Restricted compositions and permutations: From old to new Gray codes

Any Gray code for a set of combinatorial objects defines a total order relation on this set: x is less than y if and only if y occurs after x in the Gray code list. Let ≺ denote the order relation induced by the classical Gray code for the product set (the natural extension of the Binary Reflected Gray Code to k-ary tuples). The restriction of ≺ to the set of compositions and bounded compositions gives known Gray codes for those sets. Here we show that ≺ restricted to the set of bounded compositions of an interval yields still a Gray code. An n-composition of an interval is an n-tuple of integers whose sum lies between two integers; and the set of bounded n-compositions of an interval simultaneously generalizes product set and compositions of an integer, and so ≺ put under a single roof all these Gray codes. As a byproduct we obtain Gray codes for permutations with a number of inversions lying between two integers, and with even/odd number of inversions or cycles. Such particular classes of permutations are used to solve some computational difficult problems.