A241830 Number of partitions p = [x(1), ..., x(k)], where x(1) >= x(2) >= ... >= x(k), of n such that max(x(i) - x(i-1)) = number of parts of p.

0, 0, 0, 0, 1, 0, 2, 0, 3, 2, 5, 3, 9, 5, 11, 11, 18, 15, 29, 26, 41, 42, 60, 61, 91, 91, 125, 137, 182, 195, 260, 282, 364, 406, 509, 569, 715, 795, 980, 1111, 1351, 1523, 1847, 2087, 2505, 2847, 3384, 3844, 4563, 5174, 6098, 6941, 8134, 9243, 10807, 12273

Offset

0,7

Links

Table of n, a(n) for n=0..55.

Formula

a(n) = A241829(n) - A241828(n) = A241831(n) - A241832(n).

a(n) + A241828(n) + A241832(n) = A000041(n) for n >= 0.

Example

a(6) counts these 2 partitions:  42, 411.

Mathematica

z = 30; f[n_] := f[n] = IntegerPartitions[n]; g[p_] := Max[-Differences[p]]
Table[Count[f[n], p_ /; g[p] < Length[p]], {n, 0, z}]  (* A241828 *)
Table[Count[f[n], p_ /; g[p] <= Length[p]], {n, 0, z}] (* A241829 *)
Table[Count[f[n], p_ /; g[p] == Length[p]], {n, 0, z}] (* A241830 *)
Table[Count[f[n], p_ /; g[p] >= Length[p]], {n, 0, z}] (* A241831 *)
Table[Count[f[n], p_ /; g[p] > Length[p]], {n, 0, z}]  (* A241832 *)

Crossrefs

Cf. A241828, A241829, A241831, A241832, A000041.

Sequence in context: A349125 A231117 A021496 * A151929 A266691 A083236

Adjacent sequences: A241827 A241828 A241829 * A241831 A241832 A241833

Keyword

nonn,easy

Author

Clark Kimberling, Apr 30 2014

Status

approved