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

%I A357865 #8 Oct 20 2022 12:44:26
%S A357865 0,0,0,1,1,4,5,10,13,22,31,45,57,85,115,155,199,267,344,452,577,744,
%T A357865 940,1191,1486,1877,2339,2910,3595,4442,5453,6688,8162,9960,12089,
%U A357865 14662,17698,21365,25703,30869,36961,44207,52728,62801,74644,88587,104930,124113
%N A357865 Number of integer partitions of n whose run-sums are not weakly increasing.
%C A357865 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).
%H A357865 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>
%e A357865 The a(0) = 0 through a(8) = 13 partitions:
%e A357865   .  .  .  (21)  (31)  (32)   (42)    (43)     (53)
%e A357865                        (41)   (51)    (52)     (62)
%e A357865                        (221)  (321)   (61)     (71)
%e A357865                        (311)  (411)   (331)    (332)
%e A357865                               (2211)  (421)    (431)
%e A357865                                       (511)    (521)
%e A357865                                       (2221)   (611)
%e A357865                                       (3211)   (3221)
%e A357865                                       (4111)   (3311)
%e A357865                                       (22111)  (4211)
%e A357865                                                (5111)
%e A357865                                                (22211)
%e A357865                                                (32111)
%t A357865 Table[Length[Select[IntegerPartitions[n],!LessEqual@@Total/@Split[#]&]],{n,0,30}]
%Y A357865 The complement is counted by A304406, ranked by A357861.
%Y A357865 Number of rows in A354584 summing to n that are not weakly decreasing.
%Y A357865 These partitions are ranked by A357850.
%Y A357865 The opposite (not weakly decreasing) version is A357878, ranked by A357876.
%Y A357865 A000041 counts integer partitions, strict A000009.
%Y A357865 A304442 counts partitions with equal run-sums, distinct A353837.
%Y A357865 Cf. A047966, A098859, A239312, A275870, A304405, A304428, A304430, A353832, A353864, A357875.
%K A357865 nonn
%O A357865 0,6
%A A357865 _Gus Wiseman_, Oct 19 2022