A080237 -id:A080237 - cp's OEIS frontend

A085178 Array A(x,y) giving the position of the y-th x in A080237 listed by rising antidiagonals.

1, 3, 2, 8, 5, 4, 22, 13, 7, 6, 64, 36, 18, 10, 9, 196, 106, 50, 21, 12, 11, 625, 328, 148, 59, 27, 15, 14, 2055, 1054, 460, 176, 63, 32, 17, 16, 6917, 3485, 1483, 550, 190, 78, 35, 20, 19, 23713, 11779, 4915, 1780, 598, 195, 92, 41, 24, 23, 82499, 40509, 16641, 5916

Offset: 1

Antti Karttunen, Jun 18 2003

Read by upwards antidiagonals as A(1,1), A(2,1), A(1,2), A(3,1), A(2,2), A(1,3), etc.

Index entries for sequences that are permutations of the natural numbers

Transpose: A085176.
Inverse permutation: A085179.
The first row: A014138, the first column: A085223.
Variant: A085180.

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

A007001 Trajectory of 1 under the morphism 1 -> 12, 2 -> 123, 3 -> 1234, etc.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 1, 2, 3, 1, 2

Offset: 1

N. J. A. Sloane, Mira Bernstein, James Propp

Records in this sequence occur at positions: 1, 2, 5, 14, 42, 132, 429, 1430, ... (which appear to be the Catalan numbers A000108). - Robert G. Wilson v, May 07 2005
The records do occur at Catalan numbers. Of the first C(n) numbers, the number that are equal to k is A033184(n,k), with the one n last. - Franklin T. Adams-Watters, Mar 29 2009
Let (T(1) < T(2) < ... < T(A000108(m))) denote the sequence of Young tableaux of shape (2^m) ordered lexicographically with respect to their columns, and let f(T(i), T(j)) denote the first label of disagreement among T(i) and T(j). Then, empirically, if we take away the zeros from (f(T(1), T(A000108(m) - i + 1)) - f(T(A000108(m) - i), T(A000108(m) - i + 1)), i=1..A000108(m)-1), we obtain the first A000108(m - 1) - 1 terms in this sequence. This is illustrated in the below example. - John M. Campbell, Sep 07 2018
The average of the first k terms tends to 3 as k tends to infinity. - Andrew Slattery, Jan 19 2021

			From _John M. Campbell_, Sep 07 2018: (Start)
Letting m = 5, as above let (T(1) < T(2) < ... < T(42)) denote the lexicographic sequence of Young tableaux of shape (2, 2, 2, 2, 2). In this case, the sequence (f(T(1), T(43 - i)) - f(T(42 - i), T(43 - i)), i=1..41) is equal to (0, 1, 0, 0, 2, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 2, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0). Removing the zeroes from this tuple, we obtain (1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 3), which gives us the first 13 = A000108(m - 1) - 1 terms in this sequence. For example, the first term in the preceding tuple is 0 since T(1) and T(42) are respectively
   [ 5 10] [ 9 10]
   [ 4 9 ] [ 7 8 ]
   [ 3 8 ] [ 5 6 ]
   [ 2 7 ] [ 3 4 ]
   [ 1 6 ] [ 1 2 ]
and T(41) is equal to
   [ 9 10]
   [ 7 8 ]
   [ 5 6 ]
   [ 2 4 ]
   [ 1 3 ]
so that the first letter of disagreement between T(1) and T(42) is 2, and that between T(41) and T(42) is also 2. (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. West, Generating trees and forbidden subsequences, Proc. 6th FPSAC [ Conference on Formal Power Series and Algebraic Combinatorics ] (1994), pp. 441-450 (see p. 443).

T. D. Noe, Table of n, a(n) for n = 1..4862 (8 iterations)
C. Banderier, A. Denise, P. Flajolet, M. Bousquet-Mélou et al., Generating Functions for Generating Trees, Discrete Mathematics 246(1-3), March 2002, pp. 29-55.
Antti Karttunen, Notes concerning A080237-tree and related sequences.
S. Lehr, J. Shallit and J. Tromp, On the vector space of the automatic reals, Theoret. Comput. Sci. 163 (1996), no. 1-2, 193-210.
James Propp and N. J. A. Sloane, Email, March 1994
Index entries for sequences that are fixed points of mappings

Cf. A000245, A085182. a(n)=A076050(n)-1. Partial sums: A080336. Positions of ones: A085197. The first occurrence of each n is at A000108(n). See A085180.

