A080300 -id:A080300 - cp's OEIS frontend

A080320 a(n) = A057118(A080295(n)) = A080300(A080318(n)). Positions of A080318 in A014486.

1, 8, 625, 82461, 2414517826, 465696894874, 19586243923520645, 911881322544255111111, 2344958374133795816706574529598, 540549352213084909964319555106232

Offset: 0

Antti Karttunen, Mar 02 2003

nonn

Cf. A080321.

A014486 List of totally balanced sequences of 2n binary digits written in base 10. Binary expansion of each term contains n 0's and n 1's and reading from left to right (the most significant to the least significant bit), the number of 0's never exceeds the number of 1's.

Original entry on oeis.org

0, 2, 10, 12, 42, 44, 50, 52, 56, 170, 172, 178, 180, 184, 202, 204, 210, 212, 216, 226, 228, 232, 240, 682, 684, 690, 692, 696, 714, 716, 722, 724, 728, 738, 740, 744, 752, 810, 812, 818, 820, 824, 842, 844, 850, 852, 856, 866, 868, 872, 880, 906, 908, 914

Offset: 0

Wouter Meeussen

The binary Dyck-Language (A063171) in decimal representation.
These encode width 2n mountain ranges, rooted planar trees of n+1 vertices and n edges, planar planted trees with n nodes, rooted plane binary trees with n+1 leaves (2n edges, 2n+1 vertices, n internal nodes, the root included), Dyck words, binary bracketings, parenthesizations, non-crossing handshakes and partitions and many other combinatorial structures in Catalan family, enumerated by A000108.
Is Sum_{k=1..n} a(k) / n^(5/2) bounded? - Benoit Cloitre, Aug 18 2002
This list is the intersection of A061854 and A031443. - Jason Kimberley, Jan 18 2013
The sequence does start at n = 0, since in the binary interpretation of the Dyck language (e.g., as parenthesizations where "1" stands for "(" and "0" stands for ")") having a(0) = 0 will do since it would stand for the empty string where the "0"s and "1"s are balanced (hence the parentheses are balanced). - Daniel Forgues, Feb 17 2013
It appears that for n>=1 this sequence can be obtained by concatenating the terms of the irregular array whose n-th row length is A000108(n) and that is defined recursively by B(n,0) = A020988(n) and B(n,k) = B(n, k-1) + D(n, k-1) where D(x,y) = (2^(2*(A089309(B(x,y))-1))-1)*(2/3) + 2^A007814(B(x,y)). - Raúl Mario Torres Silva and Michel Marcus, May 01 2020
This encoding is related to the ranking by standard ordered tree numbers in that (1) the binary encoding of trees ordered by standard ranking is given by A358505, while (2) the standard ranking of trees ordered by binary encoding is given by A358523. - Gus Wiseman, Nov 21 2022

			a(19) = 226_10 = 11100010_2 = A063171(19) as bracket expression: ( ( ( ) ) )( ) and as a binary tree, proceeding from left to right in depth-first fashion, with 1's in binary expansion standing for internal (branching) nodes and 0's for leaves:
  0   0
   \ /
    1   0 0  (0)
     \ /   \ /
      1     1
       \   /
         1
Note that in this coding scheme the last leaf of the binary trees (here in parentheses) is implicit. This tree can be also converted to a particular S-expression in languages like Lisp, Scheme and Prolog, if we interpret its internal nodes (1's) as cons cells with each leftward leaning branch being the "car" and the rightward leaning branch the "cdr" part of the pair, with the terminal nodes (0's) being ()'s (NILs). Thus we have (cons (cons (cons () ()) ()) (cons () ())) = '( ( ( () . () ) . () ) . ( () . () ) ) = (((())) ()) i.e., the same bracket expression as above, but surrounded by extra parentheses. This mapping is performed by the Scheme function A014486->parenthesization given below.
From _Gus Wiseman_, Nov 21 2022: (Start)
The terms and corresponding ordered rooted trees begin:
    0: o
    2: (o)
   10: (oo)
   12: ((o))
   42: (ooo)
   44: (o(o))
   50: ((o)o)
   52: ((oo))
   56: (((o)))
  170: (oooo)
  172: (oo(o))
  178: (o(o)o)
  180: (o(oo))
  184: (o((o)))
(End)

Donald E. Knuth, The Art of Computer Programming, Vol. 4A: Combinatorial Algorithms, Part 1, Addison-Wesley, 2011, Section 7.2.1.6, pp. 443 (Algorithm P).

