A389574 Increasing partition array based on the tails of the Champernowne constant (A033307); see Comments.

1, 2, 10, 3, 12, 11, 4, 13, 16, 31, 5, 14, 18, 33, 51, 6, 15, 20, 38, 53, 71, 7, 17, 22, 40, 55, 73, 91, 8, 19, 24, 42, 60, 75, 93, 111, 9, 21, 26, 44, 62, 77, 95, 113, 131, 29, 23, 28, 46, 64, 82, 97, 115, 133, 151, 49, 25, 30, 48, 66, 84, 99, 117, 135, 153

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 Champernowne constant c is the fractional part of c*10^n, for n>=0. With c written (in any base) as .d1 d2 d3 d4 ..., the dense sequence S is simply (.d1 d2 d3 ... , .d2 d3 d4 ..., .d3 d4 d5 ..., d4 d5 d6... ); i.e., for base 10: (.12345..., .23456..., .34567..., .45678...).

For a guide to related partition arrays, see A388853.

Links

Table of n, a(n) for n=1..65.

Example

First few terms of S are t(0) = 0.123456789101112..., t(1) = 0.23456789101112..., t(2) = 0.3456789101112..., . . . , t(11) = 0.101112..., etc. from which the first few terms of the increasing and decreasing partition arrays can be checked.
Corner
    1    2    3    4    5    6    7    8    9   29
   10   12   13   14   15   17   19   21   23   25
   11   16   18   20   22   24   26   28   30   32
   31   33   38   40   42   44   46   48   50   52
   51   53   55   60   62   64   66   68   70   72
   71   73   75   77   82   84   86   88   90   92
   91   93   95   97   99  119  139  159  179  207
  111  113  115  117  137  157  177  204  234  264
  131  133  135  155  175  201  231  261  287  290
  151  153  173  198  228  257  260  263  266  269

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[10], 50000];
r[n_] := FractionalPart[c*10^n];
seqS = Table[r[n], {n, 0, 48000}];  (* 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. A389575, A389576, A389577.

Sequence in context: A322000 A061196 A239136 * A239137 A379401 A252758

Adjacent sequences: A389571 A389572 A389573 * A389575 A389576 A389577

Keyword

nonn,tabl,base

Author

Clark Kimberling, Oct 14 2025

Status

approved