Compositions of random transpositions

LetY=(y1,y2, ...),y1≥y2≥..., be the list of sizes of the cycles in the composition ofcn transpositions on the set {1, 2, ...,n}. We prove that ifc>1/2 is constant andn → ∞, the distribution off(c)Y/n converges toPD(1), the Poisson-Dirichlet distribution with parameter 1, where the functionf is known explicitly. A new proof is presented of the theorem by Diaconis, Mayer-Wolf, Zeitouni and Zerner stating that thePD(1) measure is the unique invariant measure for the uniform coagulation-fragmentation process.