Paolo Xausa, Table of n, a(n) for n = 0..23713 (terms 0..2500 from Franklin T. Adams-Watters).
Jason Bell, Thomas Finn Lidbetter, and Jeffrey Shallit, Additive Number Theory via Approximation by Regular Languages, arXiv:1804.07996 [cs.FL], 2018.
N. G. De Bruijn and B. J. M. Morselt, A note on plane trees, J. Combinatorial Theory 2 (1967), 27-34.
Gennady Eremin, Dynamics of balanced parentheses, lexicographic series and Dyck polynomials, arXiv:1909.07675 [math.CO], 2019.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
Antti Karttunen, Catalan ranking and unranking functions, OEIS Wiki.
Antti Karttunen, Illustration of 626 initial terms (up to size n=7) with various combinatorial interpretations of Catalan numbers encoded by this sequence.
Antti Karttunen, a089408.c - C program for computing this sequence and many of the related automorphisms.
Antti Karttunen, Some notes on Catalan's Triangle.
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625 [math.CO], 2020-2021.
D. L. Kreher and D. R. Stinson, Combinatorial Algorithms, Generation, Enumeration and Search, CRC Press, 1998.
Thomas Finn Lidbetter, Counting, Adding, and Regular Languages, Master's Thesis, University of Waterloo, Ontario, Canada, 2018.
R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016.
OEIS Wiki, Combinatorial interpretations of Catalan numbers
Frank Ruskey, Algorithmic Solution of Two Combinatorial Problems.
R. P. Stanley, Hipparchus, Plutarch, Schroeder and Hough, Am. Math. Monthly, Vol. 104, No. 4, p. 344, 1997.
R. P. Stanley, Exercises on Catalan and Related Numbers.
Paolo Xausa, A Mathematica implementation of Knuth's algorithms P and U.
Index entries for encodings of plane rooted trees (various subsets of this sequence).
Index entries for sequences related to parenthesizing
Index entries for signature-permutations induced by Catalan automorphisms (permutations of natural numbers induced by various bijective operations acting on these structures)
Index entries for the sequences induced by list functions of Lisp (sequences induced by various other operations on these codes or the corresponding structures).

Characteristic function: A080116. Inverse function: A080300.
The terms of binary width 2n are counted by A000108(n). Subset of A036990. Number of peaks in each mountain (number of leaves in rooted plane general trees): A057514. Number of trailing zeros in the binary expansion: A080237. First differences: A085192.
Cf. also A009766, A014137, A071156, A079436, A085184, A213704.
Cf. A020988, A089309, A007814.
Branches of the ordered tree are counted by A057515.
Edges of the ordered tree are counted by A072643.
The Matula-Goebel number of the ordered tree is A127301.
The standard ranking of the ordered tree is A358523.
The depth of the ordered tree is A358550.
Nodes of the ordered tree are counted by A358551.
Cf. A000081, A001263, A358505, A358524.

