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 multiplicat
ion 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 $mathcal{H}^d_q$.
We confirm a conjecture of Monical, Tokcan and Yong on a characterization of the lattice points in the Newton polytopes of key polynomials.
The Springer numbers are defined in connection with the irreducible root systems of type $B_n$, which also arise as the generalized Euler and class numbers introduced by Shanks. Combinatorial interpretations of the Springer numbers have been found by
Purtill in terms of Andre signed permutations, and by Arnold in terms of snakes of type $B_n$. We introduce the inversion code of a snake of type $B_n$ and establish a bijection between labeled ballot paths of length n and snakes of type $B_n$. Moreover, we obtain the bivariate generating function for the number B(n,k) of labeled ballot paths starting at (0,0) and ending at (n,k). Using our bijection, we find a statistic $alpha$ such that the number of snakes $pi$ of type $B_n$ with $alpha(pi)=k$ equals B(n,k). We also show that our bijection specializes to a bijection between labeled Dyck paths of length 2n and alternating permutations on [2n].
The Dirichlet series $L_m(s)$ are of fundamental importance in number theory. Shanks defined the generalized Euler and class numbers in connection with these Dirichlet series, denoted by ${s_{m,n}}_{ngeq 0}$. We obtain a formula for the exponential g
enerating function $s_m(x)$ of $s_{m,n}$, where m is an arbitrary positive integer. In particular, for m>1, say, $m=bu^2$, where b is square-free and u>1, we prove that $s_m(x)$ can be expressed as a linear combination of the four functions $w(b,t)sec (btx)(pm cos ((b-p)tx)pm sin (ptx))$, where p is an integer satisfying $0leq pleq b$, $t|u^2$ and $w(b,t)=K_bt/u$ with $K_b$ being a constant depending on b. Moreover, the Dirichlet series $L_m(s)$ can be easily computed from the generating function formula for $s_m(x)$. Finally, we show that the main ingredient in the formula for $s_{m,n}$ has a combinatorial interpretation in terms of the m-signed permutations defined by Ehrenborg and Readdy. In principle, this answers a question posed by Shanks concerning a combinatorial interpretation for the numbers $s_{m,n}$.
