Exact Nearest Neighbor Statistics for One-Dimensional Lattice Spaces

It is shown that A(n₁₁,q,N), the number of ways of arranging q indistinguishable particles on a one‐dimensional lattice space of N compartments in such a way as to create n₁₁ nearest neighbor pairs is ${\mathrm{A(n}}_{11}\mathrm{,q,N)=}\left(\begin{array}{c}\text{N−q+1}\\ {\mathrm{q-n}}_{11}\end{array}\right)\times \left(\begin{array}{c}\text{q−1}\\ n_{11}\end{array}\right)$ . A similar expression is also derived for n₀₀, the number of pairs of vacant nearest neighbors. The normalization, first moment, and most probable value of these statistics are also discussed.