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For every pattern p over the alphabet {x, y, x R , y R }, we specify the least k such that p is k-avoidable.

Suppose V is a finite set and C a collection of subsets of V that contains ∅ and V and is closed under taking intersections. Then C is called a convexity and the ordered pair (V, C) is called an aligned space and the elements of C are referred to as convex sets. For a set S ⊆ V , the convex hull of S relative to C, denoted by CHC(S), is the smallest convex… (More)

We introduce the sequence (i n) n≥0 defined by i n = (−1) inv 2 (n) , where inv 2 (n) denotes the number of inversions (i.e., occurrences of 10 as a scattered subsequence) in the binary representation of n. We show that this sequence has many similarities to the classical Rudin–Shapiro sequence. In particular, if S(N) denotes the N-th partial sum of the… (More)

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