We follow the same line of argument as in the preceding proof. The left-hand side is now , and a term occurs on the left-hand since once if all the are distinct, and not at all otherwise. Thus, it suffices to show (1) To prove this, note first that the sign of is positive if the permutations of cycle type are even, and negative if they are odd: thus, the left-hand side of (1) is the signed sum of the number of even and odd permutations in the isotropy group . But such an isotropy group has equal numbers of even and odd permutations unless it is trivial, i.e. unless the associated partition is . The sum and duality conjectures We first state the sum conjecture, which is due to C. Moen. Sum conjecture (Hoffman). For positive integers k and n, , where the sum is extended over k-tuples of positive integers with . Three remarks concerning this conjecture are in order. First, it implies . Second, in the case it says that , or using the relation between the and and Theorem 1, This was proved by Euler and has been rediscovered several times, in particular by Williams. Finally, C. Moen has proved the same conjecture for k=3 by lengthy but elementary arguments. For the duality conjecture, we first define an involution on the set of finite sequences of positive integers whose first element is greater than 1. Let be the set of strictly increasing finite sequences of positive integers, and let be the function that sends a sequence in to its sequence of partial sums. If is the set of sequences in whose last element is at most , we have two commuting involutions and on defined by and = complement of in arranged in increasing order. The our definition of is for with . For example, We shall say the sequences and are dual to each other, and refer to a sequence fixed by as self-dual. Duality conjecture (Hoffman). If is dual to , then . This sum conjecture is also known as Sum Theorem, and it may be expressed as follows: the Riemann zeta value of an integer n ≥ 2 is equal to the sum of all the valid (i.e. with s1 > 1) MZVs of the partitions of length k and weight n, with 1 ≤ k ≤ n − 1. In formula: For