Nest[ Flatten[ # /. a_Integer -> Range[a + 1]] &, {1}, 6] (* Robert G. Wilson v, Jan 24 2006 *)

a(n)=local(v,w); if(n<1,0,v=[1]; while(#v

From n > 1 onward a(n) = A080237(A081291(n-1)). - Antti Karttunen, Jul 31 2003

More terms from Larry Reeves (larryr(AT)acm.org), Sep 22 2000

A057503 Signature-permutation of a Catalan Automorphism: Deutsch's 1998 bijection on Dyck paths.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Sep 03 2000

nonn

Deutsch shows in his 1998 paper that this automorphism maps the number of returns of Dyck path to the height of the last peak, i.e., that A057515(n) = A080237(A057503(n)) holds for all n, thus the two parameters have the same distribution.

From the recursive forms of A057161 and A057503 it is seen that both can be viewed as a convergent limits of a process where either the left or right side argument of A085201 in formula for A057501 is "iteratively recursivized", and on the other hand, both of these can then in turn be made to converge towards A057505, when the other side of the formula is also "recursivized" in the same way. - Antti Karttunen, Jun 06 2014

Antti Karttunen, Table of n, a(n) for n = 0..2055
Emeric Deutsch, A bijection on Dyck paths and its consequences, Discrete Math., 179 (1998), 253-256.
Antti Karttunen, C program which computes this sequence..
Antti Karttunen, Introductory Survey of Catalan Automorphisms and Bijections (an unfinished draft), pp. 53-54.
Index entries for signature-permutations of Catalan automorphisms

Inverse: A057504. Row 17 of A122285. Cf. A057501, A057161, A057505.

The number of cycles, count of the fixed points, maximum cycle sizes and LCM's of all cycle sizes in range [A014137(n-1)..A014138(n)] of this permutation are given by LEFT(LEFT(A001683)), LEFT(A019590), A057544 and A057544, the same sequences as for A057162 because this is a conjugate of it (cf. the Formula section).

a(0) = 0, and for n >= 1, a(n) = A085201(A072771(n), A057548(a(A072772(n)))). [This formula 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 the unary form of function 'list'].
a(n) = A057164(A057162(A057164(n))). [For the proof, see pp. 53-54 in the "Introductory survey ..." draft, eq. 144.]
Other identities:
A057515(n) = A080237(a(n)) holds for all n. [See the Comments section.]

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

A057504 Signature-permutation of the inverse of Deutsch's 1998 bijection on Dyck paths.

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 8, 5, 4, 17, 16, 18, 15, 14, 20, 19, 21, 12, 11, 22, 13, 10, 9, 45, 44, 46, 43, 42, 48, 47, 49, 40, 39, 50, 41, 38, 37, 54, 53, 55, 52, 51, 57, 56, 58, 31, 30, 59, 32, 29, 28, 61, 60, 62, 34, 33, 63, 35, 26, 25, 64, 36, 27, 24, 23, 129, 128, 130, 127, 126

Offset: 0

Antti Karttunen, Sep 03 2000

nonn

Emeric Deutsch, A bijection on Dyck paths and its consequences, Discrete Math., 179 (1998), 253-256.
Antti Karttunen, C program which computes this sequence.
Index entries for signature-permutations of Catalan automorphisms

Inverse: A057503. Row 12 of A122286.

A080237(n) = A057515(a(n)) holds for all n. See comment at A057503.

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

A085197 Positions of ones in A007001. Repeating part in each sub-permutation A082315[A014137(n-1)..A014138(n-1)] normalized to begin from 1.

Original entry on oeis.org

1, 3, 6, 8, 11, 15, 17, 20, 22, 25, 29, 31, 34, 38, 43, 45, 48, 50, 53, 57, 59, 62, 64, 67, 71, 73, 76, 80, 85, 87, 90, 92, 95, 99, 101, 104, 108, 113, 115, 118, 122, 127, 133, 135, 138, 140, 143, 147, 149, 152, 154, 157, 161, 163, 166, 170, 175, 177, 180, 182, 185, 189

Offset: 1

Antti Karttunen, Jun 14 2003. Proposed by Wouter Meeussen Mar 15 2003

nonn

From the second term 3 onward also one more than the partial sums of A076050.

Paolo Xausa, Table of n, a(n) for n = 1..16796
Antti Karttunen, Notes concerning A080237-tree and related sequences.

Cf. A085196. First column of A085180.

PositionIndex[Nest[Flatten[Map[Range[#+1] &, #]] &, {1}, 6]][[1]] (* Paolo Xausa, Mar 04 2024 *)

a(n) = A080336(n-1) + n = A082854(A082315(A072795(A081291(n-1)))).

a(n) = n if n < 2, otherwise a(n-1)+A076050(n-1).

A126307 a(n) is the length of the leftmost ascent (i.e., height of the first peak) in the n-th Dyck path encoded by A014486(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 2, 2, 3, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Antti Karttunen, Jan 02 2007

nonn

In other words, this sequence gives the number of leading 1's in the terms of A063171.