# Maple procedure CatalanUnrank is adapted from the algorithm 3.24 of the CAGES book and the Scheme function CatalanUnrank from Ruskey's thesis. See the a089408.c program for the corresponding C procedures.
CatalanSequences := proc(upto_n) local n,a,r; a := []; for n from 0 to upto_n do for r from 0 to (binomial(2*n,n)/(n+1))-1 do a := [op(a),CatalanUnrank(n,r)]; od; od; return a; end;
CatalanUnrank := proc(n,rr) local r,x,y,lo,m,a; r := (binomial(2*n,n)/(n+1))-(rr+1); y := 0; lo := 0; a := 0; for x from 1 to 2*n do m := Mn(n,x,y+1); if(r <= lo+m-1) then y := y+1; a := 2*a + 1; else lo := lo+m; y := y-1; a := 2*a; fi; od; return a; end;
Mn := (n,x,y) -> binomial(2*n-x,n-((x+y)/2)) - binomial(2*n-x,n-1-((x+y)/2));
# Alternative:
bin := n -> ListTools:-Reverse(convert(n, base, 2)):
isA014486 := proc(n): local B, s, b; s := 0;
    if n > 0 then
      for b in bin(n) do
          s := s + ifelse(b = 1, 1, -1);
           if 0 > s then return false fi;
      od fi;
  s = 0 end:
select(isA014486, [seq(0..240)]);  # Peter Luschny, Mar 13 2024

cat[ n_ ] := (2 n)!/n!/(n+1)!; b2d[li_List] := Fold[2#1+#2&, 0, li]
d2b[n_Integer] := IntegerDigits[n, 2]
tree[n_] := Join[Table[1, {i, 1, n}], Table[0, {i, 1, n}]]
nexttree[t_] := Flatten[Reverse[t]/. {a___, 0, 0, 1, b___}:> Reverse[{Sort[{a, 0}]//Reverse, 1, 0, b}]]
wood[ n_ /; n<8 ] := NestList[ nexttree, tree[ n ], cat[ n ]-1 ]
Table[ Reverse[ b2d/@wood[ j ] ], {j, 0, 6} ]//Flatten
(* Alternative code *)
tbQ[n_]:=Module[{idn2=IntegerDigits[n,2]},Count[idn2,1]==Length[idn2]/2&&Min[Accumulate[idn2/.{0->-1}]]>=0]; Join[{0},Select[Range[900],tbQ]] (* Harvey P. Dale, Jul 04 2013 *)
balancedQ[0] = True; balancedQ[n_] := Module[{s = 0}, Do[s += If[b == 1, 1, -1]; If[s < 0, Return[False]], {b, IntegerDigits[n, 2]}]; Return[s == 0] ]; A014486 = FromDigits /@ IntegerDigits[Select[Range[0, 1000], balancedQ ]] (* Jean-François Alcover, Mar 05 2016 *)
A014486Q[0] = True; A014486Q[n_] := Catch[Fold[If[# < 0, Throw[False], If[#2 == 0, # - 1, # + 1]] &, 0, IntegerDigits[n, 2]] == 0]; Select[Range[0, 880], A014486Q] (* JungHwan Min, Dec 11 2016 *)
(* Uses Algorithm P from Knuth's TAOCP section 7.2.1.6 - see References and Links. *)
alist[n_] := Block[{a = Flatten[Table[{1, 0}, n]], m = 2*n - 1, j, k},
    FromDigits[#, 2]& /@ Reap[
    While[True,
        Sow[a]; a[[m]] = 0;
        If[a[[m - 1]] == 0,
            a[[--m]] = 1, j = m - 1; k = 2*n - 1;
            While[j > 1 && a[[j]] == 1, a[[j--]] = 0; a[[k]] = 1; k -= 2];
            If[j == 1, Break[]];
            a[[j]] = 1; m = 2*n - 1]
    ]][[2, 1]]];
Join[{{0}, {2}}, Array[alist, 4, 2]] (* Paolo Xausa, Mar 16 2024 *)

isA014486(n)=my(v=binary(n),t=0);for(i=1,#v,t+=if(v[i],1,-1);if(t<0,return(0))); t==0 \\ Charles R Greathouse IV, Jun 10 2011

a_rows(N) = my(a=Vec([[0]], N)); for(r=1, N-1, my(b=a[r], c=List()); foreach(b, t, my(v=if(t, valuation(t, 2), 0)); for(i=0, v, listput(~c, (t<<2)+(2<Ruud H.G. van Tol, May 16 2024

