A240302 Number of partitions of n such that (maximal multiplicity of parts) > (multiplicity of the maximal part).

0, 0, 0, 0, 1, 2, 3, 7, 10, 16, 23, 35, 47, 70, 93, 126, 169, 228, 294, 391, 501, 648, 827, 1057, 1329, 1683, 2105, 2631, 3266, 4056, 4992, 6156, 7538, 9221, 11234, 13664, 16549, 20033, 24152, 29077, 34904, 41844, 50012, 59710, 71100, 84541, 100318, 118869

Offset: 0

Author

Clark Kimberling, Apr 04 2014

Keywords

• nonn
• easy

Examples

			a(7) counts these 7 partitions: 511, 4111, 322, 3211, 31111, 22111, 211111.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A240221, A000041.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
      `if`(i<1, 0, b(n, i-1, k) +add(b(n-i*j, i-1, `if`(k=-1, j,
      `if`(k=0, 0, `if`(j>k, 0, k)))), j=1..n/i)))
    end:
a:= n-> b(n$2, -1):
seq(a(n), n=0..70);  # Alois P. Heinz, Apr 12 2014

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] == Count[p, Max[p]]], {n, 0, z}] (* A171979 *)
Table[Count[f[n], p_ /; m[p] > Count[p, Max[p]]], {n, 0, z}] (* A240302 *)
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, If[k == 0, 1, 0],
     If[i < 1, 0, b[n, i - 1, k] + Sum[b[n - i*j, i - 1, If[k == -1, j,
     If[k == 0, 0, If[j > k, 0, k]]]], {j, 1, n/i}]]];
a[n_] := b[n, n, -1];
a /@ Range[0, 70] (* Jean-François Alcover, Jun 05 2021, after Alois P. Heinz *)

Formula

a(n) + A171979(n) = A000041(n) for n >= 1.