The distribution of the square sum of Dirichlet random variables and a table with quantiles of Greenwood's statistic

The exact distribution of the square sum of Dirichlet random variables is given by two different univariate integral representations. Alternatively, three representations by orthogonal series with Jacobi or Legendre polynomials are derived. As a special case the distribution of the square sum of spacings - also called Greenwood’s statistic - is obtained. Nine quantiles of this statistic are tabulated with eight digits where the number of squares ranges from 10 to 100.