I think this is actually an extremely common situation in algebraic combinatorics (where the $q$ in a $q$-analog has meaning as both a "deformation parameter" and a prime power corresponding to a finite field). Here is just one very simple example.

For commuting variables $x$ and $y$ the binomial theorem $(x+y)^n = \sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^{k}$. We can "deform" this theory by making the variables $q$-commute in the sense that $xy = qyx$. Then the $q$-binomial theorem says $(x+y)^n = \sum_{k=0}^{n}\binom{n}{k}_q x^{n-k}y^{k}$ where $\binom{n}{k}_q$ is the usual $q$-binomial coefficient. (So $\binom{n}{k}_q= \frac{[n]_q!}{[n-k]_q![k]_q!}$ with $[n]_q = \frac{1-q^n}{1-q}$ and $[n]_q!=[n]_q[n-1]_q\cdots[1]_q$.)

On the other hand, $\binom{n}{k}_q$ for $q$ a prime power can also be interpreted as the number of $k$-dimensional subspaces of an $n$-dimensional vector space $V$ over $\mathbb{F}_q$. Sending $q\to 1$ we recover the interpretation of $\binom{n}{k}$ as the number of $k$ element subsets of an $n$ element set.

Like I said, this binomial coefficient example is very simple, but it is typical of many examples in algebraic combinatorics. See for example the desiderata (i)-(vi) for a $q$-analog listed in the Reiner-Stanton-White AMS Notices paper "What is... cyclic sieving?" (https://www.ams.org/notices/201402/rnoti-p169.pdf).

I think this is actually an extremely common situation in algebraic combinatorics (where the $q$ in a $q$-analog has meaning as both a "deformation parameter" and a prime power corresponding to a finite field). Here is just one very simple example.

For commuting variables $x$ and $y$ the binomial theorem says $(x+y)^n = \sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^{k}$. We can "deform" this theory by making the variables $q$-commute in the sense that $xy = qyx$. Then the $q$-binomial theorem says $(x+y)^n = \sum_{k=0}^{n}\binom{n}{k}_q x^{n-k}y^{k}$, where $\binom{n}{k}_q$ is the usual $q$-binomial coefficient. (So $\binom{n}{k}_q= \frac{[n]_q!}{[n-k]_q![k]_q!}$ with $[n]_q = \frac{1-q^n}{1-q}$ and $[n]_q!=[n]_q[n-1]_q\cdots[1]_q$.)

On the other hand, $\binom{n}{k}_q$ for $q$ a prime power can also be interpreted as the number of $k$-dimensional subspaces of an $n$-dimensional vector space $V$ over $\mathbb{F}_q$. Sending $q\to 1$ we recover the interpretation of $\binom{n}{k}$ as the number of $k$ element subsets of an $n$ element set.

Like I said, this binomial coefficient example is very simple, but it is typical of many examples in algebraic combinatorics. See for example the desiderata (i)-(vi) for a $q$-analog listed in the Reiner-Stanton-White AMS Notices paper "What is... cyclic sieving?" (https://www.ams.org/notices/201402/rnoti-p169.pdf).

I think this is actually an extremely common situation in algebraic combinatorics (where the $q$ in a $q$-analog has meaning as both a "deformation parameter" and a prime power corresponding to a finite field). Here is just one very simple example.

For commuting variables $x$ and $y$ the binomial theorem $(x+y)^n = \sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^{k}$. We can "deform" this theory by making the variables $q$-commute in the sense that $xy = qyx$. Then the $q$-binomial theorem says $(x+y)^n = \sum_{k=0}^{n}\binom{n}{k}_q x^{n-k}y^{k}$ where $\binom{n}{k}_q$ is the usual $q$-binomial coefficient. (So $\binom{n}{k}_q= \frac{[n]_q!}{[n-k]_q![k]_q!}$ with $[n]_q = \frac{1-q^n}{1-q}$ and $[n]_q!=[n]_q[n-1]_q\cdots[1]_1$.)

On the other hand, $\binom{n}{k}_q$ for $q$ a prime power can also be interpreted as the number of $k$-dimensional subspaces of an $n$-dimensional vector space $V$ over $\mathbb{F}_q$. Sending $q\to 1$ we recover the interpretation of $\binom{n}{k}$ as the number of $k$ element subsets of an $n$ element set.

Like I said, this binomial coefficient example is very simple, but it is typical of many examples in algebraic combinatorics. See for example the desiderata (i)-(vi) for a $q$-analog listed in the Reiner-Stanton-White AMS Notices paper "What is... cyclic sieving?" (https://www.ams.org/notices/201402/rnoti-p169.pdf).

I think this is actually an extremely common situation in algebraic combinatorics (where the $q$ in a $q$-analog has meaning as both a "deformation parameter" and a prime power corresponding to a finite field). Here is just one very simple example.

For commuting variables $x$ and $y$ the binomial theorem $(x+y)^n = \sum_{k=0}^{n}\binom{n}{k}x^{n-k}y^{k}$. We can "deform" this theory by making the variables $q$-commute in the sense that $xy = qyx$. Then the $q$-binomial theorem says $(x+y)^n = \sum_{k=0}^{n}\binom{n}{k}_q x^{n-k}y^{k}$ where $\binom{n}{k}_q$ is the usual $q$-binomial coefficient. (So $\binom{n}{k}_q= \frac{[n]_q!}{[n-k]_q![k]_q!}$ with $[n]_q = \frac{1-q^n}{1-q}$ and $[n]_q!=[n]_q[n-1]_q\cdots[1]_q$.)

On the other hand, $\binom{n}{k}_q$ for $q$ a prime power can also be interpreted as the number of $k$-dimensional subspaces of an $n$-dimensional vector space $V$ over $\mathbb{F}_q$. Sending $q\to 1$ we recover the interpretation of $\binom{n}{k}$ as the number of $k$ element subsets of an $n$ element set.

Like I said, this binomial coefficient example is very simple, but it is typical of many examples in algebraic combinatorics. See for example the desiderata (i)-(vi) for a $q$-analog listed in the Reiner-Stanton-White AMS Notices paper "What is... cyclic sieving?" (https://www.ams.org/notices/201402/rnoti-p169.pdf).