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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 29 2018
EXTENSIONS
Updated with corrected terminology by Gus Wiseman, Mar 07 2020
STATUS
approved