A380362 -id:A380362 - cp's OEIS frontend

A346779 Number of Halin graphs on n unlabeled nodes.

0, 0, 0, 1, 1, 2, 2, 4, 6, 13, 22, 50, 106, 252, 589, 1475, 3669, 9435, 24345, 63837, 168234, 447562, 1196390, 3218221, 8694411, 23598318, 64292975, 175820236, 482391019, 1327680919, 3664644419, 10142533143, 28141900501, 78269260312, 218170198957, 609414024190

Offset: 1

Eric W. Weisstein, Aug 03 2021

nonn

			a(4) = 1 (K_4)
a(5) = 1 (W_5)
a(6) = 2 (3-prism graph, W_6)
a(7) = 2 (W_7 and one other)
a(8) = 4 (W_8 and 3 others)

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Peter J. Taylor, Python program
Eric Weisstein's World of Mathematics, Halin Graph.
Wikipedia, Halin graph.

Row sums of A380362.
Antidiagonal sums of A295634.
Cf. A380360.

A346779seq(36) \\ See PARI Link in A380362 for program code. - Andrew Howroyd, Jan 26 2025

Python
```
# See Taylor link
```

a(13) from Eric W. Weisstein, Aug 16 2021
a(14) from Eric W. Weisstein, Sep 29 2021
a(15)-a(24) from Peter J. Taylor, May 20 2023
a(25) onwards from Andrew Howroyd, Jan 25 2025

A380360 Number of embeddings on the sphere of Halin graphs on n unlabeled nodes up to orientation-preserving homeomorphisms.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 4, 7, 16, 32, 76, 181, 443, 1098, 2793, 7127, 18458, 48128, 126580, 334955, 892187, 2388674, 6428489, 17377599, 47174939, 128555088, 351580903, 964696719, 2655197386, 7329051870, 20284610084, 56283140111, 156537249660, 436338547904, 1218824493990, 3411297202411

Offset: 1

Andrew Howroyd, Jan 25 2025

nonn

Halin graphs are planar and 3-connected and can be embedding in the sphere in essentially one way up to mirror symmetry. This sequence counts each graph as either 1 or 2 depending on if it is mirror symmetric.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Halin Graph.
Wikipedia, Halin graph.

Row sums of A380361.
Antidiagonal sums of A295633.
Cf. A005043, A346779, A380362.

A380360seq(36) \\ See PARI Link in A380362 for program code.

A380361 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of Halin graphs on n unlabeled nodes with circuit rank k up to orientation-preserving homeomorphisms, 3 <= k <= n-1.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 0, 4, 2, 1, 0, 0, 0, 4, 8, 3, 1, 0, 0, 0, 0, 12, 16, 3, 1, 0, 0, 0, 0, 6, 40, 25, 4, 1, 0, 0, 0, 0, 0, 43, 93, 40, 4, 1, 0, 0, 0, 0, 0, 19, 165, 197, 56, 5, 1, 0, 0, 0, 0, 0, 0, 143, 505, 364, 80, 5, 1

Offset: 4

Andrew Howroyd, Jan 25 2025

The circuit rank is equal to the number of leaves on the tree before it is extended into a Halin graph by joining up the leaves.
The main diagonal of the graph corresponds with the wheel graphs which have the greatest circuit rank of all Halin graphs.
T(n,k) is also the number of nonequivalent dissections of a k-gon into n-k polygons by nonintersecting diagonals up to rotation.

			Triangle begins:
  n\k| 3  4  5  6   7   8    9   10  11  12  13
-----+-----------------------------------------
   4 | 1;
   5 | 0, 1;
   6 | 0, 1, 1;
   7 | 0, 0, 1, 1;
   8 | 0, 0, 1, 2,  1;
   9 | 0, 0, 0, 4,  2,  1;
  10 | 0, 0, 0, 4,  8,  3,   1;
  11 | 0, 0, 0, 0, 12, 16,   3,   1;
  12 | 0, 0, 0, 0,  6, 40,  25,   4,  1;
  13 | 0, 0, 0, 0,  0, 43,  93,  40,  4,  1;
  14 | 0, 0, 0, 0,  0, 19, 165, 197, 56,  5,  1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 4..1278 (first 50 rows)
Eric Weisstein's World of Mathematics, Halin Graph.
Wikipedia, Circuit rank.
Wikipedia, Halin graph.

Row sums are A380360.
Column sums are A003455.
Main diagonal is A000012.
Central coefficients are A001683.
Cf. A295633, A380362.

\\ See PARI Link in A380362 for program code.
{ my(T=A380361rows(12)); for(i=1, #T, print(T[i])) }

T(n,k) = A295633(k, n-k).

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A346779 Number of Halin graphs on n unlabeled nodes.

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A380360 Number of embeddings on the sphere of Halin graphs on n unlabeled nodes up to orientation-preserving homeomorphisms.

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A380361 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of Halin graphs on n unlabeled nodes with circuit rank k up to orientation-preserving homeomorphisms, 3 <= k <= n-1.

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