A384880 Number of strict integer partitions of n with all distinct lengths of maximal anti-runs (decreasing by more than 1).

1, 1, 1, 1, 2, 2, 3, 4, 6, 6, 9, 10, 12, 15, 18, 21, 25, 30, 34, 41, 46, 55, 63, 75, 85, 99, 114, 133, 152, 178, 201, 236, 269, 308, 352, 404, 460, 525, 594, 674, 763, 865, 974, 1098, 1236, 1385, 1558, 1745, 1952, 2181, 2435, 2712, 3026, 3363, 3740, 4151, 4612

Offset

0,5

Links

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

Example

The strict partition y = (10,7,6,4,2,1) has maximal anti-runs ((10,7),(6,4,2),(1)), with lengths (2,3,1), so y is counted under a(30).
The a(1) = 1 through a(14) = 18 partitions (A-E = 10-14):
  1  2  3  4   5   6   7    8    9    A    B    C    D     E
           31  41  42  52   53   63   64   74   75   85    86
                   51  61   62   72   73   83   84   94    95
                       421  71   81   82   92   93   A3    A4
                            431  531  91   A1   A2   B2    B3
                            521  621  532  542  B1   C1    C2
                                      541  632  642  643   D1
                                      631  641  651  652   653
                                      721  731  732  742   743
                                           821  741  751   752
                                                831  832   761
                                                921  841   842
                                                     931   851
                                                     A21   932
                                                     6421  941
                                                           A31
                                                           B21
                                                           7421

Mathematica

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&UnsameQ@@Length/@Split[#, #2<#1-1&]&]], {n, 0, 30}]

Crossrefs

For subsets instead of strict partitions we have A384177.

For runs instead of anti-runs we have A384178.

This is the strict case of A384885.

A000041 counts integer partitions, strict A000009.

A047993 counts partitions with max part = length.

A098859 counts Wilf partitions (complement A336866), compositions A242882.

A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.

A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.

Cf. A008289, A089259, A325324, A325325, A329739, A357982, A384175, A384176, A384884, A384886.

Sequence in context: A046934 A093594 A008806 * A356607 A366843 A370805

Adjacent sequences: A384877 A384878 A384879 * A384881 A384882 A384883

Keyword

nonn

Author

Gus Wiseman, Jun 13 2025

Status

approved