A073201 -id:A073201 - cp's OEIS frontend

A073200 Number of simple Catalan bijections of type B.

0, 1, 0, 3, 1, 0, 2, 2, 1, 0, 7, 3, 3, 1, 0, 8, 4, 2, 3, 1, 0, 6, 6, 8, 2, 3, 1, 0, 4, 5, 7, 7, 2, 3, 1, 0, 5, 7, 6, 6, 8, 2, 3, 1, 0, 17, 8, 5, 8, 7, 7, 2, 2, 1, 0, 18, 9, 4, 4, 6, 8, 7, 3, 3, 1, 0, 20, 10, 22, 5, 5, 5, 8, 4, 2, 2, 1, 0, 21, 14, 21, 17, 4, 4, 6, 5, 8, 3, 3, 1, 0

Offset: 0

Antti Karttunen, Jun 25 2002

Each row is a permutation of nonnegative integers induced by a Catalan bijection (constructed as explained below) acting on the parenthesizations/plane binary trees as encoded and ordered by A014486/A063171.
The construction process is akin to the constructive mapping of primitive recursive functions to N: we have two basic primitives, A069770 (row 0) and A072796 (row 1), of which the former swaps the left and the right subtree of a binary tree and the latter exchanges the positions of the two leftmost subtrees of plane general trees, unless the tree's degree is less than 2, in which case it just fixes it. From then on, the even rows are constructed recursively from any other Catalan bijection in this table, using one of the five allowed recursion types:
0 - Apply the given Catalan bijection and then recurse down to both subtrees of the new binary tree obtained. (last decimal digit of row number = 2)
1 - First recurse down to both subtrees of the old binary tree and only after that apply the given Catalan bijection. (last digit = 4)
2 - Apply the given Catalan bijection and then recurse down to the right subtree of the new binary tree obtained. (last digit = 6)
3 - First recurse down to the right subtree of old binary tree and only after that apply the given Catalan bijection. (last digit = 8)
4 - First recurse down to the left subtree of old binary tree, after that apply the given Catalan bijection and then recurse down to the right subtree of the new binary tree. (last digit = 0)
The odd rows > 2 are compositions of the rows 0, 1, 2, 4, 6, 8, ... (i.e. either one of the primitives A069770 or A072796, or one of the recursive compositions) at the left hand side and any Catalan bijection from the same array at the right hand side. See the scheme-functions index-for-recursive-sgtb and index-for-composed-sgtb how to compute the positions of the recursive and ordinary compositions in this table.

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

Four other tables giving the corresponding cycle-counts: A073201, counts of the fixed elements: A073202, the lengths of the largest cycles: A073203, the LCM's of all the cycles: A073204. The ordinary compositions are encoded using the N X N -> N bijection A054238 (which in turn uses the bit-interleaving function A000695).
The first 21 rows of this table:.
Row 0: A069770. Row 1: A072796. Row 2: A057163. Row 3: A073269, Row 4: A057163 (duplicate), Row 5: A073270, Row 6: A069767, Row 7: A001477 (identity perm.), Row 8: A069768, Row 9: A073280.
Row 10: A069770 (dupl.), Row 11: A072796 (dupl.), Row 12: A057511, Row 13: A073282, Row 14: A057512, Row 15: A073281, Row 16: A057509, Row 17: A073280 (dupl.), Row 18: A057510, Row 19: A073283, Row 20: A073284.
Other Catalan bijection-induced EIS-permutations which occur in this table. Only the first known occurrence is given. Involutions are marked with *, others paired with their inverse:.
Row 164: A057164*, Row 168: A057508*, Row 179: A072797*.
Row 41: A073286 - Row 69: A073287. Row 105: A073290 - Row 197: A073291. Row 416: A073288 - Row 696: A073289.
Row 261: A057501 - Row 521: A057502. Row 2618: A057503 - Row 5216: A057504. Row 2614: A057505 - Row 5212: A057506.
Row 10435: A073292 - Row ...: A073293. Row 17517: A057161 - Row ...: A057162.
For a more practical enumeration system of (some) Catalan automorphisms see table A089840 and its various "recursive derivations".

