be a square lattice, R be a set of all compact simply connected regions

Abstract. Let T be a nite set of tiles, B be a set of regions tileable by T. We introduce a tile counting group G (T; B) as a group of all linear relations for the number of times each tile 2 T can occur in a tiling of a region 2 B. We compute the tile counting group for a large set of ribbon tiles also known as rim hooks in a context of representation theory of the symmetric group. The tile counting group is presented by its set of generators which consists of cer-tain new tile invariants. In a special case these invariants generalize Conway-Lagarias invariant for trimino tilings and a height invariant which is related to computation of characters of the symmetric group and goes back to G. de B. Robinson. The heart of the proof is the rim hook bijection betwen rim hook tableaux of a given skew Young shape and certain standard skew Young tableaux. We also discuss signed tilings by the ribbon tiles and apply our results to the tileability problem. The full version is available fro