A072503 Number of ways to lace a shoe with n eyelet pairs such that there is no direct "horizontal" connection between any adjacent eyelet pair.

3, 45, 1824, 109560, 9520560, 1145057760, 181091917440

Offset: 3

Author

Hugo Pfoertner, Jan 27 2003

Keywords

• nonn
• more

Comments

The lacing must not have any "straight connections" between adjacent eyelet pairs (e.g. 2<->2*n-1, 3<->2*n-2, 4<->2*n-3,....). There are no symmetric solutions.

From Sean A. Irvine, Oct 06 2024: (Start)

The lacing must begin and end with the top eyelet pair (1 and 2n in Pfoertner's numbering).

Every eyelet must be used.

Under versus over crossings are not considered distinct.

Three consecutive eyelets on the same side are not permitted.

Because there are no symmetric solutions, mirrored solutions can be handled by simply dividing the total solutions by 2. (End)

Examples

			The 6 non-straight lacings for n=3 are: 124536, 135426, 142356, 145326, 153246, 154236. Not counting mirror images we get a(3)=3.
		

Links

• Sean A. Irvine, Java program (github)
• Hugo Pfoertner, FORTRAN program to count non-straight shoe lacings and results for N=3,4
• Index entries for sequences related to shoe lacings

Crossrefs

Cf. A078602, A078698, A078702, A002866.

Programs

Fortran

See Links.

Extensions

a(9) from Sean A. Irvine, Oct 07 2024