Pattern restricted Stirling kk-ary words, the plateau statistic and the kernel method

Stirling kk-ary words resemble classical Stirling permutations except that all entries of a Stirling kk-ary word of length nn belong to the set {1,2,…,k}{1,2,…,k}. We first study generating functions for the number of pattern restricted Stirling kk-ary words with a given number of plateaux by using the kernel method. We then study the generating function for the number of Stirling kk-ary words of length nn according to the number of distinct letters.