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A357865
Number of integer partitions of n whose run-sums are not weakly increasing.
4
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
0,6
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).
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], !LessEqual@@Total/@Split[#]&]], {n, 0, 30}]
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.
Sequence in context: A114517 A283246 A236283 * A366864 A322610 A322468
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 19 2022
STATUS
approved