A332576 Number of integer partitions of n that are all 1's or whose run-lengths cover an initial interval of positive integers.

1, 1, 2, 3, 4, 6, 6, 10, 12, 17, 21, 31, 35, 51, 59, 80, 97, 130, 153, 204, 244, 308, 376, 475, 564, 708, 851, 1043, 1247, 1533, 1816, 2216, 2633, 3174, 3766, 4526, 5324, 6376, 7520, 8917, 10479, 12415, 14524, 17134, 20035, 23489, 27423, 32091, 37286, 43512

Offset

0,3

Comments

First differs from A317491 at a(11) = 31, A317491(11) = 30.

Links

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

Formula

a(n > 1) = A317081(n) + 1.

Example

The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (31)    (32)     (42)      (43)       (53)
             (111)  (211)   (41)     (51)      (52)       (62)
                    (1111)  (221)    (321)     (61)       (71)
                            (311)    (411)     (322)      (332)
                            (11111)  (111111)  (331)      (422)
                                               (421)      (431)
                                               (511)      (521)
                                               (3211)     (611)
                                               (1111111)  (3221)
                                                          (4211)
                                                          (11111111)

Mathematica

nQ[ptn_]:=Or[ptn=={}, Union[ptn]=={1}, Union[Length/@Split[ptn]]==Range[Max[Length/@Split[ptn]]]];
Table[Length[Select[IntegerPartitions[n], nQ]], {n, 0, 30}]

Crossrefs

The narrow version is A317081.

Heinz numbers of these partitions first differ from A317492 in having 420.

Not counting constant-1 sequences gives A317081.

Dominated by A332295.

Cf. A000009, A001462, A181819, A182850, A317245, A317491, A329746, A329747, A332272, A332277.

Sequence in context: A373700 A317491 A332295 * A028335 A346117 A007464

Adjacent sequences: A332573 A332574 A332575 * A332577 A332578 A332579

Keyword

nonn

Author

Gus Wiseman, Mar 05 2020

Status

approved