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

1,2

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

Links

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

Formula

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A347867 A256248 A089223 * A279467 A094861 A173473

Adjacent sequences: A240054 A240055 A240056 * A240058 A240059 A240060

Keyword

nonn,easy

Author

Clark Kimberling, Apr 02 2014

Status

approved