Search Results

I would be very surprised if it depended smoothly on the parameters. It is related to questions of block designs (and mutually orthogonal latin squares, finite planes, group divisible designs, etc.) w …

The word $1001$-$1001$-$101$-$1001$-$101$-$1001$-$1$ (separated to show how it was constructed and to ease reading) is cube-free and does not contain the subword $010$. Perhaps you can prove that no …

To extrapolate on Brendan McKay's comment for question one, here is a recurrence for $M(m,n)$. Though not exactly an inclusion exclusion argument (it's not hard to turn it into one), this essentially …

João Pedro Neto has it listed on his site, http://www.di.fc.ul.pt/~jpn/gv/soldiers.htm Though in his version, it seems like non-orthogonal lines must have slope $1$ or $-1$, so that connecting $(1,1 …

Here is a solution to your warm up problem. It uses a few known elementary identities, and some short inductions for identities I didn't recognize. I also changed your notation slightly from $l$ to $k …

This question has its genesis in a group assignment: $k$ students are to be given oral exams. Each student will be asked one distinct question from $n$ questions given to them earlier, no two students …

Here are some solutions for the largest possible set of conditions, $n=2k$. Suppose $p$ is a prime that divides ${2k\choose k}, {2k-1\choose k-1}, \ldots, {k+1\choose 1}$. Then the selection conditio …

Conway & Sloane's "Sphere Packings, Lattices and Groups" references Coxeter's "Regular Polytopes" for the phrase "halfcube", but Coxeter only uses the notation $h\Pi_n$, saying $h$ can be taken to sta …

You can get recurrence relations for the coefficients by considering, for each monomial, which lower degree monomials will contribute to its coefficient. For example, suppose we want to find the coe …

This game can be described as an impartial edge colouring game on $K_n$ where creating a monochrome $K_k$ is not allowed, and the last player to make a move wins (normal play). Hence, it is equivalent …

Partition your graph into vertices of even degree, $N$, and vertices of odd degree, $D$. If the size of both sets is even, then you can add a matching to the set with lower degree to get a regular gra …

Not an answer, but this might get you more help by phrasing it in terms of a common combinatorics problem - finding a lower bound for the size of a transversal of a hypergraph. A hypergraph is a colle …

For a one dimensional lattice, the solution with $a_n = n$ is trivial: starting from $0$, we proceed to $1, -1, 2, -2, \ldots$. After $2n$ steps, we have covered $[-n, n]$ without stepping anywhere ou …

Here is a rephrasing that may help: the graph $G$ has vertices $[c]$ and an edge between two vertices if one is a prime multiple of the other. Then the game is for each player to take some vertices, n …