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A078698 Number of ways to lace a shoe that has n pairs of eyelets such that each eyelet has at least one direct connection to the opposite side. 5
1, 2, 20, 396, 14976, 907200, 79315200, 9551001600, 1513528934400, 305106949324800, 76296489615360000, 23175289163980800000, 8404709419090575360000, 3587225703492542791680000, 1779970753996760560435200000, 1016036270188884847558656000000, 661106386935312429191528448000000 (list; graph; refs; listen; history; text; internal format)



The lace is "directed": reversing the order of eyelets along the path counts as a different solution. It must begin and end at the extreme pair of eyelets,


C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 494.


Table of n, a(n) for n=1..17.

A. Khrabrov, K. Kokhas, Points on a line, shoelace and dominoes, arXiv:1505.06309 [math.CO], (23-May-2015).

Hugo Pfoertner, FORTRAN program and results for N=2,3,4

Index entries for sequences related to shoe lacings


Conjecture: a(n) = (n-1)!^2*A051286(n). - Vladeta Jovovic, Sep 14 2005 (correct, see the Khrabrov/Kokhas reference, Joerg Arndt, May 26 2015)


a(3) = 20: label the eyelets 1,2,3 from front to back on the left side then 4,5,6 from back to front on the right side. The lacings are: 124356 154326 153426 142536 145236 135246 and the following and their mirror images: 125346 124536 125436 152346 153246 152436 154236.

Examples for n=2,3,4 can be found following the FORTRAN program at given link.


a[n_] := (n-1)!^2 Sum[Binomial[n-k, k]^2, {k, 0, n/2}];

Array[a, 17] (* Jean-Fran├žois Alcover, Jul 20 2018 *)


FORTRAN program provided at Pfoertner link


Cf. A078700, A078702, A078602.

Sequence in context: A263207 A218306 A009236 * A090728 A210896 A090309

Adjacent sequences:  A078695 A078696 A078697 * A078699 A078700 A078701




Hugo Pfoertner, Dec 18 2002


Terms a(9) and beyond (using A051286) from Joerg Arndt, May 26 2015



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Last modified December 10 16:44 EST 2018. Contains 318049 sequences. (Running on oeis4.)