Counting Subwords Occurrences in Base-b Expansions

We count the number of distinct (scattered) subwords occurring in the base-b expansion of the nonnegative integers. More precisely, we consider the sequence (Sb(n))n≥0 counting the number of positive entries on each row of a generalization of the Pascal triangle to binomial coefficients of base-b expansions. By using a convenient tree structure, we provide recurrence relations for (Sb(n))n≥0 leading to the bregularity of the latter sequence. Then we deduce the asymptotics of the summatory function of the sequence (Sb(n))n≥0.