Moving Faces to Other Places: Facet Derangements

Abstract

Derangements are a popular topic in combinatorics classes. We study a generalization to face derangements of the n-dimensional hypercube. These derangements can be classified as odd or even, depending on whether the underlying isometry is direct or indirect, providing a link to abstract algebra. We emphasize the interplay between the geometry, algebra and combinatorics of these sequences, with lots of pretty pictures.

Cyclic Derangements

“…Recently, Gordon and McMahon [11] looked at the problem of enumerating isometries of the n-dimensional hypercube that leave no facet unmoved. Algebraically, such an isometry is an element σ of the hyperoctahedral group B n for which σ(i) = i for any i. Combinatorially, the problem then is to enumerate n × n matrices with entries from {0, ±1} such that each row and column has exactly one nonzero entry and no diagonal entry equals 1.…”

“…Table 1 gives values for d (r) n for r ≤ 5 and n ≤ 6. We also have the following generalization of [11](Proposition 3.2), giving a formula relating the number of cyclic derangements with the number of permutation derangements.…”

“…To begin, we define a map ϕ that is a sort of inverse to the map dp. Say that σ i is a subcedant of σ if σ i < i with respect to the total order in (11), and let sub(σ) denote the number of subcedants of σ. For σ ∈ C r ≀ S m , let s 1 < · · · < s sub(σ) be the absolute values of subcedants of σ.…”

“…Remark 3.6. There is another total ordering on elements of (C r × [n]) ∪ {0} that is equally as natural as the order given in (11), namely (18) ζ r−1 n < · · · < ζ r−1 1 < ζ r−2 n < · · · < ζ1 < 0 < 1 < 2 < · · · < n.…”

“…where des(σ) is the number of descents of σ and exc(σ) is the number of weak excedances of σ. We generalize these statistics to C r ≀ S n by saying 1 ≤ i ≤ n is a weak excedant of σ ∈ C r ≀ S n if σ(i) = i or if |σ(i)| = i and σ 2 (i) > σ(i) with respect to the total order in (11).…”