A332339 Number of alternately co-strong reversed integer partitions of n.

1, 1, 2, 3, 4, 5, 8, 8, 12, 14, 18, 20, 29, 28, 40, 45, 54, 59, 82, 81, 108, 118, 141, 154, 204, 204, 255, 285, 339, 363, 458, 471, 580, 632, 741, 806, 983, 1015, 1225, 1341, 1562, 1667, 2003, 2107, 2491, 2712, 3101, 3344, 3962, 4182, 4860, 5270, 6022, 6482

Offset: 0

Author

Gus Wiseman, Feb 17 2020

Keywords

• nonn

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 integer partitions of n.

Examples

			The a(1) = 1 through a(8) = 12 reversed partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (111)  (22)    (23)     (24)      (25)       (26)
                    (1111)  (122)    (33)      (34)       (35)
                            (11111)  (123)     (124)      (44)
                                     (222)     (133)      (125)
                                     (1122)    (1222)     (134)
                                     (111111)  (1111111)  (233)
                                                          (1133)
                                                          (2222)
                                                          (11222)
                                                          (11111111)
For example, starting with the composition y = (1,2,3,3,4,4,4) and repeatedly taking run-lengths and reversing gives (1,2,3,3,4,4,4) -> (3,2,1,1) -> (2,1,1) -> (2,1) -> (1,1) -> (2) -> (1). All of these have weakly increasing run-lengths and the last is equal to (1), so y is counted under a(21).
		

Crossrefs

The total (instead of alternating) version is A316496.

Alternately strong partitions are A317256.

The case of ordinary (not reversed) partitions is (also) A317256.

The generalization to compositions is A332338.

Cf. A100883, A181819, A182850, A317257, A329744, A329746, A332275, A332289, A332292, A332340.

Programs

Mathematica

tniQ[q_]:=Or[q=={},q=={1},And[LessEqual@@Length/@Split[q],tniQ[Reverse[Length/@Split[q]]]]];
Table[Length[Select[Sort/@IntegerPartitions[n],tniQ]],{n,0,30}]