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A332338 Number of alternately co-strong compositions of n.

%I A332338 #8 Feb 18 2020 04:47:59
%S A332338 1,1,2,4,7,12,24,39,72,125,224,387,697,1205,2141,3736,6598,11516,
%T A332338 20331,35526,62507,109436,192200,336533,590582,1034187
%N A332338 Number of alternately co-strong compositions of n.
%C A332338 A sequence is alternately co-strong if either it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and, when reversed, are themselves an alternately co-strong sequence.
%e A332338 The a(1) = 1 through a(5) = 12 compositions:
%e A332338   (1)  (2)   (3)    (4)     (5)
%e A332338        (11)  (12)   (13)    (14)
%e A332338              (21)   (22)    (23)
%e A332338              (111)  (31)    (32)
%e A332338                     (112)   (41)
%e A332338                     (121)   (113)
%e A332338                     (1111)  (131)
%e A332338                             (212)
%e A332338                             (221)
%e A332338                             (1112)
%e A332338                             (1121)
%e A332338                             (11111)
%e A332338 For example, starting with the composition y = (1,6,2,2,1,1,1,1) and repeatedly taking run-lengths and reversing gives (1,6,2,2,1,1,1,1) -> (4,2,1,1) -> (2,1,1) -> (2,1) -> (1,1) -> (2). All of these have weakly increasing run-lengths and the last is a singleton, so y is counted under a(15).
%t A332338 tniQ[q_]:=Or[q=={},q=={1},And[LessEqual@@Length/@Split[q],tniQ[Reverse[Length/@Split[q]]]]];
%t A332338 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],tniQ]],{n,0,10}]
%Y A332338 The case of partitions is A317256.
%Y A332338 The recursive (rather than alternating) version is A332274.
%Y A332338 The total (rather than alternating) version is (also) A332274.
%Y A332338 The strong version is this same sequence.
%Y A332338 The case of reversed partitions is A332339.
%Y A332338 The normal version is A332340(n) + 1 for n > 1.
%Y A332338 Cf. A001462, A100883, A181819, A182850, A316496, A317257, A329744, A329746, A332275, A332292, A332296.
%K A332338 nonn,more
%O A332338 0,3
%A A332338 _Gus Wiseman_, Feb 17 2020