A388720 Number of multisets summing to n (or reversed partitions of n) with superdiagonal run-lengths.

1, 1, 2, 2, 3, 3, 5, 5, 8, 8, 12, 12, 17, 16, 25, 22, 32, 31, 44, 41, 58, 54, 77, 74, 98, 94, 129, 124, 160, 160, 203, 200, 256, 251, 317, 316, 386, 390, 479, 482, 578, 589, 709, 716, 857, 869, 1030, 1058, 1236, 1268, 1486, 1526, 1761, 1824, 2112, 2166, 2491

Offset

0,3

Comments

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

Links

Table of n, a(n) for n=0..56.

Example

The multiset y = {1,1,1,3,3} has run-lengths (3,2), which are superdiagonal, so y is counted under a(9).
The multiset y = {1,2,2,3} has run-lengths (1,2,1), which are not superdiagonal, so y is not counted under a(8).
The a(1) = 1 through a(9) = 8 multisets:
  1  2   3    4     5      6       7        8         9
     11  111  22    122    33      133      44        144
              1111  11111  222     1222     233       333
                           1122    11122    1133      11133
                           111111  1111111  2222      12222
                                            11222     111222
                                            111122    1111122
                                            11111111  111111111

Mathematica

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

Crossrefs

For elements instead of their run-lengths we have A238873.

For non-reversed partitions we have A388714, strictly A388716.

These partitions are ranked by A388715 (for elements apparently A387112).

The strictly superdiagonal case is A388721, ranks A388719.

A000041 counts integer partitions, strict A000009.

A000700 counts self-conjugate partitions, ranks A088902.

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.

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. A025157, A052335, A052337, A387577, A387578, A388710, A388711.

Sequence in context: A325768 A371736 A371794 * A239949 A103609 A237800

Adjacent sequences: A388717 A388718 A388719 * A388721 A388722 A388723

Keyword

nonn

Author

Gus Wiseman, Sep 22 2025

Status

approved