Digits and Beyond
This is a survey about digits from a personal point of view. Counting the occurrences of digits (and, more generally, subblocks) is discussed in the context of various positional number systems. The methods to achieve this are Delange's elementary method and Flajolet’s idea to use the Mellin-Perron summation (or integral) formula.
Then we move to problems from Theoretical Computer Science (register function of binary trees, number of exchanges in Baxter’s odd-questions) are also discussed. An open problem from physicists Yekutieli and Mandelbrot can also be treated in that fashion.
Furthermore, we consider representations of numbers where some digits are forbidden. As a representative example, we discuss the Cantor distribution and its moments, asymptotically analyzed by Mellin transforms. Other problems in this context lead to sums involving Bernoulli numbers which can be attacked by analytic Depoissonization.
Very briefly we mention carry propagation, mergesort parameters and jump interpolation search trees.
KeywordsBinary Tree Dirichlet Series Gray Code Probability Letter Exponential Generate Function
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