A317256 - OEIS

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

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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.

LINKS

Table of n, a(n) for n=0..49.

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