A Symmetric Functions Approach to Stockhausen's Problem

Abstract

We consider problems in sequence enumeration suggested by Stockhausen's problem, and derive a generating series for the number of sequences of length $k$ on $n$ available symbols such that adjacent symbols are distinct, the terminal symbol occurs exactly $r$ times, and all other symbols occur at most $r-1$ times. The analysis makes extensive use of techniques from the theory of symmetric functions. Each algebraic step is examined to obtain information for formulating a direct combinatorial construction for such sequences.