Can you please give me an example for the cyclic equivalence classes of multiple indices on the following set?
$$ I(w,d)=\{\ (k_1,...k_d)\mid k_1+\ldots+k_d=w, k_1,...k_d \ge 1\,\}$$
where $ w$: weight and $k$: depth of multiple zeta values.
And I want to know what is meant by the following statement.
"Two elements of $I(w,d)$ are cyclically equivalent if there are cyclic permutations of each other".
Eg: $ (k_1,...k_d) \equiv (k_2,...k_d,k_1) \equiv (k_3,...k_d,k_1,k_2) $. Is this correct for the cyclic equivalent elements?
For example, $$I(6,3)=\{(1,1,4),(1,2,3),(1,3,2),(1,4,1),(2,1,3),(2,2,2),(2,3,1),(3,1,2),(3,2,1),(4,1,1)\} $$ and herein we have the cyclic equivalences $(1,1,4)\sim (1,4,1)\sim(4,1,1)$ and $(1,2,3)\sim(2,3,1)\sim(3,1,2)$ and $(1,3,2)\sim(3,2,1)\sim(2,1,3)$, whereas $(2,2,2)$ is cyclically equivalent only to itself. Note specifically that $(1,2,3)\not\sim(1,3,2)$ as it takes a non-cyclic permutation to turn one into the other.