# Partition symmetry

Let p(M, N, n) be the number of partitions of the integer n into at most M parts, each containing integers at most N. Then

p(M, N, n) = p(N, M, n).

That is, you can swap the size of the partition multisets and the upper bound on the elements in the multisets.

For example, lets look at the partitions of 6 into multisets with at most 3 elements. The Mathematica command

`    IntegerPartitions[6, 3]`

returns

• {6}
• {5, 1}
• {4, 2}
• {4, 1, 1}
• {3, 3}
• {3, 2, 1}
• {2, 2, 2}

Now let’s look at the partitions of 6 into sets with any number of elements, but with no elements greater than 3. The Mathematica command

`    IntegerPartitions[6, All, {1, 2, 3}]`

returns

• {3, 3}
• {3, 2, 1}
• {3, 1, 1, 1}
• {2, 2, 2}
• {2, 2, 1, 1}
• {2, 1, 1, 1, 1}
• {1, 1, 1, 1, 1, 1}}

Both return a list of 7 multisets because

p(3, 6, 6) = p(6, 3, 6) = 7.

As another example, let’s look at partitions of 11. First we look at partitions with at most 3 elements, with each element less than or equal to 5. We list these with

`    IntegerPartitions[11, 3, Range[5]]`

which gives

• {5, 5, 1}
• {5, 4, 2}
• {5, 3, 3}
• {4, 4, 3}

Now let’s look at partitions of 11 into multisets with at most 5 elements, each less than or equal to 3 using

`    IntegerPartitions[11, 5,  Range[3]]`

This gives us

• {5, 5, 1}
• {5, 4, 2}
• {5, 3, 3}
• {4, 4, 3}

Not only do both lists have the same number of partitions, which we would expect because

p(3, 5, 11) = p(5, 3, 11),

in this case they actually give the same list of partitions.

The symmetry relation here follows from the symmetry of the q-binomial coefficient because p(M, N, n) equals the coefficient of qn in the q-binomial

It’s not immediately obvious from the definition that the rational function defining q-binomial coefficient is in fact a polynomial, but it is.

## 2 thoughts on “Partition symmetry”

1. There’s an easy explanation of this symmetry. This:

* * * * *
* *
*

becomes this:

* * *
* *
*
*
*

2. I was wondering if you happened to know that all these bound together determine all the specific skew Young tableaux and their assemblages making up all the hyper-polytopes?:
1. https://en.wikipedia.org/wiki/Telephone_number_(mathematics)
2. https://en.wikipedia.org/wiki/Hosoya_index
(The sums of the Hosoya indexes and the sums of the related matching-polynomials for the direct sums of complete graphs determine the breakdowns of the specific Young tableaux, if I remember right.)
3.https://en.wikipedia.org/wiki/Lenstra%E2%80%93Lenstra%E2%80%93Lov%C3%A1sz_lattice_basis_reduction_algorithm for the solution of the “Rational-coefficients-polynomial = 0”-equations.
4. http://giovanniviglietta.com/slides/bellows1.pdf
5. https://giovanniviglietta.com/slides/bellows2.pdf