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%I #8 Feb 18 2020 04:47:59
%S 1,1,2,4,7,12,24,39,72,125,224,387,697,1205,2141,3736,6598,11516,
%T 20331,35526,62507,109436,192200,336533,590582,1034187
%N Number of alternately co-strong compositions of n.
%C 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 The a(1) = 1 through a(5) = 12 compositions:
%e (1) (2) (3) (4) (5)
%e (11) (12) (13) (14)
%e (21) (22) (23)
%e (111) (31) (32)
%e (112) (41)
%e (121) (113)
%e (1111) (131)
%e (212)
%e (221)
%e (1112)
%e (1121)
%e (11111)
%e 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 tniQ[q_]:=Or[q=={},q=={1},And[LessEqual@@Length/@Split[q],tniQ[Reverse[Length/@Split[q]]]]];
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],tniQ]],{n,0,10}]
%Y The case of partitions is A317256.
%Y The recursive (rather than alternating) version is A332274.
%Y The total (rather than alternating) version is (also) A332274.
%Y The strong version is this same sequence.
%Y The case of reversed partitions is A332339.
%Y The normal version is A332340(n) + 1 for n > 1.
%Y Cf. A001462, A100883, A181819, A182850, A316496, A317257, A329744, A329746, A332275, A332292, A332296.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Feb 17 2020