A238873 Number of superdiagonal partitions: partitions (p1, p2, p3, ...) of n such that pi >= i.

1, 1, 1, 2, 3, 3, 5, 7, 9, 11, 14, 19, 25, 31, 38, 46, 59, 73, 92, 112, 135, 162, 196, 237, 289, 349, 417, 496, 587, 691, 820, 970, 1151, 1357, 1598, 1870, 2183, 2537, 2952, 3433, 3997, 4644, 5393, 6248, 7220, 8318, 9566, 10981, 12605, 14457, 16582, 19002, 21767, 24886, 28424, 32396, 36873, 41901, 47579, 53974, 61221

Offset

0,4

Comments

For the conventional ordering of partitions (weakly decreasing) we get A003114. - Gus Wiseman, Oct 06 2025

Conjecture: Also the number of integer partitions y of n with choosable initial intervals, meaning it is possible to choose distinct positive integers (z_1, z_2, ...) such that z_i <= y_i for all i. - Gus Wiseman, Sep 26 2025

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 132 terms from Joerg Arndt)

M. Archibald, A. Blecher, S. Elizalde, and A. Knopfmacher, Subdiagonal and superdiagonal partitions, Afr. Mat. 36, 77 (2025). See p. 5.

Example

The a(13) = 31 such partitions of 13 are:
  01:  [ 1 2 3 7 ]
  02:  [ 1 2 4 6 ]
  03:  [ 1 2 5 5 ]
  04:  [ 1 2 10 ]
  05:  [ 1 3 3 6 ]
  06:  [ 1 3 4 5 ]
  07:  [ 1 3 9 ]
  08:  [ 1 4 4 4 ]
  09:  [ 1 4 8 ]
  10:  [ 1 5 7 ]
  11:  [ 1 6 6 ]
  12:  [ 1 12 ]
  13:  [ 2 2 3 6 ]
  14:  [ 2 2 4 5 ]
  15:  [ 2 2 9 ]
  16:  [ 2 3 3 5 ]
  17:  [ 2 3 4 4 ]
  18:  [ 2 3 8 ]
  19:  [ 2 4 7 ]
  20:  [ 2 5 6 ]
  21:  [ 2 11 ]
  22:  [ 3 3 3 4 ]
  23:  [ 3 3 7 ]
  24:  [ 3 4 6 ]
  25:  [ 3 5 5 ]
  26:  [ 3 10 ]
  27:  [ 4 4 5 ]
  28:  [ 4 9 ]
  29:  [ 5 8 ]
  30:  [ 6 7 ]
  31:  [ 13 ]

Mathematica

suppQ[mset_]:=And@@Table[mset[[i]]>=i, {i, Length[mset]}];
Table[Length[Select[Reverse/@IntegerPartitions[n], suppQ]], {n, 0, 15}] (* Gus Wiseman, Sep 26 2025 *)

Crossrefs

Cf. A219282 (superdiagonal compositions), A238394 (strictly superdiagonal partitions), A025147 (strictly superdiagonal partitions into distinct parts).

Cf. A238875 (subdiagonal partitions), A008930 (subdiagonal compositions), A010054 (subdiagonal partitions into distinct parts).

Cf. A238859 (compositions of n with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).

Cf. A238860 (partitions with superdiagonal growth), A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).

The version for divisors is A239312, ranks A368110.

The complement for divisors is A370320, ranks A355740.

These partitions appear to have ranks A387112.

The complement appears to be counted by A387118, ranks A387113.

A000041 counts integer partitions.

A001522 counts partitions with a diagonal, complement A064428.

A003106 counts strictly superdiagonal partitions, strict A237979, ranks A352830.

A003114 counts superdiagonal partitions, strict A025157.

A114088 counts partitions by excedances.

A352833 counts partitions by fixed points, reversed A238352.

Cf. A000005, A005703, A052335, A238395, A367867, A370594, A387578.

Sequence in context: A030779 A030729 A111865 * A042955 A035553 A108961

Adjacent sequences: A238870 A238871 A238872 * A238874 A238875 A238876

Keyword

nonn

Author

Joerg Arndt, Mar 23 2014

Status

approved