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The Wang tiling is a classical problem in combinatorics. A major theoretical question is to find a (small) set of tiles which tiles the plane only aperiodically. In this case, resulting tilings are rather restrictive. On the other hand, Wang tiles are used as a tool to generate textures and patterns in computer graphics. In these applications, a set of tiles is normally chosen so that it tiles the plane or its sub-regions easily in many different ways. With computer graphics applications in mind, we introduce a class of such tileset, which we call sequentially permissive tilesets, and consider tiling problems with constrained boundary. We apply our methodology to a special set of Wang tiles, called Brick Wang tiles, introduced by Derouet-Jourdan et al. in 2015 to model wall patterns. We generalise their result by providing a linear algorithm to decide and solve the tiling problem for arbitrary planar regions with holes.

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Abstract. For positive integers m and n, we denote by BH(m, n) the set of all H ∈ M n×n (C) such that HH * = nI n and each entry of H is an m-th root of unity where H * is the adjoint matrix of H and I n is the identity matrix. For H 1 , H 2 ∈ BH(m, n) we say that H 1 is equivalent to H 2 if H 1 = P H 2 Q for some monomial matrices P, Q whose nonzero entries are m-th roots of unity. In this paper we classify BH(17, 17) up to equivalence by computer search.

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