Particle content of the (k,3)-configurations

For all $k$, we construct a bijection between the set of sequences of non-negative integers $\mathbf{a}=(a_i{)}_{i\in {\mathbf{Z}}_{\ge 0}}$ satisfying $a_i+a_{i+1}+a_{i+2}\le k$ and the set of rigged partitions $(\lambda ,\rho )$. Here $\lambda =({\lambda }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\lambda }_n)$ is a partition satisfying $k\ge {\lambda }_1\ge \mathrm{.}\mathrm{.}\mathrm{.}\ge {\lambda }_n\ge 1$ and $\rho=(\rho_1,…,\rho_n)\in{\bf Z}{\geq0}^n$issuchthat$\rho_j\geq\rho{j+1}$if$\lambda_j=\lambda_{j+1}$. One can think of $\lambda$ as the particle content of the configuration ${\bf a}$and$\rho_j$astheenergylevelofthe$j$-th particle, which has the weight ${\lambda }_j$. The total energy ${\sum }_{i}i{a}_i$ is written as the sum of the two-body interaction term ${\sum }_{j<j^{\mathrm{\prime }}}{A}_{{\lambda }_j,{\lambda }_{j^{\mathrm{\prime }}}}$ and the free part ${\sum }_j{\rho }_j$. The bijection implies a fermionic formula for the one-dimensional configuration sums ${\sum }_{\mathbf{a}}{q}^{{\sum }_{i}i{a}_i}$. 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 $i>N$.