A380361 -id:A380361 - cp's OEIS frontend

A380362 Triangle read by rows: T(n,k) is the number of Halin graphs on n unlabeled nodes with circuit rank k, 3 <= k <= n-1.

1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 0, 3, 2, 1, 0, 0, 0, 3, 6, 3, 1, 0, 0, 0, 0, 7, 11, 3, 1, 0, 0, 0, 0, 4, 24, 17, 4, 1, 0, 0, 0, 0, 0, 24, 51, 26, 4, 1, 0, 0, 0, 0, 0, 12, 89, 109, 36, 5, 1, 0, 0, 0, 0, 0, 0, 74, 265, 194, 50, 5, 1, 0, 0, 0, 0, 0, 0, 27, 371, 660, 345, 65, 6, 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 rotations and reflections.

			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, 3, 2,  1;
  10 | 0, 0, 0, 3, 6,  3,  1;
  11 | 0, 0, 0, 0, 7, 11,  3,   1;
  12 | 0, 0, 0, 0, 4, 24, 17,   4,  1;
  13 | 0, 0, 0, 0, 0, 24, 51,  26,  4,  1;
  14 | 0, 0, 0, 0, 0, 12, 89, 109, 36,  5,  1;
   ...

Andrew Howroyd, Table of n, a(n) for n = 4..1278 (first 50 rows)
Andrew Howroyd, Formulas and PARI Program, Jan 2025.
Eric Weisstein's World of Mathematics, Halin Graph.
Wikipedia, Circuit rank.
Wikipedia, Halin graph.

Row sums are A346779.
Column sums are A001004.
Main diagonal is A000012.
Central coefficients are A000207.
Cf. A295634, A380361.

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

T(n,k) = A295634(k, n-k).

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.

cp's OEIS Frontend

A380362 Triangle read by rows: T(n,k) is the number of Halin graphs on n unlabeled nodes with circuit rank k, 3 <= k <= n-1.

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