A241413 Number of partitions p of n such that the number of numbers having multiplicity 1 in p is a part of p.

0, 1, 0, 1, 1, 4, 5, 8, 10, 17, 21, 29, 38, 59, 68, 100, 124, 170, 214, 288, 351, 470, 576, 743, 921, 1176, 1430, 1816, 2214, 2753, 3364, 4176, 5015, 6215, 7478, 9120, 10966, 13351, 15916, 19301, 22982, 27618, 32846, 39354, 46515, 55570, 65598, 77842, 91730

Offset

0,6

Links

Table of n, a(n) for n=0..48.

Example

a(6) counts these 5 partitions:  42, 411, 321, 3111, 21111; e.g., 411 is counted because 1 part of 411 has multiplicity 1, and 1 is a part of 411.

Mathematica

z = 30; f[n_] := f[n] = IntegerPartitions[n]; u[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] ==  &]]]; e[q_] := Length[DeleteDuplicates[Select[q, Count[q, #] > 1 &]]]
Table[Count[f[n], p_ /; MemberQ[p, u[p]]], {n, 0, z}]  (* A241413 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && MemberQ[p, e[p]]], {n, 0, z}]  (* A241414 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && MemberQ[p, e[p]] ], {n, 0, z}] (* A241415 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241416 *)
Table[Count[f[n], p_ /; ! MemberQ[p, u[p]] && ! MemberQ[p, e[p]] ], {n, 0, z}] (* A241417 *)
Table[Count[f[n], p_ /; MemberQ[p, u[p]] || MemberQ[p, e[p]] ], {n, 0, z}] (* A239737 *)

Crossrefs

Cf. A241414, A241415, A241416, A241417, A239737.

Sequence in context: A022435 A190394 A116050 * A056721 A057479 A050140

Adjacent sequences: A241410 A241411 A241412 * A241414 A241415 A241416

Keyword

nonn,easy

Author

Clark Kimberling, Apr 23 2014

Status

approved