A343092 Triangle read by rows: T(n,k) is the number of rooted toroidal maps with n edges and k faces and without isthmuses, n >= 2, k = 1..n-1.

1, 4, 10, 10, 79, 70, 20, 340, 900, 420, 35, 1071, 5846, 7885, 2310, 56, 2772, 26320, 71372, 59080, 12012, 84, 6258, 93436, 431739, 706068, 398846, 60060, 120, 12768, 280120, 2000280, 5494896, 6052840, 2499096, 291720, 165, 24090, 739420, 7643265, 32055391, 58677420, 46759630, 14805705, 1385670

Offset

2,2

Comments

The number of vertices is n - k.

Column k is a polynomial of degree 3*k. This is because adding a face can increase the number of vertices whose degree is greater than two by at most two.

Links

Andrew Howroyd, Table of n, a(n) for n = 2..1276

T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259, Table VId.

Example

Triangle begins:
   1;
   4,   10;
  10,   79,    70;
  20,  340,   900,    420;
  35, 1071,  5846,   7885,   2310;
  56, 2772, 26320,  71372,  59080,  12012;
  84, 6258, 93436, 431739, 706068, 398846, 60060;
  ...

Prog

(PARI) \\ Needs F from A342989.
G(n, m, y, z)={my(p=F(n, m, y, z)); subst(p, x, serreverse(x*p^2))}
H(n, g=1)={my(q=G(n, g, 'y, 'z)-x, v=Vec(polcoef(sqrt(serreverse(x/q^2)/x), g, 'y))); [Vecrev(t) | t<-v]}
{ my(T=H(10)); for(n=1, #T, print(T[n])) }

Crossrefs

Columns 1..2 are A000292, A006469.

Diagonals are A002802, A006425, A006426, A006427.

Row sums are A343093.

Cf. A269921, A342981, A342989, A343090.

Sequence in context: A264163 A223165 A263558 * A346751 A292910 A292954

Adjacent sequences: A343089 A343090 A343091 * A343093 A343094 A343095

Keyword

nonn,tabl

Author

Andrew Howroyd, Apr 04 2021

Status

approved