OFFSET
0,3
COMMENTS
A sequence is alternately co-strong if either it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and, when reversed, are themselves an alternately co-strong sequence.
Also the number of alternately strong reversed integer partitions of n.
EXAMPLE
The a(1) = 1 through a(7) = 13 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) (3211)
(3111) (4111)
(21111) (22111)
(111111) (31111)
(211111)
(1111111)
For example, starting with the partition y = (3,2,2,1,1) and repeatedly taking run-lengths and reversing gives (3,2,2,1,1) -> (2,2,1) -> (1,2), which is not weakly decreasing, so y is not alternately co-strong. On the other hand, we have (3,3,2,2,1,1,1) -> (3,2,2) -> (2,1) -> (1,1) -> (2) -> (1), so (3,3,2,2,1,1,1) is counted under a(13).
MATHEMATICA
tniQ[q_]:=Or[q=={}, q=={1}, And[LessEqual@@Length/@Split[q], tniQ[Reverse[Length/@Split[q]]]]];
Table[Length[Select[IntegerPartitions[n], tniQ]], {n, 0, 30}]
CROSSREFS
Cf. A000041, A100883, A181819, A182850, A182857, A304660, A305563, A317081, A317086, A317245, A317258.
The Heinz numbers of these partitions are given by A317257.
The total (instead of alternating) version is A332275.
Dominates A332289 (the normal version).
The generalization to compositions is A332338.
The dual version is A332339.
The case of reversed partitions is (also) A332339.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 25 2018
EXTENSIONS
Updated with corrected terminology by Gus Wiseman, Mar 08 2020
STATUS
approved