Combinatorics of Singly-Repairable Families

Combinatorics of Singly-Repairable Families

A non-empty set F of n-bit vectors over alphabet {0, 1} is called singly repairable, if every vector u ∈ F satisfies the following conditions: (i) if any bit of u is changed (from 0 to 1 or vice versa), the new vector does not belong to F (ii) there is a unique choice of a different bit that can then be changed to give another vector 6= u in F . Such families F exist only for even n and we show...

Combinatorics of Unique Maximal Factorization Families (UMFFs)

Suppose a set W of strings contains exactly one rotation (cyclic shift) of every primitive string on some alphabet Σ. Then W is a circ-UMFF if and only if every word in Σ has a unique maximal factorization over W . The classic circ-UMFF is the set of Lyndon words based on lexicographic ordering (1958). Duval (1983) designed a linear sequential Lyndon factorization algorithm; a corresponding PRA...

The Order Steps of an Analytic Combinatorics

‎Analytic combinatorics aims to enable precise quantitative predictions of the properties of large combinatorial structures‎. ‎This theory has emerged over recent decades as essential both for the analysis of algorithms and for the study of scientific models in many disciplines‎, ‎including probability theory‎, ‎statistical physics‎, ‎computational biology and information theory‎. ‎With a caref...

Combinatorics Enumerative Combinatorics 05

Maintenance of Repairable Systems

A commonly used definition of a repairable system (Ascher and Feingold [3]) states that this is a system which, after failing to perform one or more of its functions satisfactorily, can be restored to fully satisfactory performance by any method other than replacement of the entire system. In order to cover more realistic applications, and to cover much recent literature on the subject, we need...