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The mean number of fixed points is 1. This is very elementary. Consider the operation of rotating three values around: $p(i)\to p(j)\to p(k)\to p(i)$. Given a permutation with no fixed points, there …

Suppose we have a permutation $\pi$ on $1,2,\ldots,n$ and want to determine the parity (odd or even) of $\pi$. The standard method is find the cycles of $\pi$ and recall that the parity of $\pi$ equa …

McKay and Wormald conjectured that the number of simple $d$-regular graphs of order $n$ is asymptotically $$\sqrt 2 e^{1/4} (\lambda^\lambda(1-\lambda)^{1-\lambda})^{\binom n2}\binom{n-1}{d}^n,$$ whe …

Let $M(n,k)$ be the set of $n\times n$ matrices of nonnegative integers such that every row and every column sums to $k$. Let $P(n,k)$ be the fraction of such matrices which have no zero entries, equ …

Since nobody seems to have addressed question 3, I will. The proofs of the 4-colour theorem are effective in the sense that they can be turned into polynomial-time algorithms. So there are no planar …

Almost 17 years ago, I asked the following question on USENET, motivated by a method in numerology (I kid you not). Pick integers $n \ge 2$, $k \ge 1$. Toss $n$ $k$-sided dice. The sides of each die …

An explicit formula for this was published about 30 years ago, but it was wrong. As the matter stands, there is no explicit formula. The values up to m=15 are here. The value for m=16 is known too, …

Consider infinite digraphs whose vertices are the integers $\mathbb Z$, with the property that there are exactly two arcs coming out of each vertex. (There is no restriction on the number of in-coming …

An array of $k$ permutations of $n$ letters such that each pair are derangements of each other is a $k\times n$ Latin rectangle. If $L(k,n)$ is the number of $k\times n$ Latin rectangles, then the pro …

(1) In this paper (published J. Combinatorial Designs, 15 (2007) 98-119), in the history section starting page 3, we cite many published errors in counting Latin squares and related objects. Some, but …

The following arose as a side issue in a project on graph reconstruction. Problem: Let $a(n)$ be the greatest order of the automorphism group of a 3-connected cubic graph with $n$ vertices. Find a g …

I'm not sure we could find $R(5,5)$ in one year, because exhaustive search is infeasible and one year is probably not enough time to develop the extra theory that would make it possible. I'll dispose …

There isn't any good general computational method for determining whether a permutation group has a regular subgroup. It was recently described to me by an authority on permutation group algorithms a …

All questions about Ramsey numbers for small graphs should be first checked in Staszek Radziszowki's amazing frequently updated survey. On page 40 we find the upper bound $(e-\frac16)k!+1\approx 2.55 …

I'll only consider the case of $S_N$ to start with. Given $(g_1,\ldots,g_d)\in S_N^d$, define a directed graph $G$ on vertices $\lbrace 1,\ldots,N\rbrace$ whose edges are coloured with elements of $Z …