cp's OEIS Frontend

A332292 Number of widely alternately strongly normal integer partitions of n.

%I A332292 #11 Jun 26 2020 12:57:36
%S A332292 1,1,1,2,1,2,2,1,1,2,2,1,1,1,2,2,1,1,1,1,1,3,1,1,1,1,1,1,2,1,2,1,1,1,
%T A332292 1,1,2,1,1,1,1,1,2,1,1,2,1,1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,1,2,1,
%U A332292 1,1,2,1,1,1,1,1,1,1
%N A332292 Number of widely alternately strongly normal integer partitions of n.
%C A332292 An integer partition is widely alternately strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly decreasing run-lengths (strong) which, if reversed, are themselves a widely alternately strongly normal partition.
%C A332292 Also the number of widely alternately co-strongly normal reversed integer partitions of n.
%e A332292 The a(1) = 1, a(3) = 2, and a(21) = 3 partitions:
%e A332292   (1)  (21)   (654321)
%e A332292        (111)  (4443321)
%e A332292               (111111111111111111111)
%e A332292 For example, starting with the partition y = (4,4,4,3,3,2,1) and repeatedly taking run-lengths and reversing gives (4,4,4,3,3,2,1) -> (1,1,2,3) -> (1,1,2) -> (1,2) -> (1,1). All of these are normal with weakly decreasing run-lengths, and the last is all 1's, so y is counted under a(21).
%t A332292 totnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[ptn]==Range[Max[ptn]],GreaterEqual@@Length/@Split[ptn],totnQ[Reverse[Length/@Split[ptn]]]]];
%t A332292 Table[Length[Select[IntegerPartitions[n],totnQ]],{n,0,30}]
%Y A332292 Normal partitions are A000009.
%Y A332292 The non-strong version is A332277.
%Y A332292 The co-strong version is A332289.
%Y A332292 The case of reversed partitions is (also) A332289.
%Y A332292 The case of compositions is A332340.
%Y A332292 Cf. A100883, A181819, A317081, A317245, A317256, A317491, A329746, A329747, A332278, A332290, A332291, A332297, A332337.
%K A332292 nonn,more
%O A332292 0,4
%A A332292 _Gus Wiseman_, Feb 16 2020
%E A332292 a(71)-a(77) from _Jinyuan Wang_, Jun 26 2020