A241825 Number of partitions p = [x(1), ..., x(k)], where x(1) >= x(2) >= ... >= x(k), of n such that min(x(i) - x(i-1)) = number of distinct parts of p.

0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, 8, 7, 9, 8, 11, 10, 13, 13, 17, 17, 21, 22, 27, 28, 33, 35, 41, 43, 49, 52, 59, 62, 69, 73, 81, 85, 93, 98, 107, 112, 121, 128, 138, 145, 156, 165, 178, 188, 202

Offset

0,13

Links

Table of n, a(n) for n=0..62.

Formula

a(n) = A241824(n) - A241823(n) = A241826(n) - A241827(n) for n >= 0.

a(n) + A241823(n) + A241827(n) = A000041(n) for n >= 0.

Example

a(6) counts this single partition:  42.

Mathematica

z = 30; f[n_] := f[n] = IntegerPartitions[n]; d[p_] := d[p] = Length[DeleteDuplicates[p]]; g1[p_] := Min[-Differences[p]]
Table[Count[f[n], p_ /; g1[p] < d[p]], {n, 0, z}]  (* A241823 *)
Table[Count[f[n], p_ /; g1[p] <= d[p]], {n, 0, z}] (* A241824 *)
Table[Count[f[n], p_ /; g1[p] == d[p]], {n, 0, z}] (* A241825 *)
Table[Count[f[n], p_ /; g1[p] >= d[p]], {n, 0, z}] (* A241826 *)
Table[Count[f[n], p_ /; g1[p] > d[p]], {n, 0, z}]  (* A241827 *)

Crossrefs

Cf. A241823, A241824, A241826, A241827, A000041.

Sequence in context: A097140 A028242 A030451 * A029162 A325132 A225854

Adjacent sequences: A241822 A241823 A241824 * A241826 A241827 A241828

Keyword

nonn,easy

Author

Clark Kimberling, Apr 30 2014

Status

approved