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%I #8 Feb 14 2020 08:02:38
%S 1,1,2,3,5,6,11,12,17,22,30,32,49,53,70,82,108,119,156,171,219,250,
%T 305,336,424,468,562,637,754,835,1011,1108,1304,1461,1692,1873,2212,
%U 2417,2787,3109,3562,3911,4536,4947,5653,6265,7076,7758,8883,9669,10945,12040
%N Number of totally co-strong integer partitions of n.
%C A sequence is totally co-strong if it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and are themselves a totally co-strong sequence.
%C Also the number of totally strong reversed integer partitions of n.
%e The a(1) = 1 through a(7) = 12 partitions:
%e (1) (2) (3) (4) (5) (6) (7)
%e (11) (21) (22) (32) (33) (43)
%e (111) (31) (41) (42) (52)
%e (211) (311) (51) (61)
%e (1111) (2111) (222) (322)
%e (11111) (321) (421)
%e (411) (511)
%e (2211) (4111)
%e (3111) (22111)
%e (21111) (31111)
%e (111111) (211111)
%e (1111111)
%e For example, the partition y = (5,4,4,4,3,3,3,2,2,2,2,2,2,1,1,1,1,1,1) has run-lengths (1,3,3,6,6), with run-lengths (1,2,2), with run-lengths (1,2), with run-lengths (1,1), with run-lengths (2), with run-lengths (1). All of these having weakly increasing run-lengths, and the last is (1), so y is counted under a(44).
%t totincQ[q_]:=Or[q=={},q=={1},And[LessEqual@@Length/@Split[q],totincQ[Length/@Split[q]]]];
%t Table[Length[Select[IntegerPartitions[n],totincQ]],{n,0,30}]
%Y The strong version is A316496.
%Y The version for reversed partitions is (also) A316496.
%Y The alternating version is A317256.
%Y The generalization to compositions is A332274.
%Y Cf. A001462, A100883, A181819, A182850, A317491, A329746, A332289, A332297, A332336, A332337, A332338, A332339, A332340.
%K nonn
%O 0,3
%A _Gus Wiseman_, Feb 12 2020