cp's OEIS Frontend

A332275 Number of totally co-strong integer partitions of n.

%I A332275 #8 Feb 14 2020 08:02:38
%S A332275 1,1,2,3,5,6,11,12,17,22,30,32,49,53,70,82,108,119,156,171,219,250,
%T A332275 305,336,424,468,562,637,754,835,1011,1108,1304,1461,1692,1873,2212,
%U A332275 2417,2787,3109,3562,3911,4536,4947,5653,6265,7076,7758,8883,9669,10945,12040
%N A332275 Number of totally co-strong integer partitions of n.
%C A332275 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 A332275 Also the number of totally strong reversed integer partitions of n.
%e A332275 The a(1) = 1 through a(7) = 12 partitions:
%e A332275   (1)  (2)   (3)    (4)     (5)      (6)       (7)
%e A332275        (11)  (21)   (22)    (32)     (33)      (43)
%e A332275              (111)  (31)    (41)     (42)      (52)
%e A332275                     (211)   (311)    (51)      (61)
%e A332275                     (1111)  (2111)   (222)     (322)
%e A332275                             (11111)  (321)     (421)
%e A332275                                      (411)     (511)
%e A332275                                      (2211)    (4111)
%e A332275                                      (3111)    (22111)
%e A332275                                      (21111)   (31111)
%e A332275                                      (111111)  (211111)
%e A332275                                                (1111111)
%e A332275 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 A332275 totincQ[q_]:=Or[q=={},q=={1},And[LessEqual@@Length/@Split[q],totincQ[Length/@Split[q]]]];
%t A332275 Table[Length[Select[IntegerPartitions[n],totincQ]],{n,0,30}]
%Y A332275 The strong version is A316496.
%Y A332275 The version for reversed partitions is (also) A316496.
%Y A332275 The alternating version is A317256.
%Y A332275 The generalization to compositions is A332274.
%Y A332275 Cf. A001462, A100883, A181819, A182850, A317491, A329746, A332289, A332297, A332336, A332337, A332338, A332339, A332340.
%K A332275 nonn
%O A332275 0,3
%A A332275 _Gus Wiseman_, Feb 12 2020