Table of contents |
2 Ferrers' Graph 3 Number of Partitions 4 Bibliographical Notes 5 External Links |
The partitions of 4 are listed below:
The partition 6 + 4 + 3 + 1 of the positive number 14 can be represented
by the following graph:
Claim: The number of self-conjugate partitions is the same as the
number of partitions with distinct odd parts.
Proof (sketch): The crucial observation is that every odd part can
be "folded" in the middle to form a self conjugate graph:
The number of partitions of a positive integer n is given by the partition function p(n). The number of partitions of n into
exactly k parts is denoted by pk(n).
Ferrers graph techniques also allow us to prove results like the following:
Elementary introduction to the topic of integer partition, including
discussion of Ferrers graph, can be found in the following reference:
Examples
The partitions of 8 are listed below:
Among the 22 partitions for the number 8, 6 of them contain
only odd parts:
Curiously, if we count the partitions of 8
with distinct parts, we also obtain the number 6:
Is this only coincidence, or is it true that, for all positive number,
the number of partitions with odd parts always equals the number
of partitions with distinct parts? This and other results
can be obtained by the aid of a visual tool, Ferrers' graph.Ferrers' Graph
o o o o
o o o
o o o
o o
o
o
6+4+3+1
The 14 circles are lined up in 4 columns, each having the size of a part
of the partition. The graphs for the 5 partitions of the number 4 are
listed below:
o o o o o o o o o o o o
o o o o o
o o
o
4 3+1 2+2 2+1+1 1+1+1+1
If we now flip the graph of the partition 6 + 4 + 3 + 1 along the
NW-SE axis, we obtain another partition of 14:
o o o o o o o o o o
o o o o o o o
o o o --> o o o
o o o
o
o
6+4+3+1 4+3+3+2+1+1
By turning the rows into columns, we obtain the partition 4 + 3 + 3 + 2 +
1 + 1 of the number 14. Such partitions are said to be
conjugate of one another. In the case of the number 4, partitions
4 and 1 + 1 + 1 + 1 are conjugate pairs, and partitions 3 + 1 and 2 + 1 + 1
are conjugate of each other. Of particular interest is the partition
2 + 2, which has itself as conjugate. Such a paritition is said to be
self-conjugate.o
o
o --> o o o
o o
o o
One can then obtain a bijection between the set of partitions with distinct
odd parts and the set of self-conjugate partitions, as illustrated by
the following example:
o * x o o o o o
o * x o * * * *
o * x <--> o * x x
o * o * x
o * o *
o *
o *
o
o
9+7+3 5+5+4+3+2
distinct odd self-conjugate
Similar techniques can be employed to establish, for example, the following
equalities:
Number of Partitions
Bibliographical Notes
External Links