A073202 Array of fix-count sequences for the table A073200.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 0, 3, 0, 1, 1, 2, 8, 1, 0, 1, 1, 0, 20, 0, 0, 0, 1, 1, 5, 60, 2, 0, 1, 0, 1, 1, 0, 181, 0, 0, 0, 0, 0, 1, 1, 14, 584, 5, 0, 2, 0, 1, 2, 1, 1, 0, 1916, 0, 0, 0, 0, 0, 5, 0, 1, 1, 42, 6476, 14, 0, 5, 0, 0, 14, 1, 2, 1, 1, 0, 22210, 0, 0, 0, 0, 0, 42, 0, 1, 0, 1, 1

Offset: 0

Antti Karttunen, Jun 25 2002

Each row of this table gives the counts of elements fixed by the Catalan bijection (given in the corresponding row of A073200) when it acts on A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.

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

Cf. also A073201, A073203.
Few EIS-sequences which occur in this table. Only the first known occurrence(s) given (marked with ? if not yet proved/unclear):
Rows 0, 2, 4, etc.: "Aerated Catalan numbers" shifted right and prepended with 1 (Cf. A000108), Row 1: A073190, Rows 3, 5, 261, 2614, 2618, 17517, etc: A019590 but with offset 0 instead of 1 (means that the Catalan bijections like A073269, A073270, A057501, A057505, A057503 and A057161 never fix any Catalan structure of size larger than 1).
Row 6: A036987, Row 7: A000108, Rows 12, 14, 20, ...: A057546, Rows 16, 18: A034731, Row 41: A073268, Row 105: essentially A073267, Row 57, ..., 164: A001405, Row 168: A073192, Row 416: essentially A023359 ?, Row 10435: also A036987.

A007595 a(n) = C_n / 2 if n is even or ( C_n + C_((n-1)/2) ) / 2 if n is odd, where C = Catalan numbers (A000108).

Original entry on oeis.org

1, 1, 3, 7, 22, 66, 217, 715, 2438, 8398, 29414, 104006, 371516, 1337220, 4847637, 17678835, 64823110, 238819350, 883634026, 3282060210, 12233141908, 45741281820, 171529836218, 644952073662, 2430973304732, 9183676536076, 34766775829452, 131873975875180

Offset: 1

N. J. A. Sloane

Number of necklaces of 2 colors with 2n beads and n-1 black ones. - Wouter Meeussen, Aug 03 2002
Number of rooted planar binary trees up to reflection (trees with n internal nodes, or a total of 2n+1 nodes). - Antti Karttunen, Aug 19 2002
Number of even permutations avoiding 132.
Number of Dyck paths of length 2n having an even number of peaks at even height. Example: a(3)=3 because we have UDUDUD, U(UD)(UD)D and UUUDDD, where U=(1,1), D=(1,-1) and the peaks at even height are shown between parentheses. - Emeric Deutsch, Nov 13 2004
Number of planar trees (A002995) on n edges with one distinguished edge. - David Callan, Oct 08 2005
Assuming offset 0 this is an analog of A275165: pairs of two Catalan nestings with index sum n. - R. J. Mathar, Jul 19 2016

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..200
Peter J. Cameron, Some treelike objects Quart. J. Math. Oxford Ser., Vol. 38, No. 2 (1987), pp. 155-183. Note that line 3 on p. 163 has a typo. - _N. J. A. Sloane_, Apr 18 2014
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seq., Vol. 3 (2000), Article 00.1.5.
Paul Drube and Puttipong Pongtanapaisan, Annular Non-Crossing Matchings, Journal of Integer Sequences, Vol. 19 (2016), Article 16.2.4.
Andrew Gainer-Dewar, Pólya theory for species with an equivariant group action, arXiv preprint arXiv:1401.6202 [math.CO], 2014.
Toufik Mansour, Counting occurrences of 132 in an even permutation, arXiv:math/0211205 [math.CO], 2002.
Krishna Menon and Anurag Singh, Grassmannian permutations avoiding identity, arXiv:2212.13794 [math.CO], 2022.

a(n) = A047996(2*n, n-1) for n >= 1 and a(n) = A072506(n, n-1) for n >= 2.
Occurs in A073201 as rows 0, 2, 4, etc. (with a(0)=1 included).
Cf. also A003444, A007123.
Cf. A000108, A000150, A334843.

