A330937 Number of strictly recursively normal integer partitions of n.

1, 2, 3, 5, 7, 10, 15, 20, 27, 35, 49, 58, 81, 100, 126, 160, 206, 246, 316, 374, 462, 564, 696, 813, 1006, 1195, 1441, 1701, 2058, 2394, 2896, 3367, 4007, 4670, 5542, 6368, 7540, 8702, 10199, 11734, 13760, 15734, 18384, 21008, 24441, 27893, 32380, 36841

Offset

0,2

Comments

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.

Links

Table of n, a(n) for n=0..47.

Example

The a(1) = 1 through a(9) = 15 partitions:
  (1)  (2)  (3)   (4)    (5)    (6)    (7)     (8)     (9)
            (21)  (31)   (32)   (42)   (43)    (53)    (54)
                  (211)  (41)   (51)   (52)    (62)    (63)
                         (221)  (321)  (61)    (71)    (72)
                         (311)  (411)  (322)   (332)   (81)
                                       (331)   (422)   (432)
                                       (421)   (431)   (441)
                                       (511)   (521)   (522)
                                       (3211)  (611)   (531)
                                               (3221)  (621)
                                               (4211)  (711)
                                                       (3321)
                                                       (4221)
                                                       (4311)
                                                       (5211)
                                                       (32211)

Mathematica

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
recnQ[ptn_]:=With[{qtn=Length/@Split[ptn]}, Or[ptn=={}, UnsameQ@@qtn, And[normQ[qtn], recnQ[qtn]]]];
Table[Length[Select[IntegerPartitions[n], recnQ]], {n, 0, 30}]

Crossrefs

The narrow instead of strict version is A332272.

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

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

Sequence in context: A291298 A092021 A022475 * A011972 A338553 A272402

Adjacent sequences: A330934 A330935 A330936 * A330938 A330939 A330940

Keyword

nonn

Author

Gus Wiseman, Mar 09 2020

Status

approved