A388714 Number of integer partitions of n with superdiagonal run-lengths.

1, 1, 2, 2, 4, 4, 8, 8, 13, 14, 21, 22, 33, 36, 50, 55, 76, 84, 110, 123, 157, 177, 223, 246, 309, 344, 422, 475, 574, 640, 777, 863, 1035, 1156, 1371, 1516, 1809, 1996, 2343, 2606, 3033, 3347, 3915, 4298, 4980, 5500, 6331, 6949, 8017, 8783, 10055, 11062

Offset

0,3

Comments

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

Includes all strict partitions.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Example

The partition y = (3,2,2,1,1) has run-lengths (1,2,2), which are not superdiagonal, so y is not counted under a(9).
The partition y = (3,2,2,1,1,1) has run-lengths (1,2,3), which are superdiagonal, so y is counted under a(9).
The a(1) = 1 through a(7) = 8 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (311)    (33)      (322)
                    (211)   (2111)   (222)     (511)
                    (1111)  (11111)  (411)     (4111)
                                     (2211)    (22111)
                                     (3111)    (31111)
                                     (21111)   (211111)
                                     (111111)  (1111111)

Mathematica

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

Crossrefs

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

For strictly superdiagonal run-lengths we have A388716.

For reversed partitions we have A388720.

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.

A352833 counts partitions by fixed points, reversed partitions A238352.

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

A388713 counts superdiagonal reversed partitions by maximum.

Cf. A000700, A052335, A052337, A387576, A387578, A388710, A388711, A388712, A388719.

Sequence in context: A046971 A051754 A108747 * A116931 A206558 A385635

Adjacent sequences: A388711 A388712 A388713 * A388715 A388716 A388717

Keyword

nonn

Author

Gus Wiseman, Sep 22 2025

Status

approved