A317491 Number of fully normal integer partitions of n.

1, 1, 2, 3, 4, 6, 6, 10, 12, 17, 21, 30, 33, 46, 50, 68, 77, 100, 112, 146, 167, 201, 234, 290, 326, 400, 456, 545, 622, 744, 845, 1004, 1153, 1351, 1551, 1819, 2103, 2434, 2808, 3248, 3735, 4304, 4943, 5661, 6506, 7446, 8499, 9657, 11070, 12505, 14325, 16183

Offset

0,3

Comments

An integer partition is fully normal if either it is of the form (1,1,...,1) or its multiplicities span an initial interval of positive integers and, sorted in weakly decreasing order, are themselves fully normal.

Links

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

Formula

a(n) = A317245(n) iff n is 1 or a prime number.

Example

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

Mathematica

fulnrmQ[ptn_]:=With[{qtn=Sort[Length/@Split[ptn], Greater]}, Or[ptn=={}||Union[ptn]=={1}, And[Union[qtn]==Range[Max[qtn]], fulnrmQ[qtn]]]];
Table[Length[Select[IntegerPartitions[n], fulnrmQ]], {n, 0, 30}]

Crossrefs

Cf. A181819, A182850, A182857, A275870, A304660, A305563, A317081, A317086, A317088, A317246, A317492, A317493.

Sequence in context: A009490 A243930 A064778 * A332295 A332576 A028335

Adjacent sequences: A317488 A317489 A317490 * A317492 A317493 A317494

Keyword

nonn

Author

Gus Wiseman, Jul 30 2018

Status

approved