Algebraic and Geometric Methods in Enumerative Combinatorics

Between Shi and Ish

“…Note that these results were known (and proven similarly [1]) for both the Shi and the Ish arrangements [2,8], although not for the remaining arrangements of form A (k,n) , which are, to the best of our knowledge, considered here for the first time. We represent both the Shi and Ish arrangements in dimension n = 3 on Figure 1.…”

“…. , x n ) ∈ F n q | ∀1 ≤ i < j ≤ n , x i = x j as injective labelings in F q of the elements of [n] [1,14]. For instance, for n = 5 and q = 13, the labeling represented below corresponds to x := (2,9,3,11,12) ∈ S 5 13 .…”

“…Hence, an arc of form (j, 1) may occur more than once in the multiset A X . In Figure 2 we consider the graphs G (1,3) and G ∅ thus associated with Shi 3 and Ish 3 , respectively.…”

“…Example 3.3. Consider the multiple digraphs G (1,3) and G ∅ represented in Figure 2. We represent G (1,3) and G ∅ through the list neighbors(0), .…”

The Equivariant Volumes of the Permutahedron

“…When all cycles of σ have odd length, Theorem 2.12.3 shows that Π σ n is a lattice zonotope. In this case, it is not much more difficult to give a combinatorial formula for the Ehrhart polynomial, using the fact that L Π σ n (t) is an evaluation of the arithmetic Tutte polynomial of the corresponding vector configuration [1,4]. In general, Π σ n is a half-integral zonotope.…”