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Also the number of totally strong reversed partitions of n. A sequence is totally co-strong if it is empty, equal to (1), or weakly decreasing (strong) with totally strong run-lengths.

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

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

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

A sequence is recursively totally co-strong if it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and are themselves a recursively totally co-strong sequence.

Also the number of totally co-strong compositions reversed partitions of n. A sequence is totally co-strong if it is empty, equal to (1), or weakly increasing decreasing (co-strong) with totally co-strong run-lengths.

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 (except y) are having weakly increasing, run-lengths, and the last is a singleton, (1), so y is counted under a(44).

The total strong version is A316496.

The strong version is (also) A316496.

A sequence is recursively co-strong if it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and are themselves a recursively co-strong sequence.

Also the number of totally co-strong compositions of n. A sequence is totally co-strong if it is empty, equal to (1), or weakly increasing (co-strong) with totally co-strong run-lengths.

allocated for Gus WisemanNumber of recursively 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

0,3

A sequence is recursively co-strong if it is empty, equal to (1), or its run-lengths are weakly increasing and are themselves a recursively co-strong sequence.

Also the number of totally co-strong compositions of n. A sequence is totally co-strong if it is empty, equal to (1), or weakly increasing with totally co-strong run-lengths.

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). All of these (except y) are weakly increasing, and the last is a singleton, so y is counted under a(44).

totincQ[q_]:=Or[q=={}, q=={1}, And[LessEqual@@Length/@Split[q], totincQ[Length/@Split[q]]]];

Table[Length[Select[IntegerPartitions[n], totincQ]], {n, 0, 30}]

The total version is A316496.

The strong version is (also) 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.

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Gus Wiseman, Feb 12 2020

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editing

allocated for Gus Wiseman

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