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

Author

Andrew Howroyd, Jan 25 2025

Keywords

• nonn
• tabl

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.

Examples

			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;
  ...
		

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.

Crossrefs

Row sums are A380360.

Column sums are A003455.

Main diagonal is A000012.

Central coefficients are A001683.

Cf. A295633, A380362.

Programs

PARI

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

Formula

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