from itertools import count, islice
from sympy.utilities.iterables import multiset_permutations
def A014486_gen(): # generator of terms
    yield 0
    for l in count(1):
        for s in multiset_permutations('0'*l+'1'*(l-1)):
            c, m = 0, (l<<1)-1
            for i in range(m):
                if s[i] == '1':
                    c += 2
                if cA014486_list = list(islice(A014486_gen(),30)) # Chai Wah Wu, Nov 28 2023

def is_A014486(n) :
    B = bin(n)[2::] if n != 0 else 0
    s = 0
    for b in B :
        s += 1 if b=='1' else -1
        if 0 > s : return False
    return 0 == s
def A014486_list(n): return [k for k in (1..n) if is_A014486(k) ]
A014486_list(888) # Peter Luschny, Aug 10 2012

Additional comments from Antti Karttunen, Aug 11 2000 and May 25 2004

Added a(0)=0 (which had been removed in June 2011), Joerg Arndt, Feb 27 2013

A057163 Signature-permutation of a Catalan automorphism: Reflect a rooted plane binary tree; Deutsch's 1998 involution on Dyck paths.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Aug 18 2000

nonn

Deutsch shows in his 1999 paper that this automorphism maps the number of doublerises of Dyck paths to number of valleys and height of the first peak to the number of returns, i.e., that A126306(n) = A127284(a(n)) and A126307(n) = A057515(a(n)) hold for all n.
The A000108(n-2) n-gon triangularizations can be reflected over n axes of symmetry, which all can be generated by appropriate compositions of the permutations A057161/A057162 and A057163.
Composition with A057164 gives signature permutation for Donaghey's Map M (A057505/A057506). Embeds into itself in scale n:2n+1 as a(n) = A083928(a(A080298(n))). A127302(a(n)) = A127302(n) and A057123(A057163(n)) = A057164(A057123(n)) hold for all n.

			This involution (self-inverse permutation) of natural numbers is induced when we reflect the rooted plane binary trees encoded by A014486. E.g., we have A014486(5) = 44 (101100 in binary), A014486(7) = 52 (110100 in binary) and these encode the following rooted plane binary trees, which are reflections of each other:
    0   0             0   0
     \ /               \ /
      1   0         0   1
       \ /           \ /
    0   1             1   0
     \ /               \ /
      1                 1
thus a(5)=7 and a(7)=5.

JungHwan Min, Table of n, a(n) for n = 0..10000
Emeric Deutsch, An involution on Dyck paths and its consequences, Discrete Math., 204 (1999), no. 1-3, 163-166.
Indranil Ghosh, Python program for computing this sequence, translated from Maple code.
Antti Karttunen, C program which computes this sequence.
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020.
Index entries for signature-permutations induced by Catalan automorphisms

This automorphism conjugates between the car/cdr-flipped variants of other automorphisms, e.g., A057162(n) = a(A057161(a(n))), A069768(n) = a(A069767(a(n))), A069769(n) = a(A057508(a(n))), A069773(n) = a(A057501(a(n))), A069774(n) = a(A057502(a(n))), A069775(n) = a(A057509(a(n))), A069776(n) = a(A057510(a(n))), A069787(n) = a(A057164(a(n))).

Row 1 of tables A122201 and A122202, that is, obtained with FORK (and KROF) transformation from even simpler automorphism *A069770. Cf. A122351.

a(n) = A080300(ReflectBinTree(A014486(n)))
ReflectBinTree := n -> ReflectBinTree2(n)/2; ReflectBinTree2 := n -> (`if`((0 = n),n,ReflectBinTreeAux(A030101(n))));
ReflectBinTreeAux := proc(n) local a,b; a := ReflectBinTree2(BinTreeLeftBranch(n)); b := ReflectBinTree2(BinTreeRightBranch(n)); RETURN((2^(A070939(b)+A070939(a))) + (b * (2^(A070939(a)))) + a); end;
NextSubBinTree := proc(nn) local n,z,c; n := nn; c := 0; z := 0; while(c < 1) do z := 2*z + (n mod 2); c := c + (-1)^n; n := floor(n/2); od; RETURN(z); end;
BinTreeLeftBranch := n -> NextSubBinTree(floor(n/2));
BinTreeRightBranch := n -> NextSubBinTree(floor(n/(2^(1+A070939(BinTreeLeftBranch(n))))));

