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.

More partition posts

• Partitions and pentagons
• Odd parts and distinct parts
• Estimating the number of partitions