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%I #11 Jun 25 2023 16:55:38
%S 0,0,0,5,63,476,2772,13680,60060,241472,906048,3214848,10890880,
%T 35481600,111794176,342171648,1021031424,2979102720,8520171520,
%U 23934468096,66156625920,180198047744,484304486400,1285790105600,3375480176640,8769899593728,22567515586560,57557594931200
%N Number of unordered triples of disjoint self-avoiding paths with nodes that cover all vertices of a convex n-gon.
%F a(n) = 2^(n-12)*n*(n-1)*(n-2)*(n-4)*(n-5)*(n+2)*(n+9)/90 for n > 3; 0 for n=3.
%e For n=7 we have one 3-node path and two 2-node paths. Call two paths adjacent if we can choose one node from each path so that the two nodes are adjacent vertices of the n-gon. Then either each pair of paths is adjacent, or the two 2-node paths are not adjacent, or a 2-node path is not adjacent to the 3-node path. In each of these three cases there are 7 choices for the set of nodes for the 3-node path and 3 ways to connect them, and then the 2-node paths are uniquely determined. Thus a(7) = 3*7*3 = 63.
%Y The number of unordered pairs of disjoint self-avoiding paths with nodes that cover all vertices of a convex n-gon is A308914(n). The number of unordered triples of (not necessarily disjoint) self-avoiding paths with nodes that cover all vertices of a convex n-gon is A359404(n).
%K nonn
%O 3,4
%A _Ivaylo Kortezov_, May 04 2023
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