cp's OEIS Frontend

A332278 Number of widely totally co-strongly normal integer partitions of n.

%I A332278 #16 Mar 11 2021 18:15:54
%S A332278 1,1,1,2,2,1,2,1,1,2,3,1,1,1,1,2,1,1,1,1,2,2,1,1,1,1,1,1,3,1,1,1,1,1,
%T A332278 1,2,2,1,1,1,1,1,2,1,1,2,1,1,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1,1,1,1,2,1,
%U A332278 1,1,1,1,1,1,1,1,1,1,2
%N A332278 Number of widely totally co-strongly normal integer partitions of n.
%C A332278 A sequence of integers is widely totally co-strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) with weakly increasing run-lengths (co-strong) which are themselves a widely totally co-strongly normal sequence.
%C A332278 Is this sequence bounded?
%e A332278 The a(1) = 1 through a(20) = 2 partitions:
%e A332278    1: (1)
%e A332278    2: (11)
%e A332278    3: (21),(111)
%e A332278    4: (211),(1111)
%e A332278    5: (11111)
%e A332278    6: (321),(111111)
%e A332278    7: (1111111)
%e A332278    8: (11111111)
%e A332278    9: (32211),(111111111)
%e A332278   10: (4321),(322111),(1111111111)
%e A332278   11: (11111111111)
%e A332278   12: (111111111111)
%e A332278   13: (1111111111111)
%e A332278   14: (11111111111111)
%e A332278   15: (54321),(111111111111111)
%e A332278   16: (1111111111111111)
%e A332278   17: (11111111111111111)
%e A332278   18: (111111111111111111)
%e A332278   19: (1111111111111111111)
%e A332278   20: (4332221111),(11111111111111111111)
%t A332278 totnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[ptn]==Range[Max[ptn]],LessEqual@@Length/@Split[ptn],totnQ[Length/@Split[ptn]]]];
%t A332278 Table[Length[Select[IntegerPartitions[n],totnQ]],{n,0,30}]
%Y A332278 Not requiring co-strength gives A332277.
%Y A332278 The strong version is A332297(n) - 1 for n > 1.
%Y A332278 The narrow version is a(n) - 1 for n > 1.
%Y A332278 The alternating version is A332289.
%Y A332278 The Heinz numbers of these partitions are A332293.
%Y A332278 The case of compositions is A332337.
%Y A332278 Cf. A000009, A100883, A107429, A133808, A181819, A316496, A317245, A317491, A329746, A332279, A332290, A332291, A332292, A332296, A332576.
%K A332278 nonn,more
%O A332278 0,4
%A A332278 _Gus Wiseman_, Mar 05 2020
%E A332278 a(71)-a(78) from _Jinyuan Wang_, Jun 26 2020