A332338 Number of alternately co-strong compositions of n.

1, 1, 2, 4, 7, 12, 24, 39, 72, 125, 224, 387, 697, 1205, 2141, 3736, 6598, 11516, 20331, 35526, 62507, 109436, 192200, 336533, 590582, 1034187

Offset: 0

Author

Gus Wiseman, Feb 17 2020

Keywords

• nonn
• more

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.

Examples

			The a(1) = 1 through a(5) = 12 compositions:
  (1)  (2)   (3)    (4)     (5)
       (11)  (12)   (13)    (14)
             (21)   (22)    (23)
             (111)  (31)    (32)
                    (112)   (41)
                    (121)   (113)
                    (1111)  (131)
                            (212)
                            (221)
                            (1112)
                            (1121)
                            (11111)
For example, starting with the composition y = (1,6,2,2,1,1,1,1) and repeatedly taking run-lengths and reversing gives (1,6,2,2,1,1,1,1) -> (4,2,1,1) -> (2,1,1) -> (2,1) -> (1,1) -> (2). All of these have weakly increasing run-lengths and the last is a singleton, so y is counted under a(15).
		

Crossrefs

The case of partitions is A317256.

The recursive (rather than alternating) version is A332274.

The total (rather than alternating) version is (also) A332274.

The strong version is this same sequence.

The case of reversed partitions is A332339.

The normal version is A332340(n) + 1 for n > 1.

Cf. A001462, A100883, A181819, A182850, A316496, A317257, A329744, A329746, A332275, A332292, A332296.

Programs

Mathematica

tniQ[q_]:=Or[q=={},q=={1},And[LessEqual@@Length/@Split[q],tniQ[Reverse[Length/@Split[q]]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],tniQ]],{n,0,10}]