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A240302
Number of partitions of n such that (maximal multiplicity of parts) > (multiplicity of the maximal part).
5
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
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 *)
(* Alternative: *)
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
Sequence in context: A192116 A088163 A048448 * A281611 A054060 A024832
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 04 2014
STATUS
approved