The resonance arrangement An is the arrangement of hyperplanes in R n given by all hyperplanes of the form i∈I xi = 0, where I is a nonempty subset of {1, . . . , n}. We consider the characteristic polynomial χ(An; t) of the resonance arrangement, whose value Rn at −1 is of particular interest, and corresponds to counts of generalized retarded functions in quantum field theory, among other things. No formula is known for either the characteristic polynomial or Rn, though Rn has been computed up to n = 8. By exploiting symmetry and using computational methods, we compute the characteristic polynomial of A9, and thus obtain R9. The coefficients of the characteristic polynomial are also equal to the so-called Betti numbers of the complexified hyperplane arrangement; that is, the coefficient of t n−i is denoted by the Betti number bi(An). Explicit formulas are known for the Betti numbers up to b3(An). Using computational methods, we also obtain an explicit formula for b4(An), which gives the t n−4 coefficient of the characteristic polynomial.

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A permutation σ ∈ Sn is said to be k-universal or a k-superpattern if for every π ∈ S k , there is a subsequence of σ that is order-isomorphic to π. A simple counting argument shows that σ can be a ksuperpattern only if n ≥ (1/e 2 + o(1))k 2 , and Arratia conjectured that this lower bound is best-possible. Disproving Arratia's conjecture, we improve the trivial bound by a small constant factor. We accomplish this by designing an efficient encoding scheme for the patterns that appear in σ. This approach is quite flexible and is applicable to other universality-type problems; for example, we also improve a bound by Engen and Vatter on a problem concerning (k + 1)-ary sequences which contain all k-permutations.