A317256 Number of alternately co-strong integer partitions of n.

1, 1, 2, 3, 5, 6, 11, 13, 19, 25, 35, 42, 61, 74, 98, 122, 161, 194, 254, 304, 388, 472, 589, 700, 878, 1044, 1278, 1525, 1851, 2182, 2651, 3113, 3735, 4389, 5231, 6106, 7278, 8464, 9995, 11631, 13680, 15831, 18602, 21463, 25068, 28927, 33654, 38671, 44942, 51514

Offset: 0

Author

Gus Wiseman, Jul 25 2018

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 reversed integer partitions of n.

Examples

			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).
		

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.

Cf. A316496 A317491, A329746, A332292, A332297, A332340.

Programs

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}]

Extensions

Updated with corrected terminology by Gus Wiseman, Mar 08 2020