Partitions with numbers in their gaps
1. Partitions with gaps. Bijections between various restricted partitions of integers have been extensively studied (see [2], [3]). In this paper we introduce a generalization of partitions, which are really a kind of restricted composition [3], and obtain bijections between certain classes of them and classes of ordinary partitions. Our generalization arises naturally in connection with solutions of q-difference equations and continued fractions. In fact, continued fractions provide an easy framework for analyzing these generalized partitions. Throughout this paper we write an ordinary partition as n1 + . . .+ nk, n1 ≥ . . . ≥ nk. Define n0 =∞ and nk+1 = 0. We consider the class C(S,m) of partitions with parts ni taken from a set S = {a1, a2, . . .}, where a1 < a2 < . . . We require that if ni = ar and ni+1 = as are two consecutive parts, then r−s ≥ m. For such a partition we define the gapspace gi between ni and ni+1 to be r − s−m. By convention we put g0 =∞ and gk = t−m, where nk = at. Before proceeding we give some examples of gapspaces.