Reciprocity Law of the Gaussian (or $q$-Binomial) Coefficient

Question

It is a standard exercise in combinatorics to show that the binomial coefficient satisfies the reciprocity law $\binom{-n}{k} = (-1)^k \binom{n+k-1}{k}$ for $n, k \geqslant 0$, which is the multiset coefficient up to sign. Does the $q$-binomial coefficient $\binom{-n}{k}_{q}$ satisfy a similar reciprocity law with a nice combinatorial interpretation (and proof), which specializes to the case above for $q \to 1$?

Answer 1

I don’t know that there’s really an entirely natural combinatorial interpretation of $\binom{-n}k$ or ${\binom{-n}k}_q$ for $n>0$, but perhaps this is of some interest.

For $0\le k\le n$, $\binom{n}k$ is the number of lattice paths from $\langle 0,0\rangle$ to $\langle k,n-k\rangle$ using $k$ right-steps and $n-k$ up-steps. ${\binom{n}k}_q$ is the number of such paths broken down according to the area between the path and the horizontal axis: the coefficient of $q^r$ is the number of paths for which the area is $r$.

For $k\ge 0$ and $n>0$, $\left|\binom{-n}k\right|$ is the number of lattice paths from $\langle 0,0\rangle$ to $\langle k,-n-k\rangle$ using $n$ down-steps and $k$ down-right-steps (i.e., steps $\langle i,j\rangle\to\langle i,j-1\rangle\to\langle i+1,j-1\rangle$), with the proviso that the last step must be a down-step. $\left|{\binom{-n}k}_q\right|$ is the number of such paths broken down according to the signed area between the path and the horizontal axis: the coefficient of $q^r$ is the number of paths for which the signed area is $r$ (which of course is negative). In both cases the sign of the binomial coefficient is given by $(-1)^k$.

Still with $k\ge 0$ and $n>0$, $\binom{n+k-1}k$ is the number of lattice paths from $\langle 0,0\rangle$ to $\langle k,n-1\rangle$ using $k$ right-steps and $n-1$ up-steps; ${\binom{n+k-1}k}_q$ breaks them down by area between the path and the horizontal axis, as before. Take any one of these paths and add an up-step from $\langle k,n-1\rangle$ to $\langle k,n\rangle$. Now replace each up-step by a down-step and each right-step by a down-right-step. The resulting path goes from $\langle 0,0\rangle$ to $\langle k,-n-k\rangle$ and ends with a down-step, and the correspondence is clearly bijective, so $\left|\binom{-n}k\right|=\binom{n+k-1}k$. Moreover, the transformation takes a path with signed area $r\ge 0$ between it and the horizontal axis to one with signed area $-r-\binom{k+1}2$. Thus, up to sign the coefficient of $q^r$ in ${\binom{n+k-1}k}_q$ is equal to the coefficient of $q^{-r-\binom{k+1}2}$ in ${\binom{-n}k}_q$. By an obvious symmetry $q^r$ and $q^{(n-1)k-r}$ have the same coefficient in ${\binom{n+k-1}k}_q$, so up to sign the coefficient of $q^{(n-1)k-r}$ in ${\binom{n+k-1}k}_q$ is equal to the coefficient of $q^{-r-\binom{k+1}2}$ in ${\binom{-n}k}_q$. Finally,

$$\begin{align*} \left((n-1)k-r\right)-\left(-r-\binom{k+1}2\right)&=(n-1)k+\binom{k+1}2\\ &=nk+\binom{k}2\;, \end{align*}$$

so the coefficient of $q^r$ in $\left|{\binom{-n}k}_q\right|$ is equal to the coefficient of $q^{r+nk+\binom{k}2}$ in ${\binom{n+k-1}k}_q$, and we have

$${\binom{-n}k}_q q^{nk+\binom{k}2}=(-1)^k{\binom{n+k-1}k}_q\;.$$