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 A382076 #21 Jun 24 2025 13:22:39
%S A382076 0,0,0,1,1,5,6,13,15,27,37,54,64,99,130,172,220,295,372,488,615,788,
%T A382076 997,1253,1547,1955,2431,3005,3706,4563,5586,6840,8332,10139,12305,
%U A382076 14879,17933,21635,26010,31181,37314,44581,53156,63259,75163,89124,105553,124752,147210
%N A382076 Number of integer partitions of n whose run-sums are not all equal.
%C A382076 Also the number of integer partitions of n that cannot be partitioned into distinct constant multisets with a common sum. Multiset partitions of this type are ranked by A005117 /\ A326534 /\ A355743, while twice-partitions are counted by A382524, strict case of A279789.
%e A382076 The partition (3,2,1,1,1) has runs ((3),(2),(1,1,1)) with sums (3,2,3) so is counted under a(8).
%e A382076 The a(3) = 1 through a(8) = 15 partitions:
%e A382076 (21) (31) (32) (42) (43) (53)
%e A382076 (41) (51) (52) (62)
%e A382076 (221) (321) (61) (71)
%e A382076 (311) (411) (322) (332)
%e A382076 (2111) (2211) (331) (431)
%e A382076 (21111) (421) (521)
%e A382076 (511) (611)
%e A382076 (2221) (3221)
%e A382076 (3211) (3311)
%e A382076 (4111) (4211)
%e A382076 (22111) (5111)
%e A382076 (31111) (22211)
%e A382076 (211111) (32111)
%e A382076 (311111)
%e A382076 (2111111)
%t A382076 Table[Length[Select[IntegerPartitions[n],!SameQ@@Total/@Split[#]&]],{n,0,15}]
%Y A382076 The complement is counted by A304442, ranks A353833.
%Y A382076 For distinct instead of equal block-sums we have A381717.
%Y A382076 This is the strict case of A381993, see A381995, zeros A381871.
%Y A382076 A050361 counts factorizations into distinct prime powers, see A381715.
%Y A382076 A304405 counts partitions with weakly decreasing run-sums, ranks A357875.
%Y A382076 A304406 counts partitions with weakly increasing run-sums, ranks A357861.
%Y A382076 A304428 counts partitions with strictly decreasing run-sums, ranks A357862.
%Y A382076 A304430 counts partitions with strictly increasing run-sums, ranks A357864.
%Y A382076 A317141 counts coarsenings of prime indices, refinements A300383.
%Y A382076 A326534 ranks multiset partitions with a common sum.
%Y A382076 A353837 counts partitions with distinct run-sums.
%Y A382076 A354584 lists run-sums of weakly increasing prime indices.
%Y A382076 A355743 ranks multiset partitions into constant blocks.
%Y A382076 Cf. A000688, A005117, A006171, A047966, A279784, A381453, A381455, A381635, A381636, A381994, A382204.
%K A382076 nonn
%O A382076 0,6
%A A382076 _Gus Wiseman_, Apr 02 2025
%E A382076 More terms from _Bert Dobbelaere_, Apr 26 2025