Independently prescriable sets of n-letter words

Abstract

A set of words {A 1, A 2, …, A t} can be independently prescribed if for every set of t nonnegative integers ν i ( i = 1, 2, …, t), there exists a word S containing exactly ν i occurrences of A i ( i = 1, 2, …, t). In this paper we consider n-letter sets of independently prescribable words, where an n-letter set is a set of n-letter words. We show that the maximum size for an independently prescribable n-letter set is q n − q n−1 , where q is the number of letters in the alphabet (assuming q > 2 or n > 2). An independently prescribable n-letter set J such that | J| = q n − q n−1 can be constructed in the following way: Let D be an ( n − 1)- de Bruijn sequence, a circular sequence with exactly one occurrence of every ( n − 1)-letter word froom a q-letter alphabet. Then an independently prescribable n-letter set with cardinality q n − q n−1 consists of all n-letter words except those occurring in D. We show that each de Bruijn sequence corresponds to a unique independently prescribable n-letter set J such that | J| = q n − q n−1 . If q > 2, the number of such sets is the number of ( n − 1)-de Bruijn sequences (( n, q) ≠ (2, 3)). If q = 2 and n ⩾ 5, the number of such sets is 7 times the number of ( n − 1)-de Bruijn sequences. Some of the results in this paper are generalized in a theorem about independently prescribable sets of edges of Hamiltonian graphs.