An integer partition is just what it sounds like: a way to split a positive integer *n* into a set of smaller positive integers that add up to *n*.
For example, 4 can be partitioned in a few different ways:

- 4
- 3 + 1
- 2 + 2
- 2 + 1 + 1
- 1 + 1 + 1 + 1

The order of the parts doesn’t matter. By convention, the elements of the partition are sorted from greatest to least.

One representation of an integer partition is as a *Ferrers board*. In a Ferrers board,
each integer in a partition is represented as a row of that many boxes—3 is represented by a
row of 3 boxes—with each part of the partition placed in a different row.

So the Ferrers boards for the different partitions of 4 look like this:

4: | |

3 + 1: | |

2 + 2: | |

2 + 1 + 1: | |

1 + 1 + 1 + 1: |

(Sometimes you’ll hear this kind of diagram, or this kind of diagram flipped vertically, called a
*Young diagram*, not to be confused with a Young tableau, which has numbers in the boxes.
I don’t really understand the difference, myself. While we’re at it, a Ferrers diagram, as opposed to Ferrers board, is
similar but consists of dots instead of boxes.)

Ferrers boards are useful for proving various identities about partitions. For example, a Ferrers board for a partition can be interpreted as a different partition by reading columns instead of rows. Take this board:

The standard interpretation, reading by rows, is 3 + 2:

But reading it by columns gives us 2 + 2 + 1.

Partitions related in this way are called *conjugate* partitions.

In general, if an integer’s partition into *k* parts has greatest part *m*, in one board
interpretation there are *k* rows, the longest of which has *m* boxes; and in the other interpretation
has *m* columns, the longest of which has *k* boxes. These two interpretations can then be translated
into an identity to the effect of “An integer *n* can be partitioned into *m* parts in the same number
of ways it can be partitioned so that its greatest part is *m*.” Likewise for *k*.

Another interpretation based on Ferrers boards is, “The number of partitions of *n* is the same as the
number of paths through an *n* × *n* lattice,
starting at the southwest corner and ending at the northeast corner, taking only steps north and east,
for which there are *n* boxes above and to the left of the path.”

Ferrers boards can be partially ordered based on whether one board fits inside another one. For example, this board:

is “less than” this board:

but not this one:

The following is a Hasse diagram of the partitions of 1 through 7, where an arrow connects a Ferrers board to all of the boards that can be created by adding a single box to it:

I guess it’s really an upside-down Hasse diagram. Call the whole thing a lattice if that’s a problem.

See G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge: Cambridge University Press, 2004, especially p. 108.

Designed and rendered using *Mathematica* 7.0. Embarrassingly, I noticed only later that *Mathematica* 6
had added a built-in *IntegerPartitions* function, which might have saved some trouble.

Copyright © 2012–2018 by Robert Dickau.

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