A389577 Decreasing partition array based on the tails of the binary Champernowne constant (A030190); see Comments.

1, 2, 4, 3, 5, 9, 7, 6, 10, 11, 19, 8, 21, 12, 15, 51, 20, 23, 13, 16, 22, 131, 52, 53, 14, 17, 31, 25, 323, 132, 56, 24, 18, 37, 26, 28, 771, 324, 133, 27, 29, 58, 35, 45, 30, 1795, 772, 137, 36, 54, 62, 40, 76, 50, 32, 4099, 1796, 325, 57, 66, 64, 67, 113

Offset

1,2

Comments

Suppose that S = (s(m)), for m >= 1, is a sequence of distinct real numbers that is dense in an open interval (a,b), such as the numbers (sin(n)), for n>=1, dense in (-1,1). The increasing partition array (p(n,k)) of the set N of positive integers is defined inductively as follows: p(1,1) = 1, and for k >= 2, p(1,k) = least m such that s(m) > s(p(1,k-1)). For n>=2, p(n,1) = least new m (that is, m is not p(h,k) for any h<=n-1 and k>=1), and for k>=2, p(n,k) = least new m such that s(m) > s(p(n,k-1)).

The decreasing partition array (p(n,k)) of N is defined as follows: p(1,1)=s(1), and for k>=2, p(1,k) = least new m such that s(m) < s(p(1,k-1)). For n>=2, p(n,1) = least new m, and for k>=2, p(n,k) = least new m such that s(m) < (p(n,k-1)).

The n-th tail of the binary Champernowne constant c is the fractional part of c*2^n, for n>=0.

For a guide to related partition arrays, see A388853.

Links

Table of n, a(n) for n=1..63.

Example

The binary Champernowne constant c is 0.110111001011101111000100110101100..., with binary tails t(0) = c, t(1) = .10111001..., t(2) = .01110010...; In base 10, these tails are t(0) = 0.8622401259..., t(1) = 0.622401259..., t(2) = 0.22401259..., etc., from which the first few terms of the increasing and decreasing partition arrays can be checked.
Corner:
    1    2    3    7   19   51  131  323  771
    4    5    6    8   20   52  132  324  772
    9   10   21   23   53   56  133  137  325
   11   12   13   14   24   27   36   57   61
   15   16   17   18   29   54   66   73  134
   22   31   37   58   62   64   71   93  139
   25   26   35   40   67   74   97  150  158
   28   45   76  113  161  182  277  365  438

Mathematica

highs := {Map[First, #], Most[FoldList[Plus, 1, Map[Length, #]]]} &[
Split[Rest[FoldList[Max, -\[Infinity], #]]]] &;
lows := {Map[First, #], Most[FoldList[Plus, 1, Map[Length, #]]]} &[
Split[Rest[FoldList[Min, +\[Infinity], #]]]] &;
c = N[ChampernowneNumber[2], 500];
r[n_] := FractionalPart[c*2^n];
seqS = Table[r[n], {n, 0, 480}];  (* User: put your dense sequence S after seqS *)
indices = Range[Length[seqS]];
arrI = {};   (* start accumulating increasing partition array *)
Until[Last[arrI] == {}, AppendTo[arrI, Flatten[Map[Position[seqS, #] &,
highs[seqS[[Complement[indices, Flatten[arrI]]]]][[1]]]]]];
Grid[Take[arrI, 12]]
arrD = {};   (* start accumulating decreasing partition array *)
Until[Last[arrD] == {}, AppendTo[arrD, Flatten[Map[Position[seqS, #] &,
lows[seqS[[Complement[indices, Flatten[arrD]]]]][[1]]]]]];
Grid[Take[arrD, 12]]
(* Peter J. C. Moses, Sep 04 2025 *)

Crossrefs

Cf. A388853, A389574, A389575, A389576.

Sequence in context: A351652 A333087 A379176 * A091451 A365389 A246161

Adjacent sequences: A389574 A389575 A389576 * A389578 A389579 A389580

Keyword

nonn,tabl,base

Author

Clark Kimberling, Oct 14 2025

Status

approved