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%I A317256 #12 Mar 08 2020 21:48:30
%S A317256 1,1,2,3,5,6,11,13,19,25,35,42,61,74,98,122,161,194,254,304,388,472,
%T A317256 589,700,878,1044,1278,1525,1851,2182,2651,3113,3735,4389,5231,6106,
%U A317256 7278,8464,9995,11631,13680,15831,18602,21463,25068,28927,33654,38671,44942,51514
%N A317256 Number of alternately co-strong integer partitions of n.
%C A317256 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 A317256 Also the number of alternately strong reversed integer partitions of n.
%e A317256 The a(1) = 1 through a(7) = 13 partitions:
%e A317256   (1)  (2)   (3)    (4)     (5)      (6)       (7)
%e A317256        (11)  (21)   (22)    (32)     (33)      (43)
%e A317256              (111)  (31)    (41)     (42)      (52)
%e A317256                     (211)   (311)    (51)      (61)
%e A317256                     (1111)  (2111)   (222)     (322)
%e A317256                             (11111)  (321)     (421)
%e A317256                                      (411)     (511)
%e A317256                                      (2211)    (3211)
%e A317256                                      (3111)    (4111)
%e A317256                                      (21111)   (22111)
%e A317256                                      (111111)  (31111)
%e A317256                                                (211111)
%e A317256                                                (1111111)
%e A317256 For example, starting with the partition y = (3,2,2,1,1) and repeatedly taking run-lengths and reversing gives (3,2,2,1,1) -> (2,2,1) -> (1,2), which is not weakly decreasing, so y is not  alternately co-strong. On the other hand, we have (3,3,2,2,1,1,1) -> (3,2,2) -> (2,1) -> (1,1) -> (2) -> (1), so (3,3,2,2,1,1,1) is counted under a(13).
%t A317256 tniQ[q_]:=Or[q=={},q=={1},And[LessEqual@@Length/@Split[q],tniQ[Reverse[Length/@Split[q]]]]];
%t A317256 Table[Length[Select[IntegerPartitions[n],tniQ]],{n,0,30}]
%Y A317256 Cf. A000041, A100883, A181819, A182850, A182857, A304660, A305563, A317081, A317086, A317245, A317258.
%Y A317256 The Heinz numbers of these partitions are given by A317257.
%Y A317256 The total (instead of alternating) version is A332275.
%Y A317256 Dominates A332289 (the normal version).
%Y A317256 The generalization to compositions is A332338.
%Y A317256 The dual version is A332339.
%Y A317256 The case of reversed partitions is (also) A332339.
%Y A317256 Cf. A316496  A317491, A329746, A332292, A332297, A332340.
%K A317256 nonn
%O A317256 0,3
%A A317256 _Gus Wiseman_, Jul 25 2018
%E A317256 Updated with corrected terminology by _Gus Wiseman_, Mar 08 2020

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