Particle content of the (k,3)-configurations

For all $k$, we construct a bijection between the set of sequences of non-negative integers ${\bf a}=(a_i)_{i\in{\bf Z}_{\geq0}}$ satisfying $a_i+a_{i+1}+a_{i+2}\leq k$ and the set of rigged partitions $(λ,ρ)$. Here $λ=(λ_1,...,λ_n)$ is a partition satisfying $k\geqλ_1\geq...\geqλ_n\geq1$ and $ρ=(ρ_1,...,ρ_n)\in{\bf Z}_{\geq0}^n$ is such that $ρ_j\geqρ_{j+1}$ if $λ_j=λ_{j+1}$. One can think of $λ$ as the particle content of the configuration ${\bf a}$ and $ρ_j$ as the energy level of the $j$-th particle, which has the weight $λ_j$. The total energy $\sum_iia_i$ is written as the sum of the two-body interaction term $\sum_{j<j'}A_{λ_j,λ_{j'}}$ and the free part $\sum_jρ_j$. The bijection implies a fermionic formula for the one-dimensional configuration sums $\sum_{\bf a}q^{\sum_iia_i}$. We also derive the polynomial identities which describe the configuration sums corresponding to the configurations with prescribed values for $a_0$ and $a_1$, and such that $a_i=0$ for all $i>N$.