A240057 Number of partitions of n such that (greatest part) is not = (multiplicity of greatest part).

0, 2, 3, 4, 6, 10, 14, 21, 28, 40, 53, 74, 97, 131, 171, 225, 290, 377, 480, 616, 779, 987, 1238, 1556, 1935, 2411, 2981, 3685, 4527, 5562, 6793, 8295, 10081, 12241, 14805, 17890, 21538, 25906, 31062, 37201, 44429, 53004, 63070, 74964, 88898, 105297

Offset: 1

Author

Clark Kimberling, Apr 02 2014

Keywords

• nonn
• easy

Comments

Let # denote "number of" and c(p) = conjugate of partitionp. Then

A240057(n) = # p such that min(p) not = max(c(p));

A039899(n) = # p such that min(p) < max(c(p));

A039900(n) = # p such that min(p) <= max(c(p));

A006141(n) = # p such that min(p) = max(c(p));

A003114(n) = # p such that min(p) > max(c(p));

A003016(n) = # p such that min(p) >= max(c(p));

A064173(n) = # p such that max(p) < max(c(p));

A064174(n) = # p such that max(p) <= max(c(p));

A047993(n) = # p such that max(p) = max(c(p)).

See A240178 for related sequences. - Clark Kimberling, Apr 11 2014

Examples

			a(9) = 28 counts all the 30 partitions of 9 except 333 and 2211111.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A003106, A003114, A006141, A039899, A039900.

Programs

Maple

b:= proc(n, i) option remember; `if`(n<0, 0, `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))))
    end:
a:= n->combinat[numbpart](n)-add(b(n-j^2, j-1), j=0..isqrt(n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Apr 03 2014

Mathematica

z = 60; f[n_] := f[n] = IntegerPartitions[n];
t1 = Table[Count[f[n], p_ /; Max[p] < Count[p, Max[p]]], {n, 0, z}]  (* A003106 *)
t2 = Table[Count[f[n], p_ /; Max[p] <= Count[p, Max[p]]], {n, 0, z}] (* A003114 *)
t3 = Table[Count[f[n], p_ /; Max[p] == Count[p, Max[p]]], {n, 0, z}] (* A006141 *)
tt = Table[Count[f[n], p_ /; Max[p] != Count[p, Max[p]]], {n, 0, z}] (* A240057 *)
t4 = Table[Count[f[n], p_ /; Max[p] > Count[p, Max[p]]], {n, 0, z}] (* A039899 *)
t5 = Table[Count[f[n], p_ /; Max[p] >= Count[p, Max[p]]], {n, 0, z}] (* A039900 *)
(* second program: *)
b[n_, i_] := b[n, i] = If[n < 0, 0, If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i]]]]];
a[n_] := PartitionsP[n] - Sum[b[n - j^2, j - 1], {j, 0, Sqrt[n]}];
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *)

Formula

a(n) + A006141(n) = A000041(n) for n > 0.