# Tagged Questions

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### Infinite sum involving $q$-adic representations of whole numbers and $q$-factorial numbers

Let $q \in \mathbb{N}_{\geq 2}$. For $n \in \mathbb{N}_0$, let $$\mathrm{fac}_q(n) := \prod_{i=1}^n (1+q+\dots+q^{i-1}) = \prod_{i=1}^n \frac{q^i-1}{q-1}$$ be the $q$-factorial of $n$. In particular, ...
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### Intuition behind the Jacobi triple product

Jacobi's triple product identity states that: $\displaystyle \sum_{n = -\infty}^{\infty}z^{n}q^{n^{2}} = \prod_{n = 1}^{\infty}(1 - q^{2n})(1 + zq^{2n - 1})(1 + z^{-1}q^{2n - 1})$ I've seen a messy ...
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### Example of something easier to count with $q$-analog?

Are there any known examples of combinatorial objects that become easier to count by considering some kind of $q$-analog? It seems to me that it might be impossible for the problem of computing the ...
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### What does the $q$-Catalan Numbers count?

I had completed a paper describing the $q$-Catalan numbers, which is the $q$-analog of the Catalan numbers. The $n$-th Catalan numbers can be represented by: $$C_n=\frac{1}{n+1}{2n \choose n}$$ and ...
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### Combinatorial identity. Using echelon matrices.

Determine the exponents $e_i$ s.t. the following identity is correct. $$\sum\limits_{i=0}^k q^{e_i} {\binom mi}_q {\binom{n}{k-i}}_q = {\binom{n+m}{k}}_q$$ Note: When $q=1$ the equation reduces to a ...
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### How to prove it? (one of the Rogers-Ramanujan identities)

Prove the following identity (one of the Rogers-Ramanujan identities) on formal power series by interpreting each side as a generating function for partitions: ...
I was solving a combinatorics problem when I encountered difficulties. The problem was: $x_1 \in \{0,1\}$ $x_2 \in \{0,1,2\}$ . . $x_{n-1}\in\{0,1,2..,n-1\}$ We have to find the number of ways ...