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 $(\lambda,\rho)$. Here $\lambda=(\lambda_1,...,\lambda_n)$ is a partition satisfying $k\geq\lambda_1\geq...\geq\lambda_n\geq1$ and $\rho=(\rho_1,...,\rho_n)\in{\bf Z}_{\geq0}^n$ is such that $\rho_j\geq\rho_{j+1}$ if $\lambda_j=\lambda_{j+1}$. One can think of $\lambda$ as the particle content of the configuration ${\bf a}$ and $\rho_j$ as the energy level of the $j$-th particle, which has the weight $\lambda_j$. The total energy $\sum_iia_i$ is written as the sum of the two-body interaction term $\sum_{j<j'}A_{\lambda_j,\lambda_{j'}}$ and the free part $\sum_j\rho_j$. 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 $i>N$.