A360717
Number of unordered pairs of self-avoiding paths whose sets of nodes are disjoint subsets of a set of n points on a circle; one-node paths are allowed.
Original entry on oeis.org
0, 1, 6, 33, 185, 1050, 6027, 35014, 205326, 1209375, 7119860, 41744703, 243218703, 1406685280, 8073640785, 45991600860, 260131208396, 1461591509805, 8162196518322, 45327133739245, 250431036147285, 1377169337010390, 7540979990097191, 41130452834689218, 223528009015333050, 1210753768099880875, 6537995998163877312
Offset: 1
a(4) = A359405(4) + 4*A359405(3) + 4*3/2 = 15 + 12 + 6 = 33 with the three summands corresponding to the cases of 4, 3 and 2 used points.
If there is only one path, we get
A360715. If one-node paths are not allowed, we get
A360716.
A361284
Number of unordered triples of self-avoiding paths whose sets of nodes are disjoint subsets of a set of n points on a circle; one-node paths are not allowed.
Original entry on oeis.org
0, 0, 0, 0, 0, 15, 420, 7140, 95760, 1116990, 11891880, 118776900, 1132182480, 10415938533, 93207174060, 815777235000, 7011723045600, 59364660734172, 496238466573648, 4102968354298200, 33602671702168800, 272909132004479355, 2200084921469527092, 17618774018675345340, 140252152286127750000
Offset: 1
a(7) = A359404(7) + 7*A359404(6) = 315 + 7*15 = 420 since either all the 7 points are used or one is not.
If there is only one path, we get
A261064. If there is are two paths, we get
A360716. If all n points need to be used, we get
A359404.
A363964
Number of unordered pairs of non-intersecting non-self-intersecting paths, singletons included, with nodes that cover all vertices of a convex labeled n-gon.
Original entry on oeis.org
3, 14, 55, 195, 644, 2016, 6048, 17520, 49280, 135168, 362752, 955136, 2472960, 6307840, 15876096, 39481344, 97124352, 236584960, 571146240, 1367539712, 3249799168, 7669284864, 17983078400, 41916825600, 97165246464, 224076496896, 514272002048, 1174992322560
Offset: 3
a(4)=14 since if one of the paths is a singleton (4 choices), then there are A001792(3)=3 choices for the other path, and otherwise for the two paths there are A308914(4)=2 choices, so a(4)=4*3+2=14.
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