A014486Q[0] = True; A014486Q[n_] := Catch[Fold[If[# < 0, Throw[False], If[#2 == 0, # - 1, # + 1]] &, 0, IntegerDigits[n, 2]] == 0]; tree[n_] := Block[{func, num = Append[IntegerDigits[n, 2], 0]}, func := If[num[[1]] == 0, num = Drop[num, 1]; 0, num = Drop[num, 1]; 1[func, func]]; func]; A057163L[n_] := Function[x, FirstPosition[x, FromDigits[Most@Cases[tree[#] /. 1 -> Reverse@*1, 0 | 1, All, Heads -> True], 2]][[1]] - 1 & /@ x][Select[Range[0, 2^n], A014486Q]]; A057163L[11] (* JungHwan Min, Dec 11 2016 *)

a(n) = A083927(A057164(A057123(n))).

Equivalence with Deutsch's 1998 involution realized Dec 15 2006 and entry edited accordingly by Antti Karttunen, Jan 16 2007

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

Original entry on oeis.org

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
```
See Links section.
```

a(n) = A080300(A036044(A014486(n))) = A080300(A056539(A014486(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

A057548 A014486-indices of Catalan mountain ranges with no sea-level valleys, i.e., the rooted plane general trees with root degree = 1.

Original entry on oeis.org

1, 3, 7, 8, 17, 18, 20, 21, 22, 45, 46, 48, 49, 50, 54, 55, 57, 58, 59, 61, 62, 63, 64, 129, 130, 132, 133, 134, 138, 139, 141, 142, 143, 145, 146, 147, 148, 157, 158, 160, 161, 162, 166, 167, 169, 170, 171, 173, 174, 175, 176, 180, 181, 183, 184, 185, 187, 188

Offset: 0

Antti Karttunen, Sep 07 2000

nonn

This sequence is induced by the unary form of function 'list' (present in Lisp and Scheme) when it acts on symbolless S-expressions encoded by A014486/A063171.

Index entries for the sequences induced by list functions of Lisp

We have A057515(A057548(n)) = 1 for all n. Row 0 of A072764. Column 1 of A085203. Cf. A057517, A057549, A057551.

a(n) = A080300(A057547(n)) = A069770(A072795(n)).

A081291 Complement of A072795.

Original entry on oeis.org

0, 3, 6, 7, 8, 14, 15, 16, 17, 18, 19, 20, 21, 22, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128

Offset: 0

Antti Karttunen, Mar 17 2003

nonn

This gives positions of those terms in A063171 which begin with digits 11... Also the elements of table A072764 which do not occur in the leftmost column. See the comment at A081292.

Cf. A072795, A014137, A081288.

Cf. A080300, A081289, A081290, A081292.

A014137(n) = sum(k=0, n, binomial(2*k, k)/(k+1););
A081288(n) = my(i=0); while(binomial(2*i, i)/(i+1) <= n, i++); i;
a(n) = if (n==0, 0,  n + A014137(A081288(n)-1)); \\ Michel Marcus, Apr 28 2020

a(0)=0, a(n) = n + A014137(A081288(n)-1).

a(n) = A080300(A081292(n)) = A081289(n) + n - A081290(n).

A072795 A014486-indices of the plane binary trees AND plane general trees whose left subtree is just a stick: \. thus corresponding to the parenthesizations whose first element (of the top-level list) is an empty parenthesization: ().

Original entry on oeis.org

1, 2, 4, 5, 9, 10, 11, 12, 13, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 197, 198, 199

Offset: 0

Antti Karttunen Jun 12 2002

nonn

This sequence is induced by the 'flipped form' of the function 'list': (define (flippedlist x) (cons '() x)) when it acts on symbolless S-expressions encoded by A014486/A063171.

Gives in A063171 positions of the terms which begin with digits 10...

