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 *q*^{n} 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.

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

* * * * *

* *

*

becomes this:

* * *

* *

*

*

*

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