term 11146 is 10101110001010 in binary and can be interpreted as a parenthesization: ( )( )((( )))( )( )

A. Karttunen, Some notes on Catalan's Triangle
Index entries for sequences related to parenthesizing

Obtained by "reflecting" the terms of A061854. Cf. also A035928 (ReflectBinSeq), A061856, A069766.

map(op,[seq(PalTotBalBinSequences(j),j=1..10)]);
PalTotBalBinSequences := n -> map(ReflectBinSeq,NonDivingLatticeSequences(n), n);

a(0) = 0 and the rest with the Maple function map(op, [seq(PalTotBalBinSequences(j), j=1..10)]);

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

Original entry on oeis.org

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.

A153240 Balance of general trees as ordered by A014486, variant A.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 1, 0, 2, 2, -1, 0, -2, 0, 1, -2, -1, 0, 0, 0, 1, 1, 2, 2, -1, 1, 0, 3, 3, 0, 3, 3, 3, -1, 0, -1, 1, 1, -2, -1, -3, 0, 1, -3, 0, 2, 2, -2, -1, -3, -1, 0, -3, -2, 0, 1, -3, -2, -1, 0, 0, 0, 1, 1, 2, 2, 0, 2, 2, 3, 3, 2, 3, 3, 3, -1, 0, 0, 2, 2, -2, 1, 0, 4, 4, 1, 4

Offset: 0

Antti Karttunen, Dec 21 2008

sign

This differs from variant A153241 only in that if the degree of the tree is odd (i.e. A057515(n) = 1 mod 2), then the balance of the center-subtree is always taken into account.

Note that for all n, Sum_{i=A014137(n)}^A014138(n) a(i) = 0.

			A014486(25) encodes the following general tree:
......o
......|
o.o...o.o
.\.\././
....*..
which consists of four subtrees, of which the second from right is one larger than the others, so we have a(25) = (0+1)-(0+0) = 1.

A. Karttunen, Table of n, a(n) for n = 0..2055

Differs from variant A153241 for the first time at n=268, where A153241(268) = 1, while a(268)=2. Note that (A014486->parenthesization (A014486 268)) = (() (() (())) (())). a(A061856(n)) = 0 for all n. Cf. also A153239.

A153241 Balance of general trees as ordered by A014486, variant B.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 1, 0, 2, 2, -1, 0, -2, 0, 1, -2, -1, 0, 0, 0, 1, 1, 2, 2, -1, 1, 0, 3, 3, 0, 3, 3, 3, -1, 0, -1, 1, 1, -2, -1, -3, 0, 1, -3, 0, 2, 2, -2, -1, -3, -1, 0, -3, -2, 0, 1, -3, -2, -1, 0, 0, 0, 1, 1, 2, 2, 0, 2, 2, 3, 3, 2, 3, 3, 3, -1, 0, 0, 2, 2, -2, 1, 0, 4, 4, 1, 4

Offset: 0

Antti Karttunen, Dec 21 2008

sign

This differs from variant A153240 only in that if the degree of the tree is odd (i.e. A057515(n) = 1 mod 2), then the balance of the center-subtree is taken into account ONLY if the total weight of other subtrees at the left and the right hand side from the center were balanced against each other.

Note that for all n, Sum_{i=A014137(n)}^A014138(n) a(i) = 0.

A. Karttunen, Table of n, a(n) for n = 0..2055

Differs from variant A153240 for the first time at n=268, where A153240(268) = 2, while a(268)=1. Note that (A014486->parenthesization (A014486 268)) = (() (() (())) (())). a(A061856(n)) = 0 for all n. Cf. also A153239.

A215406 A ranking algorithm for the lexicographic ordering of the Catalan families.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4

Offset: 0

Peter Luschny, Aug 09 2012

See Antti Karttunen's code in A057117. Karttunen writes: "Maple procedure CatalanRank is adapted from the algorithm 3.23 of the CAGES (Kreher and Stinson) book."

For all n>0, a(A014486(n)) = n = A080300(A014486(n)). The sequence A080300 differs from this one in that it gives 0 for those n which are not found in A014486. - Antti Karttunen, Aug 10 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000
A. Karttunen, Catalan's Triangle and ranking of Mountain Ranges
D. L. Kreher and D. R. Stinson, Combinatorial Algorithms, Generation, Enumeration and Search, CRC Press, 1998.
F. Ruskey, Algorithmic Solution of Two Combinatorial Problems, Thesis, Department of Applied Physics and Information Science, University of Victoria, 1978.

Cf. A213704, A057117, A057164, A057505, A057506, A057501, A057502, A057511, A057512, A057123, A057117, A057509, A057510, A057161, A057162, A072766, A071654, A071652, A075161, A061856, A072635, A072788, A057518, A075168, A080119, A057120, A072773.

A215406 := proc(n) local m,a,y,t,x,u,v;
m := iquo(A070939(n), 2);
a := A030101(n);
y := 0; t := 1;
for x from 0 to 2*m-2 do
    if irem(a, 2) = 1 then y := y + 1
    else u := 2*m - x;
         v := m-1 - iquo(x+y,2);
         t := t + A037012(u,v);
         y := y - 1 fi;
    a := iquo(a, 2) od;
A014137(m) - t end:
seq(A215406(i),i=0..199); # Peter Luschny, Aug 10 2012

A215406[n_] := Module[{m, d, a, y, t, x, u, v}, m = Quotient[Length[d = IntegerDigits[n, 2]], 2]; a = FromDigits[Reverse[d], 2]; y = 0; t = 1; For[x = 0, x <= 2*m - 2, x++, If[Mod[a, 2] == 1, y++, u = 2*m - x; v = m - Quotient[x + y, 2] - 1; t = t - Binomial[u - 1, v - 1] + Binomial[u - 1, v]; y--]; a = Quotient[a, 2]]; (1 - I*Sqrt[3])/2 - 4^(m + 1)*Gamma[m + 3/2]*Hypergeometric2F1[1, m + 3/2, m + 3, 4]/(Sqrt[Pi]*Gamma[m + 3]) - t]; Table[A215406[n] // Simplify, {n, 0, 86}] (* Jean-François Alcover, Jul 25 2013, translated and adapted from Peter Luschny's Maple program *)

def A215406(n) : # CatalanRankGlobal(n)
    m = A070939(n)//2
    a = A030101(n)
    y = 0; t = 1
    for x in (1..2*m-1) :
        u = 2*m - x; v = m - (x+y+1)/2
        mn = binomial(u, v) - binomial(u, v-1)
        t += mn*(1 - a%2)
        y -= (-1)^a
        a = a//2
    return A014137(m) - t

cp's OEIS Frontend

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

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A061855 Symmetric totally balanced binary sequences: those terms of A014486 which are equal to their reversed complement.

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

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A153240 Balance of general trees as ordered by A014486, variant A.

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A153241 Balance of general trees as ordered by A014486, variant B.

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A215406 A ranking algorithm for the lexicographic ordering of the Catalan families.

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Maple

Mathematica

Sage