A240306 Number of partitions p of n such that (maximal multiplicity of the parts of p) <= (number of distinct parts of p).

1, 1, 1, 2, 3, 5, 6, 9, 13, 17, 25, 33, 44, 59, 76, 100, 131, 169, 212, 278, 352, 442, 555, 703, 871, 1088, 1342, 1664, 2046, 2517, 3064, 3758, 4574, 5548, 6718, 8119, 9797, 11784, 14150, 16935, 20263, 24179, 28798, 34237, 40677, 48122, 57008, 67291, 79400

Offset

0,4

Links

Table of n, a(n) for n=0..48.

Formula

a(n) = A240305(n) + A239964(n) for n >= 0.

a(n) + A240308(n) = A000041(n) for n >= 0.

Example

a(6) counts these 6 partitions:  6, 51, 42, 411, 321, 2211.

Mathematica

z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]]  (* maximal multiplicity *); d[p_] := d[p] = Length[DeleteDuplicates[p]] (* number of distinct terms *)
t1 = Table[Count[f[n], p_ /; m[p] < d[p]], {n, 0, z}]  (* A240305 *)
t2 = Table[Count[f[n], p_ /; m[p] <= d[p]], {n, 0, z}] (* A240306 *)
t3 = Table[Count[f[n], p_ /; m[p] == d[p]], {n, 0, z}] (* A239964 *)
t4 = Table[Count[f[n], p_ /; m[p] >= d[p]], {n, 0, z}] (* A240308 *)
t5 = Table[Count[f[n], p_ /; m[p] > d[p]], {n, 0, z}]  (* A240309 *)

Crossrefs

Cf. A240305, A239964, A240308, A240309, A000041.

Sequence in context: A212584 A330228 A166048 * A094873 A240726 A240203

Adjacent sequences: A240303 A240304 A240305 * A240307 A240308 A240309

Keyword

nonn,easy

Author

Clark Kimberling, Apr 05 2014

Status

approved