The graph of overlapping permutations is a directed graph that is an analogue to the De Bruijn graph. It consists of vertices that are permutations of length n and edges that are permutations of length n + 1 in which an edge a 1 · · · a n+1 would connect the standardization of a 1 · · · a n to the standardization of a 2 · · · a n+1 . We examine properties of this graph to determine where directed cycles can exist, to count the number of directed 2-cycles within the graph, and to enumerate the vertices that are contained within closed walks and directed cycles of more general lengths.
We continue to explore cyclotomic factors in the descent set polynomial Q n (t), which was introduced by Chebikin, Ehrenborg, Pylyavskyy and Readdy. We obtain large classes of factors of the form Φ 2s or Φ 4s where s is an odd integer, with many of these being of the form Φ 2p where p is a prime. We also show that if Φ 2 is a factor of Q 2n (t) then it is a double factor. Finally, we give conditions for an odd prime power q = p r for which Φ 2p is a double factor of Q 2q (t) and of Q q+1 (t).
The diamond product is the poset operation that when applied to the face lattices of two polytopes results in the face lattice of the Cartesian product of the polytopes. Application of the diamond product to two Eulerian posets is a bilinear operation on the cd-indices of the two posets, yielding a product on cd-polynomials. A lattice path interpretation is provided for this product of two cd-monomials.
Posets, the cd-index, and coproductsConsider the poset P to be a graded poset of rank n + 1 with rank function ρ, unique minimum element0, and unique maximum element1. For further terminology on partially ordered sets, see [9, Chapter 3].
We evaluate the hyperpfaffian of a skew-symmetric k-ary polynomial f of degree k/2·(n−1). The result is a product of the Vandermonde product and a certain expression involving the coefficients of the polynomial f . The proof utilizes a sign reversing involution on a set of weighted, oriented partitions. When restricting to the classical case when k = 2 and the polynomial is (x j − x i ) n−1 , we obtain an identity due to Torelli.
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