Defant, Engen, and Miller defined a refinement of Lassalle's sequence A k+1 by considering uniquely sorted permutations of length 2k + 1 whose first element is ℓ. They showed that each such sequence is symmetric in ℓ and conjectured that these sequences are unimodal. We prove that the sequences are unimodal.

The classical Erdős-Littlewood-Offord problem concerns the random variable X = a1ξ1 + • • • + anξn, where ai ∈ R \ {0} are fixed and ξi ∼ Ber(1/2) are independent. The Erdős-Littlewood-Offord theorem states that the maximum possible concentration probability max x∈R Pr(X = x) is n ⌊n/2⌋ /2 n , achieved when the ai are all 1. As proposed by Fox, Kwan, and Sauermann, we investigate the general case where ξi ∼ Ber(p) instead. Using purely combinatorial techniques, we show that the exact maximum concentration probability is achieved when ai ∈ {−1, 1} for each i. Then, using Fourier-analytic techniques, we investigate the optimal ratio of 1s to −1s. Surprisingly, we find that in some cases, the numbers of 1s and −1s can be far from equal.

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.

We prove that the classic falling-block video game Tetris (both survival and board clearing) remains NPcomplete even when restricted to 8 columns, or to 4 rows, settling open problems posed over 15 years ago. Our reduction is from 3-Partition, similar to the previous reduction for unrestricted board sizes, but with a better packing of buckets. On the positive side, we prove that 2-column Tetris (and 1-row Tetris) is polynomial. We also prove that the generalization of Tetris to larger k-omino pieces is NP-complete even when the board starts empty, and even when restricted to 3 columns or 2 rows or constant-size pieces. Finally, we present an animated Tetris font.

We prove that the classic falling-block video game Tetris (both survival and board clearing) remains NP-complete even when restricted to 8 columns, or to 4 rows, settling open problems posed over 15 years ago [BDH + 04]. Our reduction is from 3-Partition, similar to the previous reduction for unrestricted board sizes, but with a better packing of buckets. On the positive side, we prove that 2-column Tetris (and 1-row Tetris) is polynomial. We also prove that the generalization of Tetris to larger k-omino pieces is NP-complete even when the board starts empty, even when restricted to 3 columns or 2 rows or constant-size pieces. Finally, we present an animated Tetris font.