Scala, 141 bytes

n=>(for{i<-1 to n-1 j<-i+1 to n-1 p<-1.to(n).permutations}yield{val(a,b)=p splitAt j;val(c,d)=a splitAt i;Set(c,d,b)map(_.toSet)}).toSet.size 

Try it online!

Is it short? No. Is it efficient? No. Is it clever? No. Why did I make it? I...don't know. I'll try golfing it later, if possible.