Occupation statistics from exact recursion relations for occupation by dumbbells of a 2 × N array

Starting from an exact recursion relation of the form ${\mathrm{A(q,N)=\Sigma }}_{\mathrm{i,j}}{\mathrm{A(q-i,N-j)C}}_{\mathrm{ij}}$ , we present a recursion method for calculating the various moments ${\mu }_N{\mathrm{(m)=\Sigma }}_{\text{q=0}}^N{q}^m\mathrm{A(q,N).}$ These results are applied to obtain the occupation statistics for occupation of a 2 × N array by parallel and not necessarily parallel dumbbells.