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%I #12 Mar 30 2020 16:12:46
%S 1,1,2,3,4,5,8,8,12,14,18,20,29,28,40,45,54,59,82,81,108,118,141,154,
%T 204,204,255,285,339,363,458,471,580,632,741,806,983,1015,1225,1341,
%U 1562,1667,2003,2107,2491,2712,3101,3344,3962,4182,4860,5270,6022,6482
%N Number of alternately co-strong reversed integer partitions 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.
%C Also the number of alternately strong integer partitions of n.
%e The a(1) = 1 through a(8) = 12 reversed partitions:
%e (1) (2) (3) (4) (5) (6) (7) (8)
%e (11) (12) (13) (14) (15) (16) (17)
%e (111) (22) (23) (24) (25) (26)
%e (1111) (122) (33) (34) (35)
%e (11111) (123) (124) (44)
%e (222) (133) (125)
%e (1122) (1222) (134)
%e (111111) (1111111) (233)
%e (1133)
%e (2222)
%e (11222)
%e (11111111)
%e For example, starting with the composition y = (1,2,3,3,4,4,4) and repeatedly taking run-lengths and reversing gives (1,2,3,3,4,4,4) -> (3,2,1,1) -> (2,1,1) -> (2,1) -> (1,1) -> (2) -> (1). All of these have weakly increasing run-lengths and the last is equal to (1), so y is counted under a(21).
%t tniQ[q_]:=Or[q=={},q=={1},And[LessEqual@@Length/@Split[q],tniQ[Reverse[Length/@Split[q]]]]];
%t Table[Length[Select[Sort/@IntegerPartitions[n],tniQ]],{n,0,30}]
%Y The total (instead of alternating) version is A316496.
%Y Alternately strong partitions are A317256.
%Y The case of ordinary (not reversed) partitions is (also) A317256.
%Y The generalization to compositions is A332338.
%Y Cf. A100883, A181819, A182850, A317257, A329744, A329746, A332275, A332289, A332292, A332340.
%K nonn
%O 0,3
%A _Gus Wiseman_, Feb 17 2020
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