A240314 Number of partitions p of n such that (maximal multiplicity of the parts of p) > (maximal part of p).

0, 0, 1, 1, 1, 2, 3, 5, 6, 8, 10, 14, 18, 25, 32, 42, 52, 69, 84, 108, 133, 169, 206, 260, 316, 394, 479, 591, 713, 875, 1052, 1280, 1534, 1855, 2215, 2665, 3169, 3795, 4501, 5362, 6339, 7525, 8868, 10486, 12324, 14521, 17020, 19990, 23366, 27361, 31905

Offset

0,6

Links

Table of n, a(n) for n=0..50.

Formula

a(n) = A240313(n) - A240312(n) for n >= 0.

a(n) + A240310(n) + A240312(n) = A000041(n) for n >= 0.

Example

a(6) counts these 3 partitions:  222, 2111, 111111.

Mathematica

z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]]  (* maximal multiplicity *)
Table[Count[f[n], p_ /; m[p] < Max[p]], {n, 0, z}]  (* A240310 *)
Table[Count[f[n], p_ /; m[p] <= Max[p]], {n, 0, z}] (* A240311 *)
Table[Count[f[n], p_ /; m[p] == Max[p]], {n, 0, z}] (* A240312 *)
Table[Count[f[n], p_ /; m[p] >= Max[p]], {n, 0, z}] (* A240313 *)
Table[Count[f[n], p_ /; m[p] > Max[p]], {n, 0, z}]  (* A240314 *)

Crossrefs

Cf. A240310, A240311, A240312, A240313, A000041.

Sequence in context: A284830 A053873 A332272 * A118053 A252482 A348868

Adjacent sequences: A240311 A240312 A240313 * A240315 A240316 A240317

Keyword

nonn,easy

Author

Clark Kimberling, Apr 05 2014

Status

approved