This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A079410 #14 Aug 13 2025 09:17:55 %S A079410 2,7,54,308,2890,25764 %N A079410 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 A079410 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 A079410 Hugo Pfoertner, <a href="http://www.randomwalk.de/shoelace/lacros.txt">FORTRAN program, illustration of lacings for N=3,4</a> %H A079410 <a href="/index/La#lacings">Index entries for sequences related to shoe lacings</a> %e A079410 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 A079410 (Fortran) c Program provided at Pfoertner link (including a subroutine LPG for lexicographic permutation generation). %Y A079410 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 A079410 nonn %O A079410 3,1 %A A079410 _Hugo Pfoertner_, Jan 06 2003 %E A079410 a(6) corrected by _Sean A. Irvine_, Aug 12 2025