A114639 Number of partitions of n such that the set of parts and the set of multiplicities of parts are disjoint.

1, 0, 2, 2, 2, 3, 5, 4, 7, 7, 13, 16, 19, 23, 33, 34, 44, 58, 63, 80, 101, 112, 139, 171, 196, 234, 288, 328, 394, 478, 545, 658, 777, 881, 1050, 1236, 1414, 1666, 1936, 2216, 2592, 3018, 3428, 3992, 4604, 5243, 6069, 6986, 7951, 9139, 10447, 11892, 13625

Offset

0,3

Comments

The Heinz numbers of these partitions are given by A325131. - Gus Wiseman, Apr 02 2019

Links

Alois P. Heinz, Table of n, a(n) for n = 0..100

Example

From Gus Wiseman, Apr 02 2019: (Start)
The a(2) = 2 through a(9) = 7 partitions:
  (2)   (3)    (4)     (5)      (6)       (7)        (8)         (9)
  (11)  (111)  (1111)  (32)     (33)      (43)       (44)        (54)
                       (11111)  (42)      (52)       (53)        (63)
                                (222)     (1111111)  (62)        (72)
                                (111111)             (2222)      (432)
                                                     (3311)      (222111)
                                                     (11111111)  (111111111)
(End)

Maple

b:= proc(n, i, p, m) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1, p, select(x-> x<i, m))+
      add(`if`(i=j or i in m or j in p, 0, b(n-i*j, i-1,
         select(x-> x<=n-i*j, p union {i}),
         select(x-> x<i, m union {j}))), j=1..n/i)))
    end:
a:= n-> b(n$2, {}$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 09 2016

Mathematica

b[n_, i_, p_, m_] := b[n, i, p, m] = If[n == 0, 1, If[i<1, 0, b[n, i-1, p, Select[m, #<i&]] + Sum[If[i == j || MemberQ[m, i] || MemberQ[p, j], 0, b[n-i*j, i-1, Select[ p ~Union~ {i}, # <= n-i*j&], Select[m ~Union~ {j}, #<i&]]], {j, 1, n/i}]]]; a[n_] := b[n, n, {}, {}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n], Intersection[#, Length/@Split[#]]=={}&]], {n, 0, 30}] (* Gus Wiseman, Apr 02 2019 *)

Crossrefs

Cf. A052335, A087153, A114640, A115584, A117144, A276429, A324572, A325130, A325131, A336032.

Sequence in context: A270566 A182534 A165918 * A071867 A126337 A322261

Adjacent sequences: A114636 A114637 A114638 * A114640 A114641 A114642

Keyword

nonn

Author

Vladeta Jovovic, Feb 18 2006

Extensions

a(0)=1 prepended and more terms from Alois P. Heinz, Aug 09 2016

Status

approved