A007595 := n -> (1/2)*(Cat(n) + (`mod`(n,2)*Cat((n-1)/2))); Cat := n -> binomial(2*n,n)/(n+1);

Table[(Plus@@(EulerPhi[ # ]Binomial[2n/#, (n-1)/# ] &)/@Intersection[Divisors[2n], Divisors[n-1]])/(2n), {n, 2, 32}] (* or *) Table[If[EvenQ[n], CatalanNumber[n]/2, (CatalanNumber[n] + CatalanNumber[(n-1)/2])/2], {n, 24}]
Table[(CatalanNumber[n] + 2^n Binomial[1/2, (n + 1)/2] Sin[Pi n/2])/2, {n, 1, 20}] (* Vladimir Reshetnikov, Oct 03 2016 *)
Table[If[EvenQ[n],CatalanNumber[n]/2,(CatalanNumber[n]+CatalanNumber[(n-1)/2])/2],{n,30}] (* Harvey P. Dale, Sep 06 2021 *)

catalan(n) = binomial(2*n, n)/(n+1);
a(n) = if (n % 2, (catalan(n) + catalan((n-1)/2))/2, catalan(n)/2); \\ Michel Marcus, Jan 23 2016

G.f.: (2-2*x-sqrt(1-4*x)-sqrt(1-4*x^2))/x/4. - Vladeta Jovovic, Sep 26 2003
D-finite with recurrence: n*(n+1)*a(n) -6*n*(n-1)*a(n-1) +4*(2*n^2-10*n+9)*a(n-2) +8*(n^2+n-9)*a(n-3) -48*(n-3)*(n-4)*a(n-4) +32*(2*n-9)*(n-5)*a(n-5)=0. - R. J. Mathar, Jun 03 2014, adapted to offset Feb 20 2020
a(n) ~ 4^n /(2*sqrt(Pi)*n^(3/2)). - Ilya Gutkovskiy, Jul 19 2016
a(2n) = A000150(2n). - R. J. Mathar, Jul 19 2016
a(n) = (A000108(n) + 2^n * binomial(1/2, (n+1)/2) * sin(Pi*n/2))/2. - Vladimir Reshetnikov, Oct 03 2016
Sum_{n>=1} a(n)/4^n = (3-sqrt(3))/2 (A334843). - Amiram Eldar, Mar 20 2022

Description corrected by Reiner Martin and Wouter Meeussen, Aug 04 2002

A007123 Number of connected unit interval graphs with n nodes; also number of bracelets (turnover necklaces) with n black beads and n-1 white beads.

Original entry on oeis.org

1, 1, 2, 4, 10, 26, 76, 232, 750, 2494, 8524, 29624, 104468, 372308, 1338936, 4850640, 17685270, 64834550, 238843660, 883677784, 3282152588, 12233309868, 45741634536, 171530482864, 644953425740, 2430975800876, 9183681736376, 34766785487152, 131873995933480

Offset: 1

N. J. A. Sloane

Also number of rooted planar general trees (of n vertices or n-1 edges) up to reflection. - Antti Karttunen, Aug 09 2002 (For the correspondence with bracelets, start by considering Raney's lemma as explained by Graham, Knuth & Patashnik.)
Number of connected lattice path matroids on n elements up to isomorphism.
a(n) = number of noncrossing set partitions of [n] up to reflection (i<->n+1-i). Example: a(4) counts 123, 1-23, 13-2, 1-2-3 but not 12-3 because it is the reflection of 1-23. - David Callan, Oct 08 2005
From Vladimir Shevelev, Apr 23 2011: (Start)
Also number of non-equivalent necklaces of n beads, each of which is painted by one of 2*n-1 colors.
The sequence solves the so-called Reis problem about convex k-gons in case N=2*n-1, k=n. H. Gupta (1979) gave a full solution; I gave a short proof of Gupta's result and showed an equivalence of this problem and each of the following problems: the problem of enumerating the bracelets of n beads of 2 colors, k of them black, and the problem of enumerating the necklaces of k beads, each painted by one of n colors.
a(n) is an essentially unimprovable upper estimate for the number of distinct values of the permanent in (0,1)-circulants of order 2*n-1 with n 1's in every row. (End)
The number of Dyck paths of semilength n-1 up to reversal; that is, the number of Dyck paths of semilength n-1, treating as identical a path and that path when traveled in reverse order. - Noah A Rosenberg, Jan 28 2019

			x + x^2 + 2*x^3 + 4*x^4 + 10*x^5 + 26*x^6 + 76*x^7 + 232*x^8 + 750*x^9 + ...

