login
Number of narrowly recursively normal integer partitions of n.
6

%I #7 Mar 09 2020 18:25:24

%S 1,1,2,3,5,6,8,10,14,18,23,30,37,46,52,70,80,100,116,146,171,203,236,

%T 290,332,401,458,547,626,744,851,1004,1157,1353,1553,1821,2110,2434,

%U 2810,3250,3741,4304,4949,5661,6510,7450,8501,9657,11078,12506,14329,16185

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

%C A sequence is narrowly recursively normal if either it is constant (narrow) or its run-lengths are a narrowly recursively normal sequence covering an initial interval of positive integers (normal).

%F For n > 1, a(n) = A317491(n) + A000005(n) - 2.

%e The a(6) = 8 partitions are (6), (51), (42), (411), (33), (321), (222), (111111). Missing from this list are (3111), (2211), (21111).

%e The a(1) = 1 through a(8) = 14 partitions:

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

%e (11) (21) (22) (32) (33) (43) (44)

%e (111) (31) (41) (42) (52) (53)

%e (211) (221) (51) (61) (62)

%e (1111) (311) (222) (322) (71)

%e (11111) (321) (331) (332)

%e (411) (421) (422)

%e (111111) (511) (431)

%e (3211) (521)

%e (1111111) (611)

%e (2222)

%e (3221)

%e (4211)

%e (11111111)

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

%t recnQ[ptn_]:=With[{qtn=Length/@Split[ptn]},Or[Length[qtn]<=1,And[normQ[qtn],recnQ[qtn]]]];

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

%Y The strict instead of narrow version is A330937.

%Y The normal case is A332277.

%Y The widely normal case is A332277(n) - 1 for n > 1.

%Y The wide version is A332295(n) - 1.

%Y Cf. A000009, A107429, A181819, A316496, A317081, A317245, A317491, A329744, A329746, A329766, A332576.

%K nonn

%O 0,3

%A _Gus Wiseman_, Mar 08 2020