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Domineering is a two player game played on a checkerboard in which one player places dominoes vertically and the other places them horizontally. We give bivariate generating polynomials enumerating Domineering positions by the number of each players pieces. We enumerate all positions, maximal positions, and positions where one player has no move. Using these polynomials we count the number of positions that occur during alternating play. Our method extends to enumerating positions from mid-game positions and we include an analysis of a tournament game.
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A rack on $[n]$ can be thought of as a set of maps $(f_x)_{x in [n]}$, where each $f_x$ is a permutation of $[n]$ such that $f_{(x)f_y} = f_y^{-1}f_xf_y$ for all $x$ and $y$. In 2013, Blackburn showed that the number of isomorphism classes of racks on $[n]$ is at least $2^{(1/4 - o(1))n^2}$ and at most $2^{(c + o(1))n^2}$, where $c approx 1.557$; in this paper we improve the upper bound to $2^{(1/4 + o(1))n^2}$, matching the lower bound. The proof involves considering racks as loopless, edge-coloured directed multigraphs on $[n]$, where we have an edge of colour $y$ between $x$ and $z$ if and only if $(x)f_y = z$, and applying various combinatorial tools.
A tight Hamilton cycle in a $k$-uniform hypergraph ($k$-graph) $G$ is a cyclic ordering of the vertices of $G$ such that every set of $k$ consecutive vertices in the ordering forms an edge. R{o}dl, Ruci{n}ski, and Szemer{e}di proved that for $kgeq 3$, every $k$-graph on $n$ vertices with minimum codegree at least $n/2+o(n)$ contains a tight Hamilton cycle. We show that the number of tight Hamilton cycles in such $k$-graphs is $exp(nln n-Theta(n))$. As a corollary, we obtain a similar estimate on the number of Hamilton $ell$-cycles in such $k$-graphs for all $ellin{0,dots,k-1}$, which makes progress on a question of Ferber, Krivelevich and Sudakov.
We study the size of certain acyclic domains that arise from geometric and combinatorial constructions. These acyclic domains consist of all permutations visited by commuting equivalence classes of maximal reduced decompositions if we consider the symmetric group and, more generally, of all c-singletons of a Cambrian lattice associated to the weak order of a finite Coxeter group. For this reason, we call these sets Cambrian acyclic domains. Extending a closed formula of Galambos--Reiner for a particular acyclic domain called Fishburns alternating scheme, we provide explicit formulae for the size of any Cambrian acyclic domain and characterize the Cambrian acyclic domains of minimum or maximum size.
We count orientations of $G(n,p)$ avoiding certain classes of oriented graphs. In particular, we study $T_r(n,p)$, the number of orientations of the binomial random graph $G(n,p)$ in which every copy of $K_r$ is transitive, and $S_r(n,p)$, the number of orientations of $G(n,p)$ containing no strongly connected copy of $K_r$. We give the correct order of growth of $log T_r(n,p)$ and $log S_r(n,p)$ up to polylogarithmic factors; for orientations with no cyclic triangle, this significantly improves a result of Allen, Kohayakawa, Mota and Parente. We also discuss the problem for a single forbidden oriented graph, and state a number of open problems and conjectures.
Hakimi, Schmeichel, and Thomassen in 1979 conjectured that every $4$-connected planar triangulation $G$ on $n$ vertices has at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. In this paper, we show that every $4$-connected planar triangulation on $n$ vertices has $Omega(n^2)$ Hamiltonian cycles. Moreover, we show that if $G$ is a $4$-connected planar triangulation on $n$ vertices and the distance between any two vertices of degree $4$ in $G$ is at least $3$, then $G$ has $2^{Omega(n^{1/4})}$ Hamiltonian cycles.
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