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6.7.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 345 & 346.
R. W. Robinson, personal communication.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1980.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..1670 (first 190 terms from R. W. Robinson)
J. E. Bonin, A. de Mier, and M. Noy, Lattice path matroids: enumerative aspects and Tutte polynomials, arXiv:math/0211188 [math.CO], 2002.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs., Vol. 3 (2000), #00.1.5.
Hansraj Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., 10 (1979), no. 8, 964-999.
Z. M. Himwich and N. A. Rosenberg, Roadblocked monotonic paths and the enumeration of coalescent histories for non-matching caterpillar gene trees and species trees, arXiv:1901.04465 [qbio.PE], 2019; Adv. Appl. Math. 113 (2020), 101939. (cf. Table 1)
Claudio Procesi, Some special bases of the 2-swap algebras, arXiv:2406.18687 [math.QA], 2024. See p. 3.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Vladimir Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629-638.
Vladimir Shevelev, Spectrum of permanent's values and its extremal magnitudes in Lambda_n^3 and Lambda_n(alpha,beta,gamma), arXiv:1104.4051 [math.CO], 2011. (Cf. Section 5).
Index entries for sequences related to bracelets

Cf. A000108, A002420, A007595, A063886, A073201.
Occurs as row 164 in A073201.
Next-to-center columns of triangle A052307.
Equal to A001405 plus A006079.

f[k_Integer, n_] := (Plus @@ (EulerPhi[ # ]Binomial[n/#, k/# ] & /@ Divisors[GCD[n, k]])/n + Binomial[(n - If[OddQ@n, 1, If[OddQ@k, 2, 0]])/2, (k - If[OddQ@k, 1, 0])/2])/2 (* Robert A. Russell, Sep 27 2004 *)
Table[ f[n, 2n - 1], {n, 10}]
(* Comment from Wouter Meeussen, Feb 02 2013, added by N. J. A. Sloane, Feb 02 2013: To get lists of the necklaces in Mathematica, use (if n=4, say):
<

{a(n) = if( n<1, 0, (2 * binomial(n-1, (n-1)\2) + binomial(2*n, n) / (2*n - 1)) / 4)} /* Michael Somos, Apr 16 2012 */

from sympy import catalan, binomial, floor
def a(n): return 1 if n==1 else (catalan(n - 1) + binomial(n - 1, floor((n - 1)/2)))/2 # Indranil Ghosh, Jun 03 2017

a(n+1) = (Catalan(n) + binomial(n, floor(n/2)))/2 = (A000108(n) + A001405(n))/2. - Antti Karttunen, Aug 09 2002
G.f.: (1 + 2*x - sqrt(1 - 4*x)*sqrt(1 - 4*x^2))/(4*sqrt(1 - 4*x^2)).
G.f.: (sqrt((1 + 2*x) / (1 - 2*x)) - sqrt(1 - 4*x)) / 4. - Michael Somos, Apr 16 2012
a(n) = (A063886(n) - A002420(n)) / 4. - Michael Somos, Apr 16 2012
D-finite with recurrence n*(n-1)*(n-4)*a(n) - 4*(n-1)*(n^2-5*n+5)*a(n-1) - 4*(n-2)*(n^2-7*n+11)*a(n-2) + 8*(2*n-7)*(n-2)*(n-3)*a(n-3)=0. - R. J. Mathar, Aug 22 2018

Extended by Christian G. Bower

Edited by Jon E. Schoenfield, Feb 14 2015

A057507 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A057505/A057506.

Original entry on oeis.org

1, 1, 1, 2, 3, 10, 18, 46, 95, 236, 528, 1288, 2936, 6984, 16212, 38528, 90717, 216648, 516358, 1240818, 2979992

Offset: 0

Antti Karttunen, Sep 03 2000

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

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

a(n) = A081148(n)+A081150(n). Bisections: A081151, A081167. Cf. A057545, A060114, A081164.

