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merged product

Let us suppose a and b are letters, and that x is a sequence, possibly null and possibly empty. We define the merged product of the sequences s and t, denoted by

Likewise, define the overlap, by

These are just the combinations of indices required to work with the de Bruijn matrices. For example, the column-stochastic matrix is defined in terms of n- and -block probabilities by the formula

Correspondingly the row-stochastic matrix is defined by

Unless stated otherwise, we will assume that a probabilistic de Bruijn matrix is column-stochastic. Since it is not always convenient to show the matrix elements as a quotient, but it is essential to know which of them intrinsically vanish, let us write

and note that it vanishes unless x=y, or alternatively when

Harold V. McIntosh