A Note on Multiset Permutations

Abstract

The author studies permutations of the multiset $\{1,1,2,2,\ldots,m,m,m+1,m+2\ldots,n\}$ such that $1,2,\ldotsn$ occurs as a not-necessarily consecutive subsequence. From the theory of symmetric functions, the generating function for the number of these permutations is known [Goulden and Jackson, Combinatorial Enumeration, John Wiley, New York, 1983, p. 73]. It is used to obtain a recurrence relation and then to give a purely combinatorial proof of the recurrence.