%I #28 May 14 2019 12:25:35

%S 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,

%T 9,5,14,7,13,13,18,7,20,17,22,10,22,19,32,15,26,26,40,15,37,36,43,21,

%U 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

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

%C 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.

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

%C 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.

%C 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

%H James Allen Fill, Svante Janson, Mark Daniel Ward, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i2p18">Partitions with Distinct Multiplicities of Parts: On An "Unsolved Problem" Posed By Herbert S. Wilf</a>, The Electronic Journal of Combinatorics, vol.19, no.2, 2012.

%H Stephan Wagner, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i4p13">The Number of Fixed Points of Wilf's Partition Involution</a>, The Electronic Journal of Combinatorics, 20(4) (2013), #P13.

%e a(16) = 4 because the following partitions of 16 are fixed points:

%e 4* 2 + 2* 4 = 2 + 2 + 2 + 2 + 4 + 4

%e 4* 4 = 4 + 4 + 4 + 4

%e 6* 1 + 2* 2 + 1* 6 = 1 + 1 + 1 + 1 + 1 + 1 + 2 + 2 + 6

%e 8* 1 + 1* 8 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 8

%e From _Gus Wiseman_, May 04 2019: (Start)

%e 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.

%e 1 22 221 3111 41111 333 3331 33222 33322 333221 4444

%e 211 322111 4221111 332221 52211111 442222

%e 511111 6111111 333211 71111111 622111111

%e 811111111

%e (End)

%t winv[n_]:=Times@@Cases[FactorInteger[n],{p_,k_}:>Prime[k]^PrimePi[p]];

%t Table[Length[Select[IntegerPartitions[n],winv[Times@@Prime/@#]==Times@@Prime/@#&]],{n,0,30}] (* _Gus Wiseman_, May 04 2019 *)

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

%K nonn

%O 0,5

%A _Joerg Arndt_, Oct 08 2012