Suppose that $\lambda\_1,\lambda\_2,\lambda\_3$ are partititions of $n$. When do there exist permutations $\sigma\_1,\sigma\_2,\sigma\_3 \in S\_n$ such that


(1) $\sigma\_1\sigma\_2\sigma\_3$ is the identity;


(2) the $\sigma\_i$ generate a transitive subgroup of $S\_n$; and


(3) the cycle type of $\sigma\_i$ is $\lambda\_i$ for all $i$?


One can show that the total number of parts among the $\lambda\_i$ must be an integer that is at most $n+2$ and that is congruent to $n$ modulo 2. Is this condition sufficient? If $\lambda\_3 = (n)$ and the total number of parts of the $\lambda\_i$ is exactly $n+2$, then the existence is known.


This problem is equivalent to the existence of an irreducible curve $X$ over the complex numbers and a nonconstant morphism $X \rightarrow \mathbb{P}^1$ whose branch locus is contained in $\{0,1,\infty\}$ (a Belyi map) and whose monodromy generators have cycle types $\lambda\_1,\lambda\_2,\lambda\_3$.



Permutations of prescribed cycle types that multiply to the identity
The answer is no. No such triples exist for $n = 4$, $\lambda\_1 = (3, 1)$, $\lambda\_2 = \lambda\_3 = (2, 2)$. (Indeed, in $S\_4$, the double transpositions and the identity form the Klein four subgroup.)


For other results on this problem, see for example <http://arxiv.org/abs/math/0508434>.


