A360276 Number of unordered quadruples of self-avoiding paths with nodes that cover all vertices of a convex n-gon; one-node paths are allowed.

0, 0, 10, 105, 1015, 9625, 90972, 861420, 8191920, 78309000, 752317280, 7257522272, 70223986560, 680703296000, 6601793730560, 63984047339520, 619018056228864, 5972223901440000, 57415027394027520, 549677356175073280, 5238367168966328320, 49678823782558924800, 468783944069762252800

Offset: 3

Author

Ivaylo Kortezov, Feb 01 2023

Keywords

• nonn

Comments

Although each path is self-avoiding, the different paths are allowed to intersect.

Examples

			a(6) = 6!/(2!2!2!2!)+6!*3/(3!3!) = 45+60 = 105; the first summand corresponds to the case of 2 two-node paths and 2 one-node paths; the second to the case of 1 three-node path and 3 one-node paths.
		

Links

• Ivaylo Kortezov, Sets of Non-self-intersecting Paths Connecting the Vertices of a Convex Polygon, Mathematics and Informatics, Vol. 65, No. 6, 2022.

Crossrefs

Cf. A001792, A359405 (unordered pairs of paths), A360021 (unordered triples of paths).

Formula

a(n) = (1/3)*n*(n-1)*(n-2)*(n-3)*2^(n-15)*(4^(n-4) + 12*3^(n-4) + 54*2^(n-4) + 108) for n != 4.