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 A353836 #6 May 27 2022 14:37:05
%S A353836 1,0,1,0,2,0,0,2,1,0,0,4,1,0,0,0,2,5,0,0,0,0,5,5,1,0,0,0,0,2,12,1,0,0,
%T A353836 0,0,0,7,12,3,0,0,0,0,0,0,3,19,8,0,0,0,0,0,0,0,5,27,9,1,0,0,0,0,0,0,0,
%U A353836 2,33,20,1,0,0,0,0,0,0,0
%N A353836 Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct run-sums.
%C A353836 The run-sums of a sequence are the sums of its maximal consecutive constant subsequences (runs). For example, the run-sums of (2,2,1,1,1,3,2,2) are (4,3,3,4).
%e A353836 Triangle begins:
%e A353836 1
%e A353836 0 1
%e A353836 0 2 0
%e A353836 0 2 1 0
%e A353836 0 4 1 0 0
%e A353836 0 2 5 0 0 0
%e A353836 0 5 5 1 0 0 0
%e A353836 0 2 12 1 0 0 0 0
%e A353836 0 7 12 3 0 0 0 0 0
%e A353836 0 3 19 8 0 0 0 0 0 0
%e A353836 0 5 27 9 1 0 0 0 0 0 0
%e A353836 0 2 33 20 1 0 0 0 0 0 0 0
%e A353836 0 13 28 34 2 0 0 0 0 0 0 0 0
%e A353836 0 2 48 46 5 0 0 0 0 0 0 0 0 0
%e A353836 0 5 65 51 14 0 0 0 0 0 0 0 0 0 0
%e A353836 0 4 57 99 15 1 0 0 0 0 0 0 0 0 0 0
%e A353836 For example, row n = 8 counts the following partitions:
%e A353836 (8) (53) (431)
%e A353836 (44) (62) (521)
%e A353836 (422) (71) (3221)
%e A353836 (2222) (332)
%e A353836 (41111) (611)
%e A353836 (221111) (3311)
%e A353836 (11111111) (4211)
%e A353836 (5111)
%e A353836 (22211)
%e A353836 (32111)
%e A353836 (311111)
%e A353836 (2111111)
%t A353836 Table[Length[Select[IntegerPartitions[n], Length[Union[Total/@Split[#]]]==k&]],{n,0,15},{k,0,n}]
%Y A353836 Row sums are A000041.
%Y A353836 Counting distinct parts instead of run-sums gives A116608.
%Y A353836 Column k = 1 is A304442, ranked by A353833 (nonprime A353834).
%Y A353836 The rank statistic is A353835, weak A353861, for compositions A353849.
%Y A353836 A275870 counts collapsible partitions, ranked by A300273.
%Y A353836 A351014 counts distinct runs in standard compositions.
%Y A353836 A353832 represents the operation of taking run-sums of a partition.
%Y A353836 A353837 counts partitions with all distinct run-sums, ranked by A353838.
%Y A353836 A353840-A353846 pertain to partition run-sum trajectory.
%Y A353836 A353864 counts rucksack partitions, ranked by A353866.
%Y A353836 A353865 counts perfect rucksack partitions, ranked by A353867.
%Y A353836 Cf. A008284, A047966, A071625, A165413, A175413, A325280, A333755.
%K A353836 nonn,tabl
%O A353836 0,5
%A A353836 _Gus Wiseman_, May 26 2022