A317588 Number of uniformly normal integer partitions of n.

1, 1, 2, 3, 4, 3, 6, 3, 5, 6, 7, 5, 8, 5, 7, 10, 7, 6, 12, 7, 12, 14, 10, 11, 18, 11, 13, 16, 18, 15, 35, 16, 26, 24, 27, 26, 47, 33, 44, 48, 58, 48, 76, 63, 81, 79, 98, 94, 123, 109, 135, 131, 148, 140, 162, 149, 152, 162, 166, 175, 202, 191, 221, 232, 233

Offset

1,3

Comments

An integer partition is uniformly normal if either (1) it is of the form (x, x, ..., x) for some x > 0, or (2a) it spans an initial interval of positive integers, and (2b) its multiplicities, sorted in weakly decreasing order, are themselves a uniformly normal integer partition.

Links

Table of n, a(n) for n=1..65.

A. David Christopher and M. Davamani Christober, Relatively Prime Uniform Partitions, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp.1-12.

Example

The a(6) = 6 uniformly normal integer partitions are (6), (33), (321), (222), (2211), (111111). Missing from this list are (51), (42), (411), (3111), (21111).
The a(21) = 14 uniformly normal integer partitions (n = 21):
  (n),
  (777),
  (654321),
  (4443321), (3333333),
  (44432211), (44333211), (44332221),
  (4432221111), (4333221111), (4332222111),
  (433322211),
  (22222221111111),
  (111111111111111111111).

Mathematica

uninrmQ[q_]:=Or[q=={}||Length[Union[q]]==1, And[Union[q]==Range[Max[q]], uninrmQ[Sort[Length/@Split[q], Greater]]]];
Table[Length[Select[IntegerPartitions[n], uninrmQ]], {n, 0, 30}]

Crossrefs

Cf. A181819, A182850, A275870, A317081, A317088, A317246, A317246, A317491, A317589, A317590.

Sequence in context: A361442 A288778 A290139 * A080383 A086369 A337532

Adjacent sequences: A317585 A317586 A317587 * A317589 A317590 A317591

Keyword

nonn

Author

Gus Wiseman, Aug 01 2018

Status

approved