Suppose we have a symmetric matrix of the following form, where the diagonal is always zero:
\begin{array}{cccc} 0 & 1 & 1 & 0\\ 1 & 0 & 1 & 1\\ 1 & 1 & 0 & 0\\ 0 & 1 & 0 & 0 \end{array}
For simplicity, you can imagine that the elements above or below the diagonal can be represented with no loss of information as a vector; e.g., $(1,1,0,1,1,0)$ for $A$.
Given that their are four ones and six total "slots" to arrange them, we can say there are $\left(\begin{array}{c} 6\\ 4 \end{array}\right)=15$ possible symmetric matrices that can be rearranged from $A$. My question is: what do you call the series of operations on $A$ that generates these kinds of symmetric matrices?