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

0,6

Links

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

Formula

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

Example

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

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 *)

Crossrefs

Cf. A240221, A000041.

Sequence in context: A192116 A088163 A048448 * A281611 A054060 A024832

Adjacent sequences: A240299 A240300 A240301 * A240303 A240304 A240305

Keyword

nonn,easy

Author

Clark Kimberling, Apr 04 2014

Status

approved