Particle content of the \((k,3)\)-configurations
Summary: For all \(k\), we construct a bijection between the set of sequences of non-negative integers \({\mathbf a}=(a_i)_{i\in\mathbb Z_{\geq 0}}\) satisfying \(a_i+ a_{i+1}+a_{i+2}\leq k\) and the set of rigged partitions \((\lambda,\rho)\). Here \(\lambda=(\lambda_1,\dots, \lambda_n)\) is a partition satisfying \(k\geq \lambda_1\geq\cdots \geq \lambda_n\geq 1\) and \(\rho=(\rho_1,\dots,\rho_n)\in \mathbb{Z}^n_{\geq 0}\) 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 \({\mathbf a}\) 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 \(\sum_{\mathbf a}q^{\sum_iia_i}\). 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 \(i>N\).
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