# The Cyclic Sum Theorem

According to the sum theorem, for any integers k and n with k < n-1 the sum of all MZVs of length k and a weight n is equal to the sum of all MZVs of length k+1 and weight n since both are equal to (n). Thus, in the case k = 2 and n = 6 we have

(5,1) +(4,2) +(3,3) +(2,4) =(4,1,1) +(3,2,1) +(3,1,2) +(2,3,1) +(2,2,2) +(2,1,3).

This identity can be split into two separate identities:

(5,1) +(2,4) =(4,1,1) +(3,1,2) +(2,1,3)

and

(4,2) +(3,3) =(3,2,1) +(2,3,1) +(2,2,2).

More generally, consider the set of compositions (ordered partitions) of n-1. We call two compositions cyclically equivalent if one is a cyclic permutation of the other. Let E be a cyclic equivalence class of length-k compositions of n-1. For each (p1, p2, ..., pk) in E there is an MZV (p1+1, p2, ..., pk) of length k and weight n; let Sk be the sum of these. On the other hand, if p1 > 1 we define the ``string'' of (p1, p2, ..., pk) to be the sum

(p1, p2, ..., pk,1) + (p1-1, p2, ..., pk,2) + ... + (2, p2, ..., pk, p1-1).

Let Sk+1 be the sum of these strings over all elements (p1, p2, ..., pk) of E with p1 > 1. Then the cyclic sum theorem states that Sk = Sk+1. In the weight-6 example above, the cyclic equivalence classes of length-2 compositions of 5 are {(4,1),(1,4)} and {(3,2),(2,3)}, giving rise to the two identities above.

Here is another example. The length-3 compositions of 7 split into five cyclic equivalence classes, each with three elements. These five classes give rise to five identities between triple and quadruple MZVs of weight 8:

(6,1,1) +(2,5,1) +(2,1,5) =(5,1,1,1) +(4,1,1,2) +(3,1,1,3) +(2,1,1,4).

(5,1,2) +(3,4,1) +(2,2,4) =(4,1,2,1) +(3,1,2,2) +(2,1,2,3) +(2,4,1,1)

(5,2,1) +(3,1,4) +(2,4,2) =(4,2,1,1) +(3,2,1,2) +(2,2,1,3) +(2,1,4,1)

(4,3,1) +(4,1,3) +(2,3,3) =(3,3,1,1) +(2,3,1,2) +(3,1,3,1) +(2,1,3,2)

(4,2,2) +(3,3,2) +(3,2,3) =(3,2,2,1) +(2,2,2,2) +(2,3,2,1) +(2,2,3,1)

These identities (as well as all other cyclic sum identities where the weight is at most 12) were verified in 1999 by Michael Bigotte using his table of known relations. In February 2000 Yasuo Ohno gave a general proof.

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