A393670 Irregular triangular array T, read by rows: when the partitions of n are arranged in reverse lexicographic order, T(n,k) is the position of the k-th partition in which adjacent parts differ by at least 2.

1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 5, 1, 2, 3, 5, 9, 1, 2, 3, 5, 8, 9, 1, 2, 3, 5, 8, 9, 14, 1, 2, 3, 5, 8, 9, 13, 14, 22, 1, 2, 3, 5, 8, 9, 13, 14, 21, 22, 1, 2, 3, 5, 8, 9, 13, 14, 20, 21, 22, 33, 1, 2, 3, 5, 8, 9, 13, 14, 20, 21, 22, 32, 33, 49

Offset

1,5

Links

Table of n, a(n) for n=1..80.

Example

First 16 rows:
    1
    1
    1
    1   2
    1   2
    1   2   3
    1   2   3
    1   2   3   5
    1   2   3   5   9
    1   2   3   5   8   9
    1   2   3   5   8   9   14
    1   2   3   5   8   9   13   14   22
    1   2   3   5   8   9   13   14   21   22
    1   2   3   5   8   9   13   14   20   21   22   33
    1   2   3   5   8   9   13   14   20   21   22   32   33   49
    1   2   3   5   8   9   13   14   20   21   22   31   32   33   48   50   74
The partitions of 5 in reverse lexicographic order are 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1; those in which adjacent parts differ by at least two are 5, 4+1, which are in positions 1 and 2, so row 5 is 1  2.

Mathematica

p[n_] := p[n] = IntegerPartitions[n]
d[n_] := -Map[Differences, p[n]]
t = Table[Select[Range[Length[d[n]]], !MemberQ[d[n][[#]], 0] && !MemberQ[d[n][[#]], 1] & ], {n, 1, 20}]
Grid[t]  (* array *)
Flatten[t]  (* sequence *)

Crossrefs

Cf. A035422 (limiting row), A003114 (row lengths).

Sequence in context: A023118 A122197 A030718 * A227779 A194066 A308916

Adjacent sequences: A393667 A393668 A393669 * A393671 A393672 A393673

Keyword

nonn,tabf

Author

Clark Kimberling, Mar 01 2026

Status

approved