How to lace a shoe

[Pfoertner, Hugo]

Neil, SeqFans,

a proposal for a notation of "directed shoelace graphs". I don't know, if
something suitable already exists,
but during my attempts to count a(3) I found the following notation
convenient.

A lacing is described by a sequence s(k), k=1..2n.

If the lace enters eyhole k from the outside of the shoe set s(k) negative.
s(1)=-1 (lacing starts a eyhole 1 from outside)
s(2n)=1 (lace leaves eyhole 2n from inside). s(1) and s(2n) could be omitted
from the notation.
For n=1 s would then be empty.

If the lace crosses itsself, the crossing may be above the earlier path (A)
or below the earlier path (B).
This is indicated by A(i,j) or B(i,j), where  i,j are the indices of the two
connected eyholes of
the crossed part of the earlier path, order preserved, signs removed.

If two eyholes on the same side (1..n) or (n+1..n) are connected, the sign
of s must change, i.e. the
path must stay inside or outside:
if (k-1 in [1,n] and k in [1,n]) or (k-1 in [n+1,2n] and k in [n+1,2n]) then
s(k-1)*s(k)<0

The notation for the 5 different graphs for n=2:
http://www.randomwalk.de/shoelace/l2_1.jpg : -1 2 -3 4
http://www.randomwalk.de/shoelace/l2_2.jpg : -1 3 -2 B(1 2) 4
http://www.randomwalk.de/shoelace/l2_3.jpg : -1 -3 2 A(1 2) 4
http://www.randomwalk.de/shoelace/l2_4.jpg : -1 -3 -2 B(1 2) 4
http://www.randomwalk.de/shoelace/l2_5.jpg : -1 -3 -2 A(1 2) 4

The examples for a(4) (I hope I got it right)

http://www.randomwalk.de/shoelace/l4a.jpg :
-1 -5 4 B(1 5) -6 B(1 5) -3 B(1 5) 7 B(1 5) 2 A(1 5) 8

http://www.randomwalk.de/shoelace/l4a.jpg
-1 5 4 B(1 5) -6 A(1 5) -3 B(1 5) 7 A(1 5) 2 B(1 5) 8

Mirror arrangements should not be counted.

For a(3) there are already 28 different cases after 3 eyholes, so a(3) will
be much larger than
the a(3)s in Neil's 3 existing sequences:
-1 2 -4, -1 2 4, -1 2 -5, -1 2 5,
-1 3 -4, -1 3 4, -1 3 -5, -1 3 5,
-1 -4 -2, -1 -4 2, -1 -4 -3, -1 -4 3, -1 -4, 5,
-1 4 -2, -1 4 2, -1 4 -3, -1 4 3, -1 4 -5,
-1 -5 -2, -1 -5 2, -1 -5 -3, -1 -5 3, -1 -5 4,
-1 5 -2, -1 5 2, -1 5 -3, -1 5 3, -1 5 -4,

the crossings still to come... 

I'll try to write a Fortran program for counting during the next days,
but due to other things having priority I'll be happy of someone else
continues this investigation.

Any comments or suggestions are appreciated.

Hugo Pfoertner
http://www.pfoertner.org/