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 A384884 #8 Jun 14 2025 23:51:17
%S A384884 1,1,2,3,4,6,9,13,18,25,35,46,60,79,104,131,170,215,271,342,431,535,
%T A384884 670,830,1019,1258,1547,1881,2298,2787,3359,4061,4890,5849,7010,8361,
%U A384884 9942,11825,14021,16558,19561,23057,27084,31821,37312,43627,50999,59500,69267
%N A384884 Number of integer partitions of n with all distinct lengths of maximal gapless runs (decreasing by 0 or 1).
%e A384884 The partition y = (6,6,4,3,3,2) has maximal gapless runs ((6,6),(4,3,3,2)), with lengths (2,4), so y is counted under a(24).
%e A384884 The a(1) = 1 through a(8) = 18 partitions:
%e A384884 (1) (2) (3) (4) (5) (6) (7) (8)
%e A384884 (11) (21) (22) (32) (33) (43) (44)
%e A384884 (111) (211) (221) (222) (322) (332)
%e A384884 (1111) (311) (321) (331) (422)
%e A384884 (2111) (411) (421) (431)
%e A384884 (11111) (2211) (511) (521)
%e A384884 (3111) (2221) (611)
%e A384884 (21111) (3211) (2222)
%e A384884 (111111) (4111) (3221)
%e A384884 (22111) (4211)
%e A384884 (31111) (5111)
%e A384884 (211111) (22211)
%e A384884 (1111111) (32111)
%e A384884 (41111)
%e A384884 (221111)
%e A384884 (311111)
%e A384884 (2111111)
%e A384884 (11111111)
%t A384884 Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[#,#2>=#1-1&]&]],{n,0,15}]
%Y A384884 For subsets instead of strict partitions we have A384175.
%Y A384884 The strict case is A384178, for anti-runs A384880.
%Y A384884 For anti-runs we have A384885.
%Y A384884 For equal instead of distinct lengths we have A384887.
%Y A384884 A000041 counts integer partitions, strict A000009.
%Y A384884 A007690 counts partitions with no singletons, complement A183558.
%Y A384884 A034296 counts flat or gapless partitions, ranks A066311 or A073491.
%Y A384884 A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
%Y A384884 A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
%Y A384884 A355394 counts partitions without a neighborless part, singleton case A355393.
%Y A384884 A356236 counts partitions with a neighborless part, singleton case A356235.
%Y A384884 A356606 counts strict partitions without a neighborless part, complement A356607.
%Y A384884 Cf. A008284, A044813, A047993, A242882, A287170, A325324, A325325, A356226, A356230, A356233, A356234, A384176, A384177, A384886.
%K A384884 nonn
%O A384884 0,3
%A A384884 _Gus Wiseman_, Jun 13 2025