Enumerating De Bruijn sequences

A cycle is a sequence a(1)a(2)...a(r) taken in a circular order-that is, a(1) follows a(r), and a(2)... a(r)a(1),... a(r)a(1)... a(r-1) are all the same cycle as a(1)a(2)... a(r). Given natural numbers q greater than or equal to 1 and s greater than or equal to 2, a cycle of s(q) letters is called a complete cycle [1, 2], or De Bruijn sequence, if subsequences a(i)a(i+1)... a(i+q-1) (1 less than or equal to i less than or equal to s(q)) consist of all possible s(q) ordered sequences b(1)b(2)...b(q) over the alphabet A (\A\ = s). In 1946, De Bruijn proved [1] (see [2]) that the number of complete cycles, under s = 2, is equal to 2(2q-1-q). We propose the overall proof for s greater than or equal to 2, which determines the number of the De Bruijn sequences to be equal to (s!)(sq-1/sq). The demonstration is based on our recent results concerning the characteristic polynomial and permanent of the arc-graph [17], applied herein to some auxiliary digraphs. Wherever possible, the main subject is discussed in the wider context of related combinatorial problems, which first include counting the linear De Bruijn sequences. Obtained results can be used for calculating the number of monocyclic and linear compounds, formed from s sorts of atoms, obeying the specified combinatorial restrictions. The former is equivalent to finding the number of respective necklaces with s kinds of beads.