Occurs for first time in A073201 as row 2614.

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

A073191 Number of separate orbits/cycles to which the Catalan bijections A072796/A072797 partition each A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.

Original entry on oeis.org

1, 1, 2, 4, 11, 31, 96, 305, 1007, 3389, 11636, 40498, 142714, 507870, 1823040, 6591885, 23989419, 87795473, 322922652, 1193058230, 4425547638, 16475756738, 61539293424, 230548633954, 866095934598, 3261868457698, 12313423931624

Offset: 0

Antti Karttunen, Jun 25 2002

nonn

Occurs for first time in A073201 as row 1.

a(n) = (A000108(n)+A073190(n))/2.

A073193 Number of separate orbits/cycles to which the Catalan bijection A057508 partitions each A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.

Original entry on oeis.org

1, 1, 2, 4, 11, 30, 93, 292, 965, 3238, 11126, 38708, 136486, 485820, 1744677, 6310584, 22973793, 84103302, 309429066, 1143487428, 4242631626, 15798011604, 59018856522, 221143860936, 830895360978, 3129747395548, 11816242209260

Offset: 0

Antti Karttunen, Jun 25 2002

nonn

Occurs for first time in A073201 as row 168.

a(n) = (A000108(n)+A073192(n))/2

A073431 Number of separate orbits/cycles to which the Catalan bijections A069767/A069768 partition each A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 12, 28, 65, 160, 408, 1074, 2898, 7998, 22508, 64426, 187251, 551730, 1645840, 4964876, 15130808, 46545788, 144424944, 451715460

Offset: 0

Antti Karttunen, Jul 31 2002

nonn

Occurs for first time in A073201 as row 6 (and 8). Column sums of the square array A074079/Row sums of the triangle A074080.

A073431 := proc(n) local i; (1/2^n) * add((2^(n-i))*A073346bi(n,i),i=0..n); end;

a(0)=1, a(n) = (1/(2^(n-1))) * Sum_{i=1..(2^(n-1))} (Sum_{j=0..A007814(i)} A073346(n, j)) = (1/(2^(n-2))) * Sum_{i=1..(2^(n-1))} A073346(n, A007814(i)) - 1 = (1/2^n) * Sum_{i=0..n} (2^(n-i))*A073346(n, i) = Sum_{i=0..n} A074079(n, i)

A073203 Array of maximum cycle length sequences for the table A073200.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 3, 2, 1, 1, 2, 2, 2, 6, 2, 2, 1, 1, 2, 2, 2, 8, 2, 3, 2, 1, 1, 2, 2, 2, 10, 2, 6, 4, 1, 1, 1, 2, 2, 2, 12, 2, 8, 8, 1, 2, 1, 1, 2, 2, 2, 14, 2, 10, 16, 1, 4, 1, 1, 1, 2, 2, 2, 16, 2, 12, 32, 1, 8, 2, 2, 1, 1

Offset: 0

Antti Karttunen, Jun 25 2002

Each row of this table gives the longest cycle/orbit produced by the Catalan bijection (given in the corresponding row of A073200) when it acts on A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.

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

Cf. also A073201, A073202, A073204.
Few EIS-sequences which occur in this table. Only the first known occurrence(s) given:.
Rows 6 and 8: A011782, Row 7: A000012, Row 12, 14: A000793 (shifted right and prepended with 1), Row 261: A057543, Row 2614: A057545, Rows 2618, 17517: A057544.

cp's OEIS Frontend

A073200 Number of simple Catalan bijections of type B.

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A073202 Array of fix-count sequences for the table A073200.

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A007595 a(n) = C_n / 2 if n is even or ( C_n + C_((n-1)/2) ) / 2 if n is odd, where C = Catalan numbers (A000108).

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A007123 Number of connected unit interval graphs with n nodes; also number of bracelets (turnover necklaces) with n black beads and n-1 white beads.

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A057507 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A057505/A057506.

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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.

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A073191 Number of separate orbits/cycles to which the Catalan bijections A072796/A072797 partition each A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.

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A073193 Number of separate orbits/cycles to which the Catalan bijection A057508 partitions each A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.

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A073431 Number of separate orbits/cycles to which the Catalan bijections A069767/A069768 partition each A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.

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