Quantized cohomological Hall algebra of the $d$-loop quiver revisited

Let $\Lambda$ be the set of partitions of length $\geq 0$. We introduce an $\mathbb{N}$-graded algebra $\mathbb{A}_q^d(\Lambda)$ associated to $\Lambda$, which can be viewed as a quantization of the algebra of partitions defined by Reineke. The multiplication of $\mathbb{A}^d_q(\Lambda)$ has some kind of quasi-commutativity, and the associativity comes from combinatorial properties of certain polynomials appeared in the quantized cohomological Hall algebra $\mathcal{H}^d_q$ of the $d$-loop quiver. It turns out that $\mathbb{A}^d_q(\Lambda)$ is isomorphic to $\mathcal{H}^d_q$, thus can be viewed as a combinatorial realization for $\mathcal{H}^d_q$.