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A316496
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Number of totally strong integer partitions of n.
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20
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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totincQ[q_]:=Or[q=={}, q=={1}, And[GreaterEqual@@Length/@Split[q], totincQ[Length/@Split[q]]]];
Table[Length[Select[IntegerPartitions[n], totincQ]], {n, 0, 30}]
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CROSSREFS
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The Heinz numbers of these partitions are A316529.
The version for compositions is A332274.
The version for reversed partitions is (also) A332275.
The narrowly normal version is A332297.
Partitions with weakly decreasing run-lengths are A100882.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Updated with corrected terminology by Gus Wiseman, Mar 07 2020
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STATUS
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approved
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