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Discrete Mathematics & Theoretical Computer Science |
Vol. 4 no. 1
1. Sums of Digits, Overlaps, and Palindromes
Let s_k(n) denote the sum of the digits in the base-k representation of n. In a celebrated paper, Thue showed that the infinite word (s_2(n) \bmod 2)_n≥ 0 is \emphoverlap-free, i.e., contains no subword of the form axaxa where x is any finite word and a is a single symbol. Let k,m be integers with k>2, m≥ 1. In this paper, generalizing Thue's result, we prove that the infinite word t_k,m := (s_k(n) \bmod m)_n≥ 0 is overlap-free if and only if m≥ k. We also prove that t_k,m contains arbitrarily long squares (i.e., subwords of the form xx where x is nonempty), and contains arbitrarily long palindromes if and only if m≤ 2.
2. Unification of Higher-order Patterns modulo Simple Syntactic Equational Theories
We present an algorithm for unification of higher-order patterns modulo simple syntactic equational theories as defined by Kirchner [14]. The algorithm by Miller [17] for pattern unification, refined by Nipkow [18] is first modified in order to behave as a first-order unification algorithm. Then the mutation rule for syntactic theories of Kirchner [13,14] is adapted to pattern E-unification. If the syntactic algorithm for a theory E terminates in the first-order case, then our algorithm will also terminate for pattern E-unification. The result is a DAG-solved form plus some equations of the form λ øverlinex.F(øverlinex) = λ øverlinex. F(øverlinex^π ) where øverlinex^π is a permutation of øverlinex When all function symbols are decomposable these latter equations can be discarded, otherwise the compatibility of such equations with the solved form remains open.
3. Permutations avoiding an increasing number of length-increasing forbidden subsequences
A permutation π is said to be τ -avoiding if it does not contain any subsequence having all the same pairwise comparisons as τ . This paper concerns the characterization and enumeration of permutations which avoid a set F^j of subsequences increasing both in number and in length at the same time. Let F^j be the set of subsequences of the form σ (j+1)(j+2), σ being any permutation on \1,...,j\. For j=1 the only subsequence in F^1 is 123 and the 123-avoiding permutations are enumerated by the Catalan numbers; for j=2 the subsequences in F^2 are 1234 2134 and the (1234,2134)avoiding permutations are enumerated by the Schröder numbers; for each other value of j greater than 2 the subsequences in F^j are j! and their length is (j+2) the permutations avoiding these j! subsequences are enumerated by a number sequence \a_n\ such that C_n ≤ a_n ≤ n!, C_n being the nth Catalan number. For each j we determine the generating function of permutations avoiding the subsequences in F^j according to the length, to the number of left minima and of non-inversions.
4. Ordered Vertex Partitioning
A transitive orientation of a graph is an orientation of the edges that produces a transitive digraph. The modular decomposition of a graph is a canonical representation of all of its modules. Finding a transitive orientation and finding the modular decomposition are in some sense dual problems. In this paper, we describe a simple O(n + m \log n) algorithm that uses this duality to find both a transitive orientation and the modular decomposition. Though the running time is not optimal, this algorithm is much simpler than any previous algorithms that are not Ω (n^2). The best known time bounds for the problems are O(n+m) but they involve sophisticated techniques.
5. Improved inclusion-exclusion identities via closure operators
Let (A_v)_v ∈ V be a finite family of sets. We establish an improved inclusion-exclusion identity for each closure operator on the power set of V having the unique base property. The result generalizes three improvements of the inclusion-exclusion principle as well as Whitney's broken circuit theorem on the chromatic polynomial of a graph.
6. Avoiding maximal parabolic subgroups of S_k
We find an explicit expression for the generating function of the number of permutations in S_n avoiding a subgroup of S_k generated by all but one simple transpositions. The generating function turns out to be rational, and its denominator is a rook polynomial for a rectangular board.