# Multivariable Gaussian polynomials

Let the (homogeneous) Gaussian numbers be defined as $[n]_{x,y} = \frac{x^n-y^n}{x-y}$, and define Gaussian factorials as $[n]_{x,y}! = [n]_{x,y}[n-1]_{x,y}\dots [1]_{x,y}$, and the Gaussian binomials as ${n \brack m}_{x,y} = \frac{[n]_{x,y}!}{[m]_{x,y}![n-m]_{x,y}!}$. (Letting $x=q$ and $y=1$ or $x=q$ and $y=q^{-1}$ recovers the more standard definitions.)

The coefficient of $x^ay^{nm-a}$ in ${n+m \brack m}_{x,y}$ is equal to the number of multiset partitions of $\{1^a, 2^{nm-a}\}$ into $m$ parts of size $n$. (Ordering the parts by how many $1$'s they have and arranging things into a rectangle this is counting partitions of size $a$ fitting in an $n \times m$ rectangle).

This interpretation has a natural analog with more variables. We can define the homogeneous multivariate Gaussian polynomials to be the polynomials $p_{n,m}(x_1,x_2,\dots x_k)$ where the coefficient of $x_1^{a_1}x_2^{a_2}\dots x_k^{a_k}$ is equal to the number of multiset partitions of $\{1^{a_1},2^{a_2}, \dots, k^{a_k} \}$ into $m$ parts of size $n$.

For example $p_{2,3}(x,y,z) =$

$$x^6+$$

$$x^5y +x^5z+$$

$$2x^4y^2 + 2x^4yz + 2x^4z^2+$$

$$2x^3y^3 + 3x^3y^2z + 3x^3yz^2 +2x^3z^3+$$

$$2x^2y^4 + 3x^2y^3z + 5x^2y^2z^2 + 3x^2yz^3 +2x^2z^4+$$

$$xy^5 + \hspace{.2cm} 2xy^4z + \hspace{.2cm} 3xy^3z^2 +\hspace{.2cm} 3xy^2z^3 +\hspace{.2cm} 2xyz^4 + \hspace{.2cm} xz^5$$

$$y^6 +\hspace{.35cm} y^5z + \hspace{.35cm} 2y^4z^2 +\hspace{.35cm} 2y^3z^3+\hspace{.35cm}2y^2z^4+\hspace{.35cm} yz^5+ \hspace{.35cm}z^6$$

Where for example the coefficient $4$ of $x^2y^2z^2$ counts the five multiset partions $\{\{1,1\},\{2,2\},\{3,3\}\}$, $\{\{1,1\},\{2,3\},\{2,3\}\}$, $\{\{1,2\},\{1,2\},\{3,3\}\}$, $\{\{1,2\},\{1,3\},\{2,3\}\}$, and $\{\{1,3\},\{1,3\},\{2,2\}\}$ of $\{1^2,2^2,3^2\}$ into $3$ parts of size $2$.

These showed up for me doing some calculations in a representation theoretic context, and I can say some things about them from that perspective. For example, I can see they satisfy a "higher rank" version of the unimodality property for the coefficients for Gaussian polynomials.

Have these been studied before? Are there combinatorial proofs of these unimodality type results in the literature? Do these polynomials have interpretations say in terms of subspaces of finite dimensional vector spaces over finite fields? Do they have simpler formulas like the original one I gave in the two variable case?

EDIT: After thinking more and fixing some sage code that had been misleading me, these polynomials $p_{n,m}(x_1, \dots, x_k)$ are just the characters of $Sym^m(Sym^n(\mathbb{C}^k))$. Oddly enough that's not the original representation theoretic context I was looking at, which was more convoluted to state, but it's probably related by something like Howe duality. Anyway I'm still interested in what's known about them combinatorially.

• Why not $[n]_{x,y}=\frac{x^n-y^n}{x-y}$? Then $[n]_{1,1}=n$, instead of $0=1^n-1^n$. – Alexander Burstein Apr 18 '18 at 5:57
• Sure, that's probably a better convention. I was mostly just caring about the Gaussian binomials, in which case the $(x-y)$ factors all cancel. I've edited it appropriately though. – Nate Apr 18 '18 at 11:09
• This upcoming AMS Sectional talk sounds potentially relevant?: ams.org/amsmtgs/2252_abstracts/1139-05-609.pdf – Sam Hopkins Apr 18 '18 at 23:36
• Yea Rosa and Mike's stuff definitely relates to the stuff I work on, too bad I'm not in Boston anymore though. – Nate Apr 19 '18 at 15:33