In How Many Ways Can You Reassemble Several Russian Dolls?

When my brilliant student Thotsaporn "Aek" Thanatipanonda asked me about how many ways can one cover n identical twins, it rang a Bell back to 1981 (see the article). As usual, the method of teaching the computer how to do its own combinatorics, and to use "analysis" (that is really part of discrete math) to enumerate challenging combinatorial objects is even more interesting than the theorems and even the algorithm. Of course, formulas are dead, long live algorithms! But even algorithms are dead, long live meta-algorithms!!