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 A383096 #12 Apr 26 2025 11:26:53
%S A383096 0,0,0,1,1,5,4,13,15,25,35,54,58,99,128,168,217,295,358,488,603,784,
%T A383096 995,1253,1517,1953,2429,2997,3688,4563,5532,6840,8311,10135,12303,
%U A383096 14875,17842,21635,26008,31177,37247,44581,53062,63259,75130,89096,105551,124752,147015,173520
%N A383096 Number of integer partitions of n having no permutation with all equal run-sums.
%e A383096 The a(3) = 1 through a(8) = 15 partitions:
%e A383096 (21) (31) (32) (42) (43) (53)
%e A383096 (41) (51) (52) (62)
%e A383096 (221) (321) (61) (71)
%e A383096 (311) (411) (322) (332)
%e A383096 (2111) (331) (431)
%e A383096 (421) (521)
%e A383096 (511) (611)
%e A383096 (2221) (3221)
%e A383096 (3211) (3311)
%e A383096 (4111) (4211)
%e A383096 (22111) (5111)
%e A383096 (31111) (22211)
%e A383096 (211111) (32111)
%e A383096 (311111)
%e A383096 (2111111)
%t A383096 Table[Length[Select[IntegerPartitions[n],Length[Select[Permutations[#],SameQ@@Total/@Split[#]&]]==0&]],{n,0,15}]
%Y A383096 For distinct instead of equal run-sums we appear to have A381717, q.v.
%Y A383096 For run-lengths instead of sums we have A382915, ranks A382879, by signature A382914.
%Y A383096 For more than one permutation we have A383097, ranks A383015.
%Y A383096 The complement is counted by A383098, ranks A383110
%Y A383096 These partitions are ranked by A383100, positions of 0 in A382877.
%Y A383096 Counting and ranking partitions by run-lengths and run-sums:
%Y A383096 - constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
%Y A383096 - distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
%Y A383096 - weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
%Y A383096 - weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
%Y A383096 - strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
%Y A383096 - strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
%Y A383096 A275870 counts collapsible partitions, ranks A300273.
%Y A383096 A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
%Y A383096 A353851 counts compositions with all equal run-sums, ranks A353848.
%Y A383096 A382876 counts permutations of prime indices with distinct run-sums, zeros A381636.
%Y A383096 A383095 counts partitions having a unique permutation with equal run-sums, ranks A383099.
%Y A383096 Cf. A006171, A329738, A353832, A353839, A353932, A354584, A382076, A382857, A383094.
%K A383096 nonn
%O A383096 0,6
%A A383096 _Gus Wiseman_, Apr 17 2025
%E A383096 More terms from _Bert Dobbelaere_, Apr 26 2025