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A217605 Number of partitions that are fixed points of a certain map (see comment). 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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.

REFERENCES

Stephan Wagner, The Number of Fixed Points of Wilf's Partition Involution, The Electronic Journal of Combinatorics, 20(4) (2013), #P13

LINKS

Table of n, a(n) for n=0..88.

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.

EXAMPLE

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

CROSSREFS

Sequence in context: A156749 A039803 A147809 * A096651 A209354 A114640

Adjacent sequences:  A217602 A217603 A217604 * A217606 A217607 A217608

KEYWORD

nonn

AUTHOR

Joerg Arndt, Oct 08 2012

STATUS

approved

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Last modified November 18 17:56 EST 2017. Contains 294894 sequences.