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

0,3

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.

Links

Table of n, a(n) for n=0..25.

Example

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).

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}]

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.

Sequence in context: A141017 A357925 A190591 * A332836 A328129 A360890

Adjacent sequences: A332335 A332336 A332337 * A332339 A332340 A332341

Keyword

nonn,more

Author

Gus Wiseman, Feb 17 2020

Status

approved