This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A357878 #6 Oct 20 2022 12:44:31
%S A357878 0,0,0,0,0,1,1,3,4,8,11,19,25,40,55,79,104,150,196,270,350,467,600,
%T A357878 786,997,1293,1632,2077,2597,3283,4067,5088,6268,7769,9517,11704,
%U A357878 14238,17405,21092,25598,30861,37278,44729,53742,64226,76811,91448,108929,129174
%N A357878 Number of integer partitions of n whose run-sums are not weakly decreasing.
%C A357878 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 A357878 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>
%e A357878 The a(0) = 0 through a(9) = 8 partitions:
%e A357878 . . . . . (2111) (21111) (322) (3221) (3222)
%e A357878 (31111) (32111) (32211)
%e A357878 (211111) (311111) (42111)
%e A357878 (2111111) (321111)
%e A357878 (411111)
%e A357878 (2211111)
%e A357878 (3111111)
%e A357878 (21111111)
%t A357878 Table[Length[Select[IntegerPartitions[n],!LessEqual@@Total/@Split[Reverse[#]]&]],{n,0,30}]
%Y A357878 The complement is counted by A304405, ranked by A357875.
%Y A357878 Number of rows in A354584 summing to n that are weakly increasing.
%Y A357878 The opposite (not weakly increasing) version is A357865, ranked by A357850.
%Y A357878 These partitions are ranked by A357876.
%Y A357878 A000041 counts integer partitions, strict A000009.
%Y A357878 A304442 counts partitions with equal run-sums, distinct A353837.
%Y A357878 Cf. A047966, A098859, A239312, A275870, A304406, A304428, A304430, A353832, A353864, A353932.
%K A357878 nonn
%O A357878 0,8
%A A357878 _Gus Wiseman_, Oct 18 2022