A332272 Number of narrowly recursively normal integer partitions of n.

1, 1, 2, 3, 5, 6, 8, 10, 14, 18, 23, 30, 37, 46, 52, 70, 80, 100, 116, 146, 171, 203, 236, 290, 332, 401, 458, 547, 626, 744, 851, 1004, 1157, 1353, 1553, 1821, 2110, 2434, 2810, 3250, 3741, 4304, 4949, 5661, 6510, 7450, 8501, 9657, 11078, 12506, 14329, 16185

Offset

0,3

Comments

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).

Links

Table of n, a(n) for n=0..51.

Formula

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

Example

The a(6) = 8 partitions are (6), (51), (42), (411), (33), (321), (222), (111111). Missing from this list are (3111), (2211), (21111).
The a(1) = 1 through a(8) = 14 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (211)   (221)    (51)      (61)       (62)
                    (1111)  (311)    (222)     (322)      (71)
                            (11111)  (321)     (331)      (332)
                                     (411)     (421)      (422)
                                     (111111)  (511)      (431)
                                               (3211)     (521)
                                               (1111111)  (611)
                                                          (2222)
                                                          (3221)
                                                          (4211)
                                                          (11111111)

Mathematica

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
recnQ[ptn_]:=With[{qtn=Length/@Split[ptn]}, Or[Length[qtn]<=1, And[normQ[qtn], recnQ[qtn]]]];
Table[Length[Select[IntegerPartitions[n], recnQ]], {n, 0, 30}]

Crossrefs

The strict instead of narrow version is A330937.

The normal case is A332277.

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

The wide version is A332295(n) - 1.

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

Sequence in context: A122493 A284830 A053873 * A240314 A118053 A252482

Adjacent sequences: A332269 A332270 A332271 * A332273 A332274 A332275

Keyword

nonn

Author

Gus Wiseman, Mar 08 2020

Status

approved