Let τ be a partition of the positive integer n. A partition of the set X n ={1,2,…, n} is said to be of type τ if the sizes of its classes form the partition τ of n. Given 1< r< n, an r element subset A of X n and a partition π of X n are said to be orthogonal if every class of π meets A in exactly one element. Let G n, r be the graph whose vertices are the r-element subsets of X n , with two sets being adjacent if they intersect in r−1 elements. The graph G n, r is Hamiltonian; Hamiltonian cycles of G n, r are early examples of error-correcting codes, where they came to be known as constant weight Gray codes. A Hamiltonian cycle A 1,A 2,…,A n r in G n, r is said to be orthogonally τ-labeled if there exists a list of distinct partitions π 1,π 2,…,π n r of type τ such that π i is orthogonal to both A i and A i+1 where i=1,2,…, n r taken modulo n r . For all but a finite class of partition types τ we present counting arguments to prove that any Hamiltonian cycle in G n, r can be orthogonally τ-labeled. The remaining cases, with the exception of cases which are equivalent to the celebrated Middle Levels Conjecture, are treated via constructive arguments in a sequel. A semigroup of transformations of X n is S n -normal if it is closed under conjugation by the permutations of X n . The combinatorics results here and in the sequel lead to a determination of the rank and idempotent rank of all S n -normal semigroups, thereby broadly generalizing the known result that the rank and idempotent rank of K( n, r) is S( n, r), a Stirling number of the second kind.