Subsequence Combinatorics and Applications to Microarray Production, DNA Sequencing and Chaining Algorithms

  • Sven Rahmann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4009)


We investigate combinatorial enumeration problems related to subsequences of strings; in contrast to substrings, subsequences need not be contiguous. For a finite alphabet Σ, the following three problems are solved. (1) Number of distinct subsequences: Given a sequence s ∈Σ n and a nonnegative integer kn, how many distinct subsequences of length k does s contain? A previous result by Chase states that this number is maximized by choosing s as a repeated permutation of the alphabet. This has applications in DNA microarray production. (2) Number of ρ -restricted ρ -generated sequences: Given s ∈Σ n and integers k ≥1 and ρ≥1, how many distinct sequences in Σ k contain no single nucleotide repeat longer than ρ and can be written as \(s_1^{r_1}\dots s_n^{r_n}\) with 0≤r i ρ for all i? For ρ= ∞, the question becomes how many length-k sequences match the regular expression s 1 * s 2 * ...s n *. These considerations allow a detailed analysis of a new DNA sequencing technology (“454 sequencing”). (3) Exact length distribution of the longest increasing subsequence: Given Σ= {1,...,K} and an integer n ≥1, determine the number of sequences in Σ n whose longest strictly increasing subsequence has length k, where 0 ≤kK. This has applications to significance computations for chaining algorithms.


Arithmetic Operation Regular Expression Deposition Sequence Alphabet Size Motif Occurrence 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Sven Rahmann
    • 1
  1. 1.Algorithms and Statistics for Systems Biology Group, Genome Informatics, Faculty of TechnologyBielefeld UniversityBielefeldGermany

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