The number of part sizes of a given multiplicity in a random Carlitz composition

A number of characteristics of random classical and Carlitz (adjacent parts are different) compositions of integer n have been studied by Knopfmacher and Prodinger, Hitczenko and Savage, Goh and Hitczenko, and also by Hitczenko, Rousseau and Savage. This paper is an attempt to complement their results by establishing asymptotics of the average multiplicity of a given part size in a random Carlitz composition. An extension of the Problem of Wilf to the Carlitz case is also presented.