A357865 Number of integer partitions of n whose run-sums are not weakly increasing.

0, 0, 0, 1, 1, 4, 5, 10, 13, 22, 31, 45, 57, 85, 115, 155, 199, 267, 344, 452, 577, 744, 940, 1191, 1486, 1877, 2339, 2910, 3595, 4442, 5453, 6688, 8162, 9960, 12089, 14662, 17698, 21365, 25703, 30869, 36961, 44207, 52728, 62801, 74644, 88587, 104930, 124113

Offset: 0

Author

Gus Wiseman, Oct 19 2022

Keywords

• nonn

Comments

The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).

Examples

			The a(0) = 0 through a(8) = 13 partitions:
  .  .  .  (21)  (31)  (32)   (42)    (43)     (53)
                       (41)   (51)    (52)     (62)
                       (221)  (321)   (61)     (71)
                       (311)  (411)   (331)    (332)
                              (2211)  (421)    (431)
                                      (511)    (521)
                                      (2221)   (611)
                                      (3211)   (3221)
                                      (4111)   (3311)
                                      (22111)  (4211)
                                               (5111)
                                               (22211)
                                               (32111)
		

Links

• Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Crossrefs

The complement is counted by A304406, ranked by A357861.

Number of rows in A354584 summing to n that are not weakly decreasing.

These partitions are ranked by A357850.

The opposite (not weakly decreasing) version is A357878, ranked by A357876.

A000041 counts integer partitions, strict A000009.

A304442 counts partitions with equal run-sums, distinct A353837.

Cf. A047966, A098859, A239312, A275870, A304405, A304428, A304430, A353832, A353864, A357875.

Programs

Mathematica

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