Suppose I have $n$ non-commuting objects that I want to split into two groups, where I identify symmetric groupings. How many pairs of groupings $g_n$ can I make in this case?
I.e. for $n = 4$ we might have $$ 4 = (4,0),(3,1),(2,2),~~~~~\implies ~~~~~g_4 = 3 $$ Where we have identified and excluded $(4,0)$ and $(0,4)$, for example. Is there a general formula for $g_n$?
Evaluating the first few, it looks like the formula for even $n$ is $g_n = \frac{n+2}{2}$. Is this correct, or a known/named formula?
I assume you are assuming $n$ is even. Then there are $1+n/2$ ways to choose the larger of the two summands in the expression $n=x+y$, since $x=n/2, y=0$ is allowed.
There's no particular name for this formula/argument.