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 (p_{1}, p_{2}, ..., p_{k}) in
*E* there is an MZV
(p_{1}+1, p_{2}, ..., p_{k}) of length *k*
and weight *n*; let *S _{k}* be the sum of these.
On the other hand, if p

(p_{1}, p_{2}, ...,
p_{k},1) +
(p_{1}-1, p_{2}, ...,
p_{k},2) + ... +
(2, p_{2}, ..., p_{k},
p_{1}-1).

Let *S _{k+1}* be the sum of these strings over all elements
(p

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.