Ken's Puzzle of the Week
Lacing (Short) Shoes
In how many ways can you lace shoes with 2/3/4 pairs of eyelets?
Label the eyelet pairs from top to bottom: 

Restrictions:
 Lacing must begin with eyelet A and end with eyelet B. A and B
will be tied across together.
 Each eyelet is used exactly once.
 Laces run straight from one eyelet to another.
 Over/under and top/bottom are not important. Only eyelettoeyelet
connections are important. For example, for 2 pairs, ADCB is equivalent to BCDA,
even though the "top" lace would be different between the two in actual
practice.
 Each eyelet must have at least one useful connection across the pairs to
tighten the shoe. Thus, for 3 pairs, AECFDB would not be valid because
E and D do not contribute to tightening. AEFCDB would be valid since A
crosses to B, C to D&F, and E to F.
Feel free to extend this to more pairs. However, I've read of
mathematicians analyzing this problem for seven pairs of eyelets and finding
millions of solutions. Is there a formula or recursive approach to
this?
Source: I didn't find the original source (if you find it, let me know.)
Here are some related links:
Optimum Shoe
Lacing,
Ian's
Shoelace Site.
Solution
Mail to Ken