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 A353837 #14 Sep 12 2023 02:07:25
%S A353837 1,1,2,3,4,7,10,14,17,28,35,49,62,85,107,149,174,238,305,384,476,614,
%T A353837 752,950,1148,1451,1763,2205,2654,3259,3966,4807,5773,7039,8404,10129,
%U A353837 12140,14528,17288,20668,24505,29062,34437,40704,48059,56748,66577,78228
%N A353837 Number of integer partitions of n with all distinct run-sums.
%C A353837 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). The first partition whose run-sums are not all distinct is (2,1,1).
%H A353837 Max Alekseyev, <a href="/A353837/b353837.txt">Table of n, a(n) for n = 0..100</a>
%e A353837 The a(0) = 1 through a(6) = 10 partitions:
%e A353837 () (1) (2) (3) (4) (5) (6)
%e A353837 (11) (21) (22) (32) (33)
%e A353837 (111) (31) (41) (42)
%e A353837 (1111) (221) (51)
%e A353837 (311) (222)
%e A353837 (2111) (321)
%e A353837 (11111) (411)
%e A353837 (2211)
%e A353837 (21111)
%e A353837 (111111)
%t A353837 Table[Length[Select[IntegerPartitions[n],UnsameQ@@Total/@Split[#]&]],{n,0,15}]
%o A353837 (Sage) a353837 = lambda n: sum( abs(BipartiteGraph( Matrix(len(p), len(D:=list(set.union(*map(lambda t: set(divisors(t)),p)))), lambda i,j: p[i]%D[j]==0) ).matching_polynomial()[len(D)-len(p)]) for p in Partitions(n,max_slope=-1) ) # _Max Alekseyev_, Sep 11 2023
%Y A353837 For multiplicities instead of run-sums we have A098859, ranked by A130091.
%Y A353837 For equal run-sums we have A304442, ranked by A353833 (nonprime A353834).
%Y A353837 These partitions are ranked by A353838, complement A353839.
%Y A353837 The version for compositions is A353850, ranked by A353852.
%Y A353837 The weak version (rucksack partitions) is A353864, ranked by A353866.
%Y A353837 The weak perfect version is A353865, ranked by A353867.
%Y A353837 A005811 counts runs in binary expansion.
%Y A353837 A275870 counts collapsible partitions, ranked by A300273.
%Y A353837 A351014 counts distinct runs in standard compositions.
%Y A353837 A353832 represents the operation of taking run-sums of a partition.
%Y A353837 A353840-A353846 pertain to partition run-sum trajectory.
%Y A353837 A353849 counts distinct run-sums in standard compositions.
%Y A353837 Cf. A000041, A008284, A047966, A071625, A073093, A116608, A175413, A181819, A333755, A353848, A353867.
%K A353837 nonn
%O A353837 0,3
%A A353837 _Gus Wiseman_, May 26 2022