A316496 Number of totally strong integer partitions of n.

1, 1, 2, 3, 4, 5, 8, 8, 12, 13, 18, 20, 27, 27, 38, 41, 52, 56, 73, 77, 99, 105, 129, 145, 176, 186, 229, 253, 300, 329, 395, 427, 504, 555, 648, 716, 836, 905, 1065, 1173, 1340, 1475, 1703, 1860, 2140, 2349, 2671, 2944, 3365, 3666, 4167, 4582, 5160, 5668

Offset: 0

Author

Gus Wiseman, Jul 29 2018

Keywords

• nonn

Comments

An integer partition is totally strong if either it is empty, equal to (1), or its run-lengths are weakly decreasing (strong) and are themselves a totally strong partition.

Examples

			The a(1) = 1 through a(8) = 12 totally strong partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (11111)  (222)     (331)      (71)
                                     (321)     (421)      (332)
                                     (2211)    (2221)     (431)
                                     (111111)  (1111111)  (521)
                                                          (2222)
                                                          (3311)
                                                          (22211)
                                                          (11111111)
For example, the partition (3,3,2,1) has run-lengths (2,1,1), which are weakly decreasing, but they have run-lengths (1,2), which are not weakly decreasing, so (3,3,2,1) is not totally strong.
		

Crossrefs

Cf. A181819, A182850, A182857, A304660, A305563, A316597.

The Heinz numbers of these partitions are A316529.

The version for compositions is A332274.

The dual version is A332275.

The version for reversed partitions is (also) A332275.

The narrowly normal version is A332297.

The alternating version is A332339 (see also A317256).

Partitions with weakly decreasing run-lengths are A100882.

Cf. A025487, A100882, A133808, A317245, A317491, A332278, A332291, A332336.

Programs

Mathematica

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

Extensions

Updated with corrected terminology by Gus Wiseman, Mar 07 2020