The complexity of Free-Flood-It on 2xn boards

We consider the complexity of problems related to the combinatorial game Free-Flood-It, in which players aim to make a coloured graph monochromatic with the minimum possible number of flooding operations. Our main result is that computing the length of an optimal sequence is fixed parameter tractable (with the number of colours present as a parameter) when restricted to rectangular 2xn boards. We also show that, when the number of colours is unbounded, the problem remains NP-hard on such boards. This resolves a question of Clifford, Jalsenius, Montanaro and Sach (2010)

D-branes at Toric Singularities: Model Building, Yukawa Couplings and Flavour Physics

We discuss general properties of D-brane model building at toric singularities. Using dimer techniques to obtain the gauge theory from the structure of the singularity, we extract results on the matter sector and superpotential of the corresponding gauge theory. We show that the number of families in toric phases is always less than or equal to three, with a unique exception being the zeroth Hirzebruch surface. With the physical input of three generations we find that the lightest family of quarks is massless and the masses of the other two can be hierarchically separated. We compute the CKM matrix for explicit models in this setting and find the singularities possess sufficient structure to allow for realistic mixing between generations and CP violation.Comment: 55 pages, v2: typos corrected, minor comments adde

The computational challenge of enumerating high-dimensional rook walks

We provide guessed recurrence equations for the counting sequences of rook paths on d-dimensional chess boards starting at (0..0) and ending at (n..n), where d=2,3,...,12. Our recurrences suggest refined asymptotic formulas of these sequences. Rigorous proofs of the guessed recurrences as well as the suggested asymptotic forms are posed as challenges to the reader

Elliptic rook and file numbers

Utilizing elliptic weights, we construct an elliptic analogue of rook numbers for Ferrers boards. Our elliptic rook numbers generalize Garsia and Remmel's q-rook numbers by two additional independent parameters a and b, and a nome p. These are shown to satisfy an elliptic extension of a factorization theorem which in the classical case was established by Goldman, Joichi and White and later was extended to the q-case by Garsia and Remmel. We obtain similar results for our elliptic analogues of Garsia and Remmel's q-file numbers for skyline boards. We also provide an elliptic extension of the j-attacking model introduced by Remmel and Wachs. Various applications of our results include elliptic analogues of (generalized) Stirling numbers of the first and second kind, Lah numbers, Abel numbers, and r-restricted versions thereof.Comment: 45 pages; 3rd version shortened (elliptic rook theory for matchings has been taken out to keep the length of this paper reasonable