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A316496
Number of totally strong integer partitions of n.
20
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
0,3
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.
EXAMPLE
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.
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}]
CROSSREFS
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.
Sequence in context: A297637 A325415 A331076 * A332339 A100882 A171979
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 29 2018
EXTENSIONS
Updated with corrected terminology by Gus Wiseman, Mar 07 2020
STATUS
approved