Column 0 of A072764, row 0 of A072766, column 1 of A085201. Complement: A081291. Cf. A085223.

Range[0, Length[#]-1] + CatalanNumber[#] & [Flatten[Array[Table[#, CatalanNumber[#]] &, 7, 0]]] (* Paolo Xausa, Mar 01 2024 *)

a(n) = n + A000108(A072643(n)) = A069770(A057548(n)) = A080300(A083937(n))

A072764 Tabular N X N -> N bijection induced by Lisp/Scheme function 'cons' combining the two planar binary trees/general trees/parenthesizations encoded by A014486(X) and A014486(Y).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 4, 8, 16, 14, 5, 17, 19, 42, 15, 9, 18, 44, 51, 43, 37, 10, 20, 47, 126, 52, 121, 38, 11, 21, 53, 135, 127, 149, 122, 39, 12, 22, 56, 154, 136, 385, 150, 123, 40, 13, 45, 60, 163, 155, 413, 386, 151, 124, 41, 23, 46, 128, 177, 164, 475, 414, 387, 152

Offset: 0

Antti Karttunen Jun 12 2002

Index entries for the sequences induced by list functions of Lisp
Index entries for sequences that are permutations of the natural numbers
A. Karttunen, Gatomorphisms (with the complete Scheme source)

Inverse permutation: A072765. a(n) = A069770(A072766(n)). Also transpose of A072766, i.e. a(n) = A072766(A038722(n)). The upper triangular region: A072773. Projection functions are A072771 ('car') & A072772 ('cdr'). The sizes of the corresponding Catalan structures: A072768. The first row: A057548, the first column: A072795, diagonal: A083938. Cf. also A080300, A025581, A002262.

a(0)=0 prepended by Sean A. Irvine, Oct 25 2024

A125976 Signature-permutation of Kreweras' 1970 involution on Dyck paths.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jan 02 2007

nonn

Lalanne shows in the 1992 paper that this automorphism preserves the sum of peak heights, i.e., that A126302(a(n)) = A126302(n) for all n. Furthermore, he also shows that A126306(a(n)) = A057514(n)-1 and likewise, that A057514(a(n)) = A126306(n)+1, for all n >= 1.
Like A069772, this involution keeps symmetric Dyck paths symmetric, but not necessarily same.
The number of cycles and fixed points in range [A014137(n-1)..A014138(n-1)] of this involution seem to be given by A007595 and the "aerated" Catalan numbers [1, 1, 0, 1, 0, 2, 0, 5, 0, 14, 0, 42, ...], thus this is probably a conjugate of A069770 (as well as of A057163).

A. Karttunen, Table of n, a(n) for n = 0..2055
G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationelle, Cahier no. 15, Paris, 1970, pp. 3-41.
J.-C. Lalanne, Une Involution sur les Chemins de Dyck, European J. Combin. 13 (1992), no. 6, 477-487.
Index entries for signature-permutations of Catalan automorphisms

Cf. A080300, A125974, A014486.
Cf. A057163, A069770, A007595, A014137, A014138.
Compositions and conjugations with other automorphisms: A125977-A125979, A125980, A126290.

a(n) = A080300(A125974(A014486(n))).

cp's OEIS Frontend

A080320 a(n) = A057118(A080295(n)) = A080300(A080318(n)). Positions of A080318 in A014486.

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A014486 List of totally balanced sequences of 2n binary digits written in base 10. Binary expansion of each term contains n 0's and n 1's and reading from left to right (the most significant to the least significant bit), the number of 0's never exceeds the number of 1's.

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A057163 Signature-permutation of a Catalan automorphism: Reflect a rooted plane binary tree; Deutsch's 1998 involution on Dyck paths.

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

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A057548 A014486-indices of Catalan mountain ranges with no sea-level valleys, i.e., the rooted plane general trees with root degree = 1.

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A081291 Complement of A072795.

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A072795 A014486-indices of the plane binary trees AND plane general trees whose left subtree is just a stick: \. thus corresponding to the parenthesizations whose first element (of the top-level list) is an empty parenthesization: ().

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