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

\binom{M+N}{N}_q = \binom{M+N}{M}_q

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

2 thoughts on “Partition symmetry

  1. 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

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