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For fixed t ≥ 2, we consider the class of representations of 1 as sum of unit fractions whose denominators are powers of t or equivalently the class of canonical compact t-ary Huffman codes or equivalently rooted t-ary plane " canonical " trees. We study the probabilistic behaviour of the height (limit distribution is shown to be normal), the number of… (More)

We present an average case analysis of two variants of dual-pivot quicksort, one with a non-algorithmic comparison-optimal partitioning strategy, the other with a closely related algorithmic strategy. For both we calculate the expected number of comparisons exactly as well as asymptotically, in particular, we provide exact expressions for the linear,… (More)

In this tutorial, we demonstrate how easy it is to construct finite state machines, in particular automata and transducers, within the computer algebra system Sage. As a beneficent byproduct, we calculate the asymptotic Hamming weight of a non-adjacent-form-like digit expansion, which was not known before.

In this note the precise minimum number of key comparisons any dual-pivot quickselect algorithm (without sampling) needs on average is determined. The result is in the form of exact as well as asymptotic formulae of this number of a comparison-optimal algorithm. It turns out that the main terms of these asymptotic expansions coincide with the main terms of… (More)

- Daniel Krenn, Stephan Wagner
- Algorithmica
- 2015

For a fixed integer base $$b\ge 2$$ b ≥ 2 , we consider the number of compositions of 1 into a given number of powers of b and, related, the maximum number of representations a positive integer can have as an ordered sum of powers of b. We study the asymptotic growth of those numbers and give precise asymptotic formulae for them, thereby improving on… (More)

For fixed t ≥ 2, we consider the class of representations of 1 as sum of unit fractions whose denominators are powers of t or equivalently the class of canonical compact t-ary Huffman codes or equivalently rooted t-ary plane " canonical " trees. We study the probabilistic behaviour of the height (limit distribution is shown to be normal), the number of… (More)

The new finite state machine package in the mathematics software system SageMath is presented and illustrated by many examples. Several combinatorial problems, in particular digit problems, are introduced, modeled by automata and transducers and solved using SageMath. In particular, we compute the asymptotic Hamming weight of a non-adjacent-form-like digit… (More)

We analyse the number of occurrences of a fixed non-zero digit in the width-w non-adjacent forms of all elements of a lattice in some region (e.g. a ball). Our result is an asymptotic formula, where its main term coincides with the full block length analysis. In its second order term a periodic fluctuation is exhibited. The proof follows Delange's method.… (More)

- Daniel Krenn, Jörg Thuswaldner, Volker Ziegler
- 2013

Let o be the maximal order of a number field. Belcher showed in the 1970s that every algebraic integer in o is the sum of pairwise distinct units, if the unit equation u+v = 2 has a non-trivial solution u, v ∈ o *. We generalize this result and give applications to signed double-base digit expansions.

- Daniel Krenn
- 2013

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