A217605 Number of partitions of n that are fixed points of a certain map (see comment).

1, 1, 0, 0, 2, 1, 1, 0, 1, 1, 3, 0, 3, 3, 3, 0, 4, 3, 2, 1, 6, 4, 5, 2, 5, 7, 10, 2, 10, 10, 11, 4, 9, 5, 14, 7, 13, 13, 18, 7, 20, 17, 22, 10, 22, 19, 32, 15, 26, 26, 40, 15, 37, 36, 43, 21, 44, 32, 55, 30, 46, 43, 75, 32, 67, 62, 83, 40, 82, 61, 104, 58, 89, 71, 136, 66, 114, 97, 149, 77, 143, 106, 176, 101, 160, 123, 222, 114, 190

Offset: 0

Author

Joerg Arndt, Oct 08 2012

Keywords

• nonn

Comments

Writing a partition of n in the form sum(k>=1, c(k) * k) another (in general different) partition is obtained as sum(k>=1, k * c(k)). For example, the partition 6 = 4* 1 + 1* 2 = 1 + 1 + 1 + 1 + 2 is mapped to 1* 4 + 2 *1 = 2* 1 + 1* 4 = 2 + 2 + 4. This sequence counts the fixed points of this map.

The map is not surjective. For example, all partitions into distinct parts are mapped to n* 1.

The map is an involution for partitions where the multiplicities of all parts are distinct (Wilf partitions, see A098859). If in addition the set of parts the same as the set of multiplicities then the partition is a fixed point.

The second part of the preceding comment is incorrect. For example, the partition (3,3,2,1,1,1) maps to (3,2,2,2,1,1) so is not a fixed point, even though the set of parts is identical to the set of multiplicities. - Gus Wiseman, May 04 2019

Examples

			a(16) = 4 because the following partitions of 16 are fixed points:
  4* 2 + 2* 4  =   2 + 2 + 2 + 2 + 4 + 4
  4* 4  =   4 + 4 + 4 + 4
  6* 1 + 2* 2 + 1* 6  =   1 + 1 + 1 + 1 + 1 + 1 + 2 + 2 + 6
  8* 1 + 1* 8  =   1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 8
From _Gus Wiseman_, May 04 2019: (Start)
The a(1) = 1 through a(16) = 4 partitions are the following (empty columns not shown). The Heinz numbers of these partitions are given by A048768.
  1  22   221  3111  41111  333  3331    33222    33322   333221    4444
     211                         322111  4221111  332221  52211111  442222
                                 511111  6111111  333211  71111111  622111111
                                                                    811111111
(End)
		

Links

• James Allen Fill, Svante Janson, Mark Daniel Ward, Partitions with Distinct Multiplicities of Parts: On An "Unsolved Problem" Posed By Herbert S. Wilf, The Electronic Journal of Combinatorics, vol.19, no.2, 2012.
• Stephan Wagner, The Number of Fixed Points of Wilf's Partition Involution, The Electronic Journal of Combinatorics, 20(4) (2013), #P13.

Crossrefs

Cf. A033461, A048767, A048768, A098859, A320348, A325324, A325325.

Programs

Mathematica

winv[n_]:=Times@@Cases[FactorInteger[n],{p_,k_}:>Prime[k]^PrimePi[p]];
Table[Length[Select[IntegerPartitions[n],winv[Times@@Prime/@#]==Times@@Prime/@#&]],{n,0,30}] (* Gus Wiseman, May 04 2019 *)