A375401 Number of integer partitions of n whose maximal anti-runs do not all have different maxima.

0, 0, 1, 1, 2, 3, 6, 7, 12, 16, 25, 33, 48, 63, 88, 116, 157, 204, 272, 349, 456, 581, 749, 946, 1205, 1511, 1904, 2371, 2960, 3661, 4538, 5577, 6862, 8389, 10257, 12472, 15164, 18348, 22192, 26731, 32177, 38593, 46254, 55256, 65952, 78500, 93340, 110706

Offset: 0

Author

Gus Wiseman, Aug 17 2024

Keywords

• nonn

Comments

An anti-run is a sequence with no adjacent equal terms. The maxima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the greatest term of each.

Examples

			The partition y = (3,2,2,1) has maximal ant-runs ((3,2),(2,1)), with maxima (3,2), so y is not counted under a(8).
The a(2) = 1 through a(8) = 12 partitions:
  (11)  (111)  (22)    (221)    (33)      (331)      (44)
               (1111)  (2111)   (222)     (2221)     (332)
                       (11111)  (2211)    (4111)     (2222)
                                (3111)    (22111)    (3311)
                                (21111)   (31111)    (5111)
                                (111111)  (211111)   (22211)
                                          (1111111)  (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)
		

Crossrefs

For identical instead of distinct we have A239955, ranks A073492.

The complement is counted by A375133, ranks A375402.

The complement for minima instead of maxima is A375134, ranks A375398.

These partitions have Heinz numbers A375403.

For minima instead of maxima we have A375404, ranks A375399.

The reverse for identical instead of distinct is A375405, ranks A375397.

A000041 counts integer partitions, strict A000009.

A003242 counts anti-run compositions, ranks A333489.

A055887 counts sequences of partitions with total sum n.

A375128 lists minima of maximal anti-runs of prime indices, sums A374706.

Cf. A034296, A115029, A141199, A279790, A358830, A358836, A374632, A374761, A375136, A375396, A375400.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Max/@Split[#,UnsameQ]&]],{n,0,30}]