Particle content of the (κ, 3)-configurations

For all κ, we construct a bijection between the set of sequences of non-negative integers a = (a_i)_i∈zZ_≥o satisfying a_i + a_i+1 + a_i+2 ≤ κ and the set of rigged partitions (λ, ρ). Here λ = (λ₁,..., λ_n) is a partition satisfying κ ≥ λ₁ ≥ ⋯ ≥ λ_n ≥ 1 and ρ = (ρ₁,..., ρ_n) ∈ Z _≥0ⁿ is such that ρ_j ≥ ρ_j+1 if λ_j = λ_j+1. One can think of λ as the particle content of the configuration a and p _j as the energy level of the j-th particle, which has the weight λ_j. The total energy ∑_i ia_i written as the sum of the two-body interaction term ∑_j<j′, A_λj,λj′ and the free part ∑_j ρ_j. The bijection implies a fermionic formula for the one-dimensional configuration sums ∑_a q^{∑i iai}. We also derive the polynomial identities which describe the configuration sums corresponding to the configurations with prescribed values for a₀ and a₁, and such that a_i = 0 for all i > N.