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.

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

4,14

Comments

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.

Links

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.

Formula

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

Example

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;
  ...

Prog

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

Crossrefs

Row sums are A380360.

Column sums are A003455.

Main diagonal is A000012.

Central coefficients are A001683.

Cf. A295633, A380362.

Sequence in context: A326453 A130116 A325774 * A350488 A212868 A184616

Adjacent sequences: A380356 A380357 A380360 * A380362 A380363 A380364

Keyword

nonn,tabl,new

Author

Andrew Howroyd, Jan 25 2025

Status

approved