A preferential arrangement of a set is a total ordering of the elements of that set with ties allowed. A barred preferential arrangement is one in which the tied blocks of elements are ordered not only amongst themselves but also with respect to one or more bars. We present various combinatorial identities for r m,ℓ , the number of barred preferential arrangements of ℓ elements with m bars, using both algebraic and combinatorial arguments. Our main result is an expression for r m,ℓ as a linear combination of the r k (= r 0,k , the number of unbarred preferential arrangements of k elements) for ℓ ≤ k ≤ ℓ + m. We also study those arrangements in which the sections, into which the blocks are segregated by the bars, must be nonempty. We conclude with an expression of r ℓ as an infinite series that is both convergent and asymptotic.

For a complex central essential hyperplane arrangement $\mathcal{A}$, let $F_{\mathcal{A}}$ denote its Milnor fiber. We use Tevelev’s theory of tropical compactifications to study invariants related to the mixed Hodge structure on the cohomology of $F_{\mathcal{A}}$. We prove that the map taking an arrangement $\mathcal{A}$ to the Hodge-Deligne polynomial of $F_{\mathcal{A}}$ is locally constant on the realization space of any loop-free matroid. When $\mathcal{A}$ consists of distinct hyperplanes, we also give a combinatorial description for the homotopy type of the boundary complex of any simple normal crossing compactification of $F_{\mathcal{A}}$. As a direct consequence, we obtain a combinatorial formula for the top weight cohomology of $F_{\mathcal{A}}$, recovering a result of Dimca and Lehrer.

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We propose a conjectural framework for computing Gorenstein measures and stringy Hodge numbers in terms of motivic integration over arcs of smooth Artin stacks, and we verify this framework in the case of fantastacks, which are certain toric Artin stacks that provide (non-separated) resolutions of singularities for toric varieties. Specifically, let X be a smooth Artin stack admitting a good moduli space π : X → X, and assume that X is a variety with log-terminal singularities, π induces an isomorphism over a nonempty open subset of X, and the exceptional locus of π has codimension at least 2. We conjecture a formula for the motivic measure for X in terms of the Gorenstein measure for X and a function measuring the degree to which π is non-separated. We also conjecture that if the stabilizers of X are special groups in the sense of Serre, then almost all arcs of X lift to arcs of X , and we explain how in this case, our conjectures imply a formula for the stringy Hodge numbers of X in terms of a certain motivic integral over the arcs of X . We prove these conjectures in the case where X is a fantastack. Contents 15 5. Fibers of the maps of jets 19 6. Gorenstein measure and toric varieties 26 7. Motivic measure and canonical stacks 29 8. Stringy invariants and toric Artin stacks: proof of Theorem 1.5 33 9. Fantastacks with special stabilizers: proof of Theorem 1.8 35 References 37necessarily satisfy the "strict valuative criterion", i.e., there may exist arcs of X that do not lift to arcs of X . Thus in general, we cannot use this conjecture to compute the total Gorenstein measure µ Gor X (L (X)), which specializes to the stringy Hodge numbers of X. This issue already occurs in the case where X is a Deligne-Mumford stack. For this reason, Yasuda uses a notion of "twisted arcs" of X instead of usual arcs of X , and this is why the inertia of X and orbifold Hodge numbers appear in Yasuda's setting. We take a different approach, emphasizing a setting in which the next conjecture predicts that almost all arcs of X lift to (finitely many) arcs of X .Conjecture 1.2. Let X be a finite type Artin stack over k admitting a good moduli space π : X → X. Assume X is an irreducible k-scheme and that π induces an isomorphism over a nonempty open subset of X. If the stabilizers of X are all special groups, then sep −1 π (0) ⊂ L (X) is measurable and µ X (sep −1 π (0)) = 0, where we note that µ X is the usual (non-Gorenstein) motivic measure on L (X).Remark 1.3. All special groups are connected, so if X is a Deligne-Mumford stack whose stabilizers are special groups, then the stabilizers of X are all trivial. Thus Conjecture 1.2 highlights a setting that is "orthogonal" to the setting considered by Yasuda.We now apply our conjectures to computing stringy Hodge numbers. In subsection 3.4, we introduce a function sep X : |L (X )| → Q ≥0 , which is essentially given by 1/(sep π •L (π)) except we are pedantic about dividing by 0. We think of its integral L (X ) sep X dµ X as a kind of motivic class of L (X ) corrected by sep X to acco...