Jamie Tucker-Foltz - cp's OEIS frontend

User: Jamie Tucker-Foltz

Jamie Tucker-Foltz has authored 2 sequences.

A374293 a(n)/binomial(n,2)! is the probability that the minimum spanning tree of the complete graph of n vertices with i.i.d. random edge weights is a specific path.

1, 1, 2, 44, 27120, 882241920, 2443792425984000, 846533597741050576896000, 50571850611494440562578575851520000, 686805008584962439650318114385825747697664000000, 2701735270674169239689693528384644314472371275610193920000000000, 3819958423456547324072333722421751679308286064300212197312630212725309440000000000

Offset: 1

Jamie Tucker-Foltz, Jul 02 2024

nonn

Equivalently, a(n) is the number of orderings of the edges of the complete graph of n vertices such that the minimal spanning tree (e.g., obtained by running Kruskal's algorithm with the edges in that order) is a specific path.

It appears that this is a subsequence of A253950. Specifically, a(n) appears at index m - n + 3, where m = binomial(n,2) is the number of edges of the complete graph on n vertices.

			a(3) = 2 because there are 2 orderings of the edges a, b, and c of K_3 that give the path {a, b}: (a, b, c) and (b, a, c).

Jamie Tucker-Foltz, Table of n, a(n) for n = 1..16
Eric Babson, Moon Duchin, Annina Iseli, Pietro Poggi-Corradini, Dylan Thurston, and Jamie Tucker-Foltz, Models of Random Spanning Trees, arXiv:2407.20226 [math.CO], 2024.
Jamie Tucker-Foltz, Code to compute a(n) on GitHub.

Cf. A052295, A253950.

E(p,m)={sum(k=0, m, sum(i=0, k, polcoef(p, i)*i!*(m-i)! )*x^k/(k!*(m-k)!))}
seq(n)={my(v=vector(n)); v[1]=1; for(n=2, n, my(p=sum(k=1, n-1, v[k]*v[n-k])); v[n]=E(intformal(p), binomial(n,2))); vector(n, n, my(m=binomial(n,2)); m!*polcoef(v[n], m))} \\ Andrew Howroyd, Jul 31 2024

A365862 Number of 4-fold rotationally symmetric pentomino tilings of the 5n X 5n square such that no two tiles can be replaced with two pentomominoes in a new configuration.

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 20, 54, 6464

Offset: 0

Jamie Tucker-Foltz, Sep 29 2023

a(n) is even for all n > 0 since no pentomino tiling can have both rotation and reflection symmetry.

			a(4) = 2:
._______________________________________.
|_. | ._. | ._|_. |_________|_________| |
| | |_| |_| |_. | ._|_. | ._. | ._| .___|
| |___| |___| | |_| ._| |_| |_| | |_|_. |
| | |_____| ._| | ._| |___| |___| .___| |
| |___. |___| |_|_| |_. |_____| |_| |___|
|_| ._|_| |_______| ._|___| | |_. |___. |
| | |___. |_. | |___|_______| ._|___| |_|
| |___| |_| | |_______| | ._| |_______| |
| | .___| ._| |_. | .___| | |_|_. |_. ._|
| |_| |___| |_| | |_|_. | |_. ._|_. |_| |
|_| |_. |___. | |___| | |_| |_| ._|_| | |
| .___|___| |_| | .___|___| | ._| .___| |
|_| ._____| |___|_|_____. | |_| |_|_. | |
| |_|_. |_. | ._____| ._|_|___| .___| |_|
|___. |_. |_|_|_. |_. |___. ._|_| |___| |
| ._|_| |_|_. ._|_. |_| | |_| ._|___. | |
| |___. | ._| |_. |_| ._| | ._|_. ._|_| |
|___| |_| | |_| | |___| | |_| ._| |_. | |
| .___|___|_____|___|_. |___| | |_| | |_|
|_|_________|_________|___|___|_____|___| (*2)

Jamie Tucker-Foltz, Locked Polyomino Tilings, arXiv:2307.15996 [math.CO], 2023.
Jamie Tucker-Foltz, GitHub repository with Python code and pictures of all tilings.
Wikipedia, Pentomino.

a(8) from Jamie Tucker-Foltz, Oct 30 2023

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User: Jamie Tucker-Foltz

A374293 a(n)/binomial(n,2)! is the probability that the minimum spanning tree of the complete graph of n vertices with i.i.d. random edge weights is a specific path.

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