DERANGEMENTS ON THE N-CUBE

Let Q(n) be the n-dimensional cube represented by a graph whose vertices are sequences of 0's and 1's of length n, where two vertices are adjacent if and only if they differ only at one Position. A k-dimensional subcube or a k-face of Q(n) is a subgraph of Q(n) spanned by all the vertices u1u2...u(n) with constant entries on n - k positions. For a k-face G(k) of Q(n) and a symmetry w of Q(n), we say that w fixes G(k) if w induces a symmetry of G(k); in other words, the image of any vertex of G(k) is still a vertex in G(k). A symmetry w of Q(n) is said to be a k-dimensional derangement if w does not fix any k-dimensional subcube of Q(n); otherwise, w is said to be a k-dimensional rearrangement. In this paper, we find a necessary and sufficient condition for a symmetry of Q(n) to have a fixed k-dimensional subcube. We find a way to compute the generating function for the number of k-dimensional rearrangements on Q(n). This makes it possible to compute explicitly such generating functions for small k. Especially, for k = 0, we have that there are 1.3...(2n-1) symmetries of Q(n) with at least one fixed vertex. A combinatorial proof of this formula is also given.