Suppose I have a collection of objects $\{X(i, j)\}_{i,j=1}^{n}$ (which could be anything – numbers, functions, matrices, $\ldots$) indexed by $1 \leq i,j \leq n$, such that the following holds:
$$ X(i,j) = X(j,i).$$
Up to the above constraint, all of the objects are different. If I want to index every object, such that no equal object is indexed more than once, I can impose the condition that $i \leq j$ in my indexing.
My question is, how do I generalize this to sets of objects with more than two indices. For example, if I have $\{X(i,j,k)\}$ such that
$$ X(i,j,k) = X(k,j,i),$$
or $\{X(i,j,k,l)\}$ such that
$$X(i,j,k,l) = X(l,k,j,i),$$
what condition on the indices can I impose so that I index each non-equal object exactly once?
Best Answer
If you have $k$ indices $(i_1,\dots,i_k)$, then for an object $X(i_1,\dots,i_k)$, either the values of the indices are palindorimic (ie $i_j=i_{n+1-j}$ for all $1\le j\le k$) in which case there is no choice to be made; or there is some minimal $1\le d\le k/2$ such that $i_d\neq i_{n+1-d}$ but for any $1\le j<d$, $i_j=i_{n+1-j}$. Choose $X(i_1,\dots,i_k)$ if $i_d< i_{n+1-d}$, and choose $X(i_k,\dots,i_1)$ if $i_d> i_{n+1-d}$.