A388716 Number of integer partitions of n with strictly superdiagonal run-lengths.

1, 0, 1, 1, 2, 1, 3, 2, 4, 5, 6, 6, 12, 9, 13, 16, 20, 18, 30, 26, 39, 40, 52, 50, 77, 70, 94, 100, 127, 124, 171, 165, 214, 220, 274, 276, 358, 357, 440, 462, 563, 574, 712, 727, 885, 925, 1096, 1143, 1379, 1428, 1678, 1777, 2085, 2171, 2554, 2674, 3103, 3288

Offset

0,5

Comments

A sequence (y_1, ..., y_k) is strictly superdiagonal iff y_i > i for all i = 1..k.

Links

Table of n, a(n) for n=0..57.

Example

The a(2) = 1 through a(10) = 6 partitions:
  11  111  22    11111  33      22111    44        333        55
           1111         222     1111111  2222      33111      22222
                        111111           221111    222111     331111
                                         11111111  2211111    2221111
                                                   111111111  22111111
                                                              1111111111

Mathematica

suppstrQ[mset_]:=And@@Table[mset[[i]]>i, {i, Length[mset]}];
Table[Length[Select[IntegerPartitions[n], suppstrQ[Length/@Split[#]]&]], {n, 0, 10}]

Crossrefs

For parts instead of run-lengths we have A003106, non-strictly superdiagonal A003114.

The non-strictly superdiagonal version is A388714, reverse A388720.

These partitions are ranked by A388719, non-strictly superdiagonal A388715.

For reversed partitions we have A388721.

A000041 counts integer partitions, strict A000009.

A001522 (complement A064428), A238395 (complement A238394) count partitions w/ diagonal.

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

A003114 counts superdiagonal partitions, complement A387118, strict partitions A025157.

A114088 counts partitions by excedances.

A115720 and A115994 count partitions by their Durfee square.

A238349 counts compositions by fixed points, complement A352523.

A238352 counts reversed partitions by fixed points.

A352833 counts partitions by fixed points.

A370592 counts choosable partitions (ranks A368100), complement A370593 (ranks A355529).

Cf. A000700, A052335, A052337, A238873, A387330, A387576, A387578, A388710, A388711.

Sequence in context: A026272 A193564 A022447 * A117194 A340647 A318746

Adjacent sequences: A388713 A388714 A388715 * A388717 A388718 A388719

Keyword

nonn

Author

Gus Wiseman, Sep 22 2025

Status

approved