1,2

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)).

For a guide to related partition arrays, see A388853.

Fractional parts of tan(n): 0.557..., -0.185..., -0.142..., 0.157..., -0.380..., -0.291..., etc., from which the first few terms of the increasing and decreasing partition arrays can be checked.

Corner:

1 7 29 51 73 95 117 139 161 262 265

2 3 4 10 32 36 83 215 494 516 617

5 6 13 17 23 39 45 54 61 102 146

8 9 14 16 35 57 67 76 98 111 124

11 12 18 19 20 38 58 64 79 89 120

15 21 22 26 42 60 82 86 101 123 130

24 25 40 41 48 70 92 104 108 145 167

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], #]]]] &;

seqS = Table[N[FractionalPart[Tan[n]], 20], {n, 1, 1000}];

(* 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*)

Cf. A388853, A389579.

Sequence in context: A217366 A282609 A282450 * A280337 A195587 A096900

Adjacent sequences: A389575 A389576 A389577 * A389579 A389580 A389581

nonn,tabl

Clark Kimberling, Oct 14 2025

approved