A057510 -id:A057510 - cp's OEIS frontend

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

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;

A127291 Signature-permutation of Elizalde's and Deutsch's 2003 bijection for Dyck paths.

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 8, 5, 4, 15, 18, 14, 16, 17, 20, 22, 19, 11, 12, 21, 13, 10, 9, 39, 47, 40, 48, 50, 41, 49, 38, 43, 46, 37, 42, 44, 45, 53, 60, 54, 61, 63, 55, 62, 52, 29, 32, 51, 28, 30, 31, 59, 64, 57, 34, 36, 56, 33, 25, 26, 58, 35, 27, 24, 23, 113, 136, 116, 139, 146

Offset: 0

Antti Karttunen, Jan 16 2007

Deutsch and Elizalde show in their paper that this automorphism converts certain properties concerning "tunnels" of Dyck path to another set of properties concerning the number of hills, even and odd rises, as well as the number of returns (A057515), thus proving the equidistribution of the said parameters.
This automorphism is implemented with function "tau" (Scheme code given below) that takes as its arguments an S-expression and a Catalan automorphism that permutes only the top level of the list (i.e., the top-level branches of a general tree, or the whole arches of a Dyck path) and thus when the permuting automorphism is applied to a list (parenthesization) of length 2n it induces some permutation of [1..2n].
This automorphism is induced in that manner by the automorphism *A127287 and likewise, *A127289 is induced by *A127285, *A057164 by *A057508, *A057501 by *A057509 and *A057502 by *A057510.
Note that so far these examples seem to satisfy the homomorphism condition, e.g., as *A127287 = *A127285 o *A057508 so is *A127291 = *A127289 o *A057164. and likewise, as *A057510 = *A057508 o *A057509 o *A057508, so is *A057502 = *A057164 o *A057501 o *A057164.
However, it remains open what are the exact criteria of the "picking automorphism" and the corresponding permutation that this method would induce a bijection. For example, if we give *A127288 (the inverse of *A127287) to function "tau" it will not induce *A127292 and actually not a bijection at all.
Instead, we have to compute the inverse of this automorphism with another, more specific algorithm that implements Deutsch's and Elizalde's description and is given in A127300.

A. Karttunen, Table of n, a(n) for n = 0..2055
Emeric Deutsch and Sergi Elizalde, A simple and unusual bijection for Dyck paths and its consequences, Annals of Combinatorics, 7 (2003), no. 3, 281-297.
Index entries for signature-permutations of Catalan automorphisms

Inverse: A127292. a(n) = A127289(A057164(n)) = A057164(A127299(A057164(n))). A127291(A057548(n)) = A072795(A127291(n)), A127291(A072795(n)) = A127307(A127291(A057502(n))) for all n >= 1. The number of cycles, maximum cycle sizes and LCM's of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A127293, A127294 and A127295. Number of fixed points begins as 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, ...

A034731 Dirichlet convolution of b_n=1 with Catalan numbers.

Original entry on oeis.org

1, 2, 3, 7, 15, 46, 133, 436, 1433, 4878, 16797, 58837, 208013, 743034, 2674457, 9695281, 35357671, 129646266, 477638701, 1767268073, 6564120555, 24466283818, 91482563641, 343059672916, 1289904147339, 4861946609466

Offset: 1

Erich Friedman

nonn

Also number of objects fixed by permutations A057509/A057510 (induced by shallow rotation of general parenthesizations/plane trees).

G. C. Greubel, Table of n, a(n) for n = 1..1000

Occurs for first time in A073202 as row 16.

Cf. A000108, A057509, A057510, A057546.

a[n_] := DivisorSum[n, CatalanNumber[#-1]&]; Array[a, 26] (* Jean-François Alcover, Dec 05 2015 *)

a(n) = sumdiv(n, d, binomial(2*(d-1),d-1)/d) \\ Michel Marcus, Jun 07 2013

{a(n) = my(A = sum(m=1, n, (1 - sqrt(1 - 4*x^m +x*O(x^n)))/2 )); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 12 2021

{a(n) = my(A = sum(m=1, n, binomial(2*m-2,m-1)/m * x^m/(1 - x^m +x*O(x^n)) )); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 12 2021

a(n) = Sum_{d divides n} C(d-1) where C() are the Catalan numbers (A000108).
a(n) ~ 4^(n-1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Dec 05 2015
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(binomial(2*k-2,k-1)/k^2)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 23 2018
G.f.: Sum_{n>=1} (1 - sqrt(1 - 4*x^n))/2. - Paul D. Hanna, Jan 12 2021
G.f.: Sum_{n>=1} A000108(n-1) * x^n/(1 - x^n) where A000108(n) = binomial(2*n,n)/(n+1). - Paul D. Hanna, Jan 12 2021

More comments from Antti Karttunen, Jan 03 2003

A069776 Permutation of natural numbers induced by the automorphism gma069776! 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, 20, 21, 19, 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, 57, 58, 59, 53, 56, 51, 52, 61, 62, 63, 60, 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 A069775. a(n) = A057163(A057510(A057163(n))) = A069770(A069774(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).

A123718 Signature permutation of a Catalan automorphism: row 253 of table A122204.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Oct 11 2006

nonn

This is the signature-permutation of Catalan automorphism which is derived from nonrecursive Catalan automorphism *A123503 with the recursion schema ENIPS (defined in A122204). See the comments at A123717.

Index entries for signature-permutations of Catalan automorphisms

Inverse: A123717. a(n) = A089854(A057510(n)). Row 253 of A122204.

A130920 Signature permutation of a Catalan automorphism: DEEPEN-transform of automorphism *A057512.

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, 48, 32, 54, 49, 50, 33, 41, 34, 46, 55, 35, 57, 58, 62, 36, 61, 59, 63, 64, 65, 107, 66, 121, 149, 67

Offset: 0

Antti Karttunen, Jun 11 2007

nonn

*A130920 = DEEPEN(*A057512) = NEPEED(*A057512) = DEEPEN(DEEPEN(*A057510)) = NEPEED(NEPEED(*A057510)). See A122283, A122284 for the definitions of DEEPEN and NEPEED transforms.

Inverse: A130919. A122351(n) = A083927(A130920(A057123(n))).

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

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

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Maple

A127291 Signature-permutation of Elizalde's and Deutsch's 2003 bijection for Dyck paths.

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A034731 Dirichlet convolution of b_n=1 with Catalan numbers.

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Mathematica

PARI

PARI

PARI

Formula

Extensions

A069776 Permutation of natural numbers induced by the automorphism gma069776! acting on the parenthesizations encoded by A014486.

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A123718 Signature permutation of a Catalan automorphism: row 253 of table A122204.

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A130920 Signature permutation of a Catalan automorphism: DEEPEN-transform of automorphism *A057512.

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

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Maple

Mathematica

Sage