			A014486(20) = 228 (11100100 in binary), encodes the following Dyck path:
    /\
   /  \/\
  /      \
and the first rising (left-hand side) slope has length 3, thus a(20)=3.

a(n) = A099563(A071156(n)) = A057515(A125985(n)) = A080237(A057164(n)) = A057515(A057504(A057164(n))).

a(n) = A090996(A014486(n)).

A179752 Maximum depth of parenthesizations encoded by A014486, or correspondingly, maximum height for the equivalent general trees.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 2, 3, 1, 2, 2, 2, 3, 2, 2, 2, 2, 3, 3, 3, 3, 4, 1, 2, 2, 2, 3, 2, 2, 2, 2, 3, 3, 3, 3, 4, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 1, 2, 2, 2, 3, 2, 2, 2, 2, 3, 3, 3, 3, 4, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 4

Offset: 0

Antti Karttunen, Aug 03 2010

Each integer n appears first at position given by A014138.

			The terms A014486[1..8] encode the following rooted plane general trees:
.1.......2.......3.......4.......5.......6.......7.......8.
...........................................................
.........................................................o.
.........................................................|.
.................o.................o...o.......o...o.....o.
.................|.................|...|........\./......|.
.o.....o...o.....o.....o.o.o...o...o...o...o.....o.......o.
.|......\./......|......\|/.....\./.....\./......|.......|.
.*.......*.......*.......*.......*.......*.......*.......*.
and the corresponding parenthesizations:
.().....()()....(())...()()()..()(())..(())()..(()())..((()))
thus a(1)=1, a(2)=1, a(3)=2, a(4)=1, a(5)=2, a(6)=2, a(7)=2, a(8)=3.

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

Cf. A179751, A126303, A126304, A080237, A085197.

blist[m_] := Select[Map[Accumulate, Permutations[PadLeft[Table[1, m], 2*m, -1]]], Min[#] >= 0 &]; Join[{{0}}, Array[Map[Max, blist[#]] &, 6]] (* Paolo Xausa, Mar 04 2024 *)

A085193 Repeating part of A085192.

Original entry on oeis.org

2, 6, 2, 4, 18, 2, 6, 2, 4, 10, 2, 4, 8, 58, 2, 6, 2, 4, 18, 2, 6, 2, 4, 10, 2, 4, 8, 26, 2, 6, 2, 4, 10, 2, 4, 8, 18, 2, 4, 8, 16, 202, 2, 6, 2, 4, 18, 2, 6, 2, 4, 10, 2, 4, 8, 58, 2, 6, 2, 4, 18, 2, 6, 2, 4, 10, 2, 4, 8, 26, 2, 6, 2, 4, 10, 2, 4, 8, 18, 2, 4, 8, 16, 74, 2, 6, 2, 4, 18, 2, 6, 2, 4, 10

Offset: 0

Antti Karttunen, Jun 14 2003

nonn

Antti Karttunen, Notes concerning A080237-tree and related sequences.

Same sequence divided by 2: A085194. Cf. A085190.

a(n) = A085192(A081291(n+1)-1).

a(n) = 4*A085193(A085182(n+1)-1) + 2 - (2^A007001(n+1)) if A007001(n+2)==1, otherwise 2^A007001(n+1).

A080336 Partial sums of A007001.

Original entry on oeis.org

0, 1, 3, 4, 6, 9, 10, 12, 13, 15, 18, 19, 21, 24, 28, 29, 31, 32, 34, 37, 38, 40, 41, 43, 46, 47, 49, 52, 56, 57, 59, 60, 62, 65, 66, 68, 71, 75, 76, 78, 81, 85, 90, 91, 93, 94, 96, 99, 100, 102, 103, 105, 108, 109, 111, 114, 118, 119, 121

Offset: 0

Benoit Cloitre, Mar 18 2003

nonn

A. Karttunen, Notes concerning A080237-tree and related sequences.

The repeating subsequences in A085196 increase to this sequence. Difference between the position of (n+1)-th 1 in A007001 (= A085197(n)) and A007001(n+1). Also a(n) = A085196(A081291(n)).

cp's OEIS Frontend

A085178 Array A(x,y) giving the position of the y-th x in A080237 listed by rising antidiagonals.

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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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A007001 Trajectory of 1 under the morphism 1 -> 12, 2 -> 123, 3 -> 1234, etc.

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A057503 Signature-permutation of a Catalan Automorphism: Deutsch's 1998 bijection on Dyck paths.

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A057504 Signature-permutation of the inverse of Deutsch's 1998 bijection on Dyck paths.

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A085197 Positions of ones in A007001. Repeating part in each sub-permutation A082315[A014137(n-1)..A014138(n-1)] normalized to begin from 1.

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A126307 a(n) is the length of the leftmost ascent (i.e., height of the first peak) in the n-th Dyck path encoded by A014486(n).

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A179752 Maximum depth of parenthesizations encoded by A014486, or correspondingly, maximum height for the equivalent general trees.

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