login
Number of integer partitions of n that are all 1's or whose run-lengths cover an initial interval of positive integers.
4

%I #7 Mar 05 2020 22:56:44

%S 1,1,2,3,4,6,6,10,12,17,21,31,35,51,59,80,97,130,153,204,244,308,376,

%T 475,564,708,851,1043,1247,1533,1816,2216,2633,3174,3766,4526,5324,

%U 6376,7520,8917,10479,12415,14524,17134,20035,23489,27423,32091,37286,43512

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

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

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

%e The a(1) = 1 through a(8) = 12 partitions:

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (11) (21) (31) (32) (42) (43) (53)

%e (111) (211) (41) (51) (52) (62)

%e (1111) (221) (321) (61) (71)

%e (311) (411) (322) (332)

%e (11111) (111111) (331) (422)

%e (421) (431)

%e (511) (521)

%e (3211) (611)

%e (1111111) (3221)

%e (4211)

%e (11111111)

%t nQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},Union[Length/@Split[ptn]]==Range[Max[Length/@Split[ptn]]]];

%t Table[Length[Select[IntegerPartitions[n],nQ]],{n,0,30}]

%Y The narrow version is A317081.

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

%Y Not counting constant-1 sequences gives A317081.

%Y Dominated by A332295.

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

%K nonn

%O 0,3

%A _Gus Wiseman_, Mar 05 2020