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Number of strictly recursively normal integer partitions of n.
1

%I #5 Mar 09 2020 18:25:08

%S 1,2,3,5,7,10,15,20,27,35,49,58,81,100,126,160,206,246,316,374,462,

%T 564,696,813,1006,1195,1441,1701,2058,2394,2896,3367,4007,4670,5542,

%U 6368,7540,8702,10199,11734,13760,15734,18384,21008,24441,27893,32380,36841

%N Number of strictly recursively normal integer partitions of n.

%C A sequence is strictly recursively normal if either it empty, its run-lengths are distinct (strict), or its run-lengths cover an initial interval of positive integers (normal) and are themselves a strictly recursively normal sequence.

%e The a(1) = 1 through a(9) = 15 partitions:

%e (1) (2) (3) (4) (5) (6) (7) (8) (9)

%e (21) (31) (32) (42) (43) (53) (54)

%e (211) (41) (51) (52) (62) (63)

%e (221) (321) (61) (71) (72)

%e (311) (411) (322) (332) (81)

%e (331) (422) (432)

%e (421) (431) (441)

%e (511) (521) (522)

%e (3211) (611) (531)

%e (3221) (621)

%e (4211) (711)

%e (3321)

%e (4221)

%e (4311)

%e (5211)

%e (32211)

%t normQ[m_]:=m=={}||Union[m]==Range[Max[m]];

%t recnQ[ptn_]:=With[{qtn=Length/@Split[ptn]},Or[ptn=={},UnsameQ@@qtn,And[normQ[qtn],recnQ[qtn]]]];

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

%Y The narrow instead of strict version is A332272.

%Y A wide instead of strict version is A332295(n) - 1 for n > 1.

%Y Cf. A107429, A181819, A316496, A317081, A317245, A317491, A329744, A329746, A329766, A332277, A332576.

%K nonn

%O 0,2

%A _Gus Wiseman_, Mar 09 2020