A360275 - OEIS

%I #9 Feb 17 2023 20:22:25

%S 0,0,0,0,0,105,3780,81900,1386000,20207880,266666400,3277354080,

%T 38198160000,427365818880,4629059635200,48842864179200,

%U 504335346278400,5114054709319680,51064119467827200,503151159589478400,4900668252598272000,47248486914198011904,451429610841538560000

%N Number of unordered quadruples of self-avoiding paths with nodes that cover all vertices of a convex n-gon.

%C The paths considered here cover at least 2 vertices. Although each path is self-avoiding, the different paths are allowed to intersect.

%H Ivaylo Kortezov, <a href="https://doi.org/10.53656/math2022-6-4-set">Sets of Non-self-intersecting Paths Connecting the Vertices of a Convex Polygon</a>, Mathematics and Informatics, Vol. 65, No. 6, 2022.

%F a(n) = (1/3)*n*(n-1)*(n-2)*(n-3)*2^(n-15)*(4^(n-4) - 4*3^(n-4) + 6*2^(n-4) - 4) for n != 4.

%e a(9) = 9!*3/(2!2!2!3!3!) = 3780 since we have to split the 9 vertices into three pairs and one triple, the order of the three pairs is irrelevant, and there are 3 ways of connecting the triple.

%Y Cf. A001792, A332426 (unordered pairs of paths), A359404 (unordered triples of paths).

%K nonn

%O 3,6

%A _Ivaylo Kortezov_, Feb 01 2023