## Eigenvalues.

Evidently this rule gives

0one ancestor,1three. Their squares are1and9, visible respectively as the number of nonzero elements in the individual tensor products. The element sum ofNis 10; its eigenvalues are , , contrasted to eigenvalues0,1for and for .The configurations of length

nwill have ancestors for an average of4each, while the sum of the squares of the number of ancestors will eventually grow according to . The growth could be as small as a factor of2per cell or as large as a factor of4, according to whether all configurations have an equal number of ancestors, or all ancestors map into a single configuration.The quiescent state will have asymptotically ancestors (whose square is ), an increasingly negligible proportion.

Because the mean is always constant and small, the variance, , will grow asymptotically at half (square root) the rate of the second moment, or by , or 18% per additional cell.

Harold V. McIntosh

E-mail:mcintosh@servidor.unam.mx