Partial S(k-1, k, v)'s inducing Pk-decompositions of Kv

Let Pk be the path with k vertices and k − 1 edges, k ≥ 4. For every integer v, v ≡ 0 or 1 (mod 2(k − 1)) if k is odd or v ≡ 0 or 1 (mod k−1) if k is even, we produce a Pk-design (V,B) of order v such that no two blocks have k − 1 or k common vertices, i.e. (V,B) is a partial S(k − 1, k, v). AMS classification: 05B05.