%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