cp's OEIS Frontend

A332289 Number of widely alternately co-strongly normal integer partitions of n.

%I A332289 #5 Feb 15 2020 11:05:02
%S A332289 1,1,1,2,2,1,2,2,1,1,3,1,1,1,2,2,1,1,1,1,2,2,1,1,1,1,1,1,3,1,1,1,1,1,
%T A332289 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 A332289 1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,2,1,1
%N A332289 Number of widely alternately co-strongly normal integer partitions of n.
%C A332289 An integer partition is widely alternately co-strongly normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly increasing run-lengths (co-strong) which, if reversed, are themselves a widely alternately co-strongly normal partition.
%e A332289 The a(1) = 1, a(3) = 2, and a(10) = 3 partitions:
%e A332289   (1)  (21)   (4321)
%e A332289        (111)  (322111)
%e A332289               (1111111111)
%e A332289 For example, starting with y = (4,3,2,2,1,1,1) and repeatedly taking run-lengths and reversing gives y -> (3,2,1,1) -> (2,1,1) -> (2,1) -> (1,1). These are all normal, have weakly increasing run-lengths, and the last is all 1's, so y is counted a(14).
%t A332289 totnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[ptn]==Range[Max[ptn]],LessEqual@@Length/@Split[ptn],totnQ[Reverse[Length/@Split[ptn]]]]];
%t A332289 Table[Length[Select[IntegerPartitions[n],totnQ]],{n,0,30}]
%Y A332289 Normal partitions are A000009.
%Y A332289 Dominated by A317245.
%Y A332289 The non-co-strong version is A332277.
%Y A332289 The total (instead of alternate) version is A332278.
%Y A332289 The Heinz numbers of these partitions are A332290.
%Y A332289 The strong version is A332292.
%Y A332289 The case of reversed partitions is (also) A332292.
%Y A332289 The generalization to compositions is A332340.
%Y A332289 Cf. A100883, A107429, A133808, A316496, A317081, A317256, A317491, A329746, A332291, A332295, A332297, A332337, A332338, A332339.
%K A332289 nonn
%O A332289 0,4
%A A332289 _Gus Wiseman_, Feb 13 2020