login
A332275
Number of totally co-strong integer partitions of n.
11
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
OFFSET
0,3
COMMENTS
A sequence is totally co-strong if it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and are themselves a totally co-strong sequence.
Also the number of totally strong reversed integer partitions of n.
EXAMPLE
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), with run-lengths (1). All of these having weakly increasing run-lengths, and the last is (1), so y is counted under a(44).
MATHEMATICA
totincQ[q_]:=Or[q=={}, q=={1}, And[LessEqual@@Length/@Split[q], totincQ[Length/@Split[q]]]];
Table[Length[Select[IntegerPartitions[n], totincQ]], {n, 0, 30}]
CROSSREFS
The strong version is A316496.
The version for reversed partitions is (also) A316496.
The alternating version is A317256.
The generalization to compositions is A332274.
Sequence in context: A033159 A366343 A199366 * A318689 A365311 A083710
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 12 2020
STATUS
approved