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A332339 Number of alternately co-strong reversed integer partitions of n.

%I A332339 #12 Mar 30 2020 16:12:46
%S A332339 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 A332339 204,204,255,285,339,363,458,471,580,632,741,806,983,1015,1225,1341,
%U A332339 1562,1667,2003,2107,2491,2712,3101,3344,3962,4182,4860,5270,6022,6482
%N A332339 Number of alternately co-strong reversed integer partitions of n.
%C A332339 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 A332339 Also the number of alternately strong integer partitions of n.
%e A332339 The a(1) = 1 through a(8) = 12 reversed partitions:
%e A332339   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A332339        (11)  (12)   (13)    (14)     (15)      (16)       (17)
%e A332339              (111)  (22)    (23)     (24)      (25)       (26)
%e A332339                     (1111)  (122)    (33)      (34)       (35)
%e A332339                             (11111)  (123)     (124)      (44)
%e A332339                                      (222)     (133)      (125)
%e A332339                                      (1122)    (1222)     (134)
%e A332339                                      (111111)  (1111111)  (233)
%e A332339                                                           (1133)
%e A332339                                                           (2222)
%e A332339                                                           (11222)
%e A332339                                                           (11111111)
%e A332339 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 A332339 tniQ[q_]:=Or[q=={},q=={1},And[LessEqual@@Length/@Split[q],tniQ[Reverse[Length/@Split[q]]]]];
%t A332339 Table[Length[Select[Sort/@IntegerPartitions[n],tniQ]],{n,0,30}]
%Y A332339 The total (instead of alternating) version is A316496.
%Y A332339 Alternately strong partitions are A317256.
%Y A332339 The case of ordinary (not reversed) partitions is (also) A317256.
%Y A332339 The generalization to compositions is A332338.
%Y A332339 Cf. A100883, A181819, A182850, A317257, A329744, A329746, A332275, A332289, A332292, A332340.
%K A332339 nonn
%O A332339 0,3
%A A332339 _Gus Wiseman_, Feb 17 2020