Exact next nearest neighbor degeneracy

Abstract

It is shown that A [n₁₁, n₁₀₁, n₁₁₁, q, N], the arrangement degeneracy arising when q indistinguishable particles are placed on a one‐dimensional lattice of N equivalent compartments so that n₁₁ occupied nearest neighbors, n₁₀₁ next nearest neighbors of the 101‐type, and n₁₁₁ next nearest neighbors of the 111‐type are created, is given by ${\mathrm{A[n}}_{11}{\mathrm{,\hspace{0.17em}n}}_{101}{\mathrm{,\hspace{0.17em}n}}_{111}\mathrm{,\hspace{0.17em}q,\hspace{0.17em}N]=}\left(\begin{array}{c}{\text{N−2q+n}}_{11}\text{+2}\\ {\mathrm{q-n}}_{11}{\mathrm{-n}}_{101}\end{array}\right)\left(\begin{array}{c}{\mathrm{q-n}}_{11}\text{−1}\\ n_{101}\end{array}\right)\left(\begin{array}{c}{\mathrm{q-n}}_{11}\\ n_{11}{\mathrm{-n}}_{111}\end{array}\right)\left(\begin{array}{c}{n}_{11}\text{−1}\\ n_{111}\end{array}\right)$ . The normalization and first moment of the next nearest neighbor density are determined. Similar results for the vacant next nearest neighbor degeneracy are also presented.