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%I #7 Mar 31 2012 10:29:02
%S 2,7,54,313,2890,25764
%N Number of ways to lace a shoe that has n pairs of eyelets such that the lace does not cross itself between the eyelet rows.
%C The lace must pass through each eyelet exactly once, must begin and end at the extreme pair of eyelets and each eyelet must have at least one direct connection to the opposite side. The corresponding sequence including all configs where the lace crosses itself in the space between the eyelet rows is A078698. The only symmetric crossing-free lacing is 1234 for N=2.
%H Hugo Pfoertner, <a href="http://www.randomwalk.de/shoelace/lacros.txt">FORTRAN program, illustration of lacings for N=3,4</a>
%H <a href="/index/La#lacings">Index entries for sequences related to shoe lacings</a>
%e With the notation introduced in A078602, the 4 crossing-free lacings for N=3 are 125346, 134256, 134526, 152346. Not counting mirror images we get a(3)=2. Lists of all crossing-free lacings for N=3,4,5,6 and illustrations of the lacings can be found following the FORTRAN program at the Pfoertner link.
%o FORTRAN program provided at Pfoertner link (including a subroutine LPG for lexicographic permutation generation)
%Y Cf. A078602, A078698, A000384 (the maximum number of lace crossings that can occur in an n-eyelet pair shoe lacing is A000384(n-1)).
%K nonn
%O 3,1
%A _Hugo Pfoertner_, Jan 06 2003
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