In an analysis of the Jacobi method for the computation of the spectrum of a Hermitian matrix, I face the following problem.
Denote ${\cal P}_2(n)$ the set of doubletons $\{a,b\}$ in $[[1,2n]]=\{1,\ldots,2n\}$. There are $n(2n-1)$ of them. I wish to partition ${\cal P}_2(n)$ in equal parts $X_1,\ldots,X_{2n-1}$ of cardinals $n$, such that the union of the elements of any $X_\alpha$ is precisely $[[1,2n]]$ ($X_\alpha$ is a partition of $[[1,2n]]$) ? This can be viewed as a special partition of the complete graph over $2n$ vertices.
For $n=2$, there is only one solution. For $n=3$, there is a unique solution, up to a permutation in $[[1,6]]$. Is there always a solution ? Is there a natural (group theoretic ?) way to build one ?
More generally, if ${\cal P}_m(mn)$ denotes the set of parts of $[[1,mn]]$ of cardinal $m$, can it be partitionned in equal parts $X_1,\ldots,X_N$ such that each $X_\alpha$ is a partition of $[[1,mn]]$.