A360717 -id:A360717 - cp's OEIS frontend

A361285 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 allowed.

0, 0, 1, 10, 85, 695, 5600, 45080, 364854, 2973270, 24382875, 200967250, 1662197251, 13772638789, 114126098450, 944285871200, 7791140945180, 64038240953196, 523977421054245, 4266101869823850, 34554155058753505, 278417272387723315, 2231755184899383220, 17799741659621513240

Offset: 1

Ivaylo Kortezov, Mar 07 2023

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

			a(4) = A360021(4) + 4*A360021(3) = 6 + 4 = 10 since either all the 4 points are used or one is not.

Ivaylo Kortezov, Sets of Paths between Vertices of a Polygon, Mathematics Competitions, Vol. 35 (2022), No. 2, ISSN:1031-7503, pp. 35-43.

If there is only one path, we get A360715. If there is are two paths, we get A360717. If all n points need to be used, we get A360021.

a(n) = {(n*(n-1)*(n-2)/384) * (7^(n-3) + 9*5^(n-3) + 3^n + 27)} \\ Andrew Howroyd, Mar 07 2023

a(n) = (n*(n-1)*(n-2)/384)*(7^(n-3) + 9*5^(n-3) + 3^n + 27).

E.g.f.: x^3*exp(x)*(exp(2*x) + 3)^3/384. - Andrew Howroyd, Mar 07 2023

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

Ivaylo Kortezov, Jun 30 2023

nonn

For each such path there is a sequence of distinct vertices of the n-gon, each (except the last one) connected by a segment with the next vertex in the sequence; the segments have no common internal points. The path itself is the union of the set of these segments and is thus direction-independent: reversing the order of the vertices leads to the same path. If the sequence of vertices has length 1 then there are no segments; we call such a path a singleton.

			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.

Cf. A001792, A308914, A332426, A360716, A360717.

a(n) = n*(n-1)*(n^2+n+36)*2^(n-8)/3.

cp's OEIS Frontend

A361285 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 allowed.

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