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

1, 1, 2, 3, 5, 6, 11, 12, 17, 22, 30, 32, 49, 53, 70, 82, 108, 119, 156, 171, 219, 250, 305, 336, 424, 468, 562, 637, 754, 835, 1011, 1108, 1304, 1461, 1692, 1873, 2212, 2417, 2787, 3109, 3562, 3911, 4536, 4947, 5653, 6265, 7076, 7758, 8883, 9669, 10945, 12040

Offset: 0

Author

Gus Wiseman, Feb 12 2020

Keywords

• nonn

Comments

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.

Also the number of totally strong reversed integer partitions of n.

Examples

			The a(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (311)    (51)      (61)
                    (1111)  (2111)   (222)     (322)
                            (11111)  (321)     (421)
                                     (411)     (511)
                                     (2211)    (4111)
                                     (3111)    (22111)
                                     (21111)   (31111)
                                     (111111)  (211111)
                                               (1111111)
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).
		

Crossrefs

The strong version is A316496.

The version for reversed partitions is (also) A316496.

The alternating version is A317256.

The generalization to compositions is A332274.

Cf. A001462, A100883, A181819, A182850, A317491, A329746, A332289, A332297, A332336, A332337, A332338, A332339, A332340.

Programs

Mathematica

totincQ[q_]:=Or[q=={},q=={1},And[LessEqual@@Length/@Split[q],totincQ[Length/@Split[q]]]];
Table[Length[Select[IntegerPartitions[n],totincQ]],{n,0,30}]