A343090 Triangle read by rows: T(n,k) is the number of rooted toroidal maps with n edges and k faces and without separating cycles or isthmuses, n >= 2, k = 1..n-1.

1, 4, 4, 10, 47, 10, 20, 240, 240, 20, 35, 831, 2246, 831, 35, 56, 2282, 12656, 12656, 2282, 56, 84, 5362, 52164, 109075, 52164, 5362, 84, 120, 11256, 173776, 648792, 648792, 173776, 11256, 120, 165, 21690, 495820, 2978245, 5360286, 2978245, 495820, 21690, 165

Offset: 2

Author

Andrew Howroyd, Apr 04 2021

Keywords

• nonn
• tabl

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.

Examples

			Triangle begins:
    1;
    4,     4;
   10,    47,     10;
   20,   240,    240,     20;
   35,   831,   2246,    831,     35;
   56,  2282,  12656,  12656,   2282,     56;
   84,  5362,  52164, 109075,  52164,   5362,    84;
  120, 11256, 173776, 648792, 648792, 173776, 11256, 120;
  ...
		

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 VIc.

Crossrefs

Columns 1..4 are A000292, A006422, A006423, A006424.

Row sums are A343091.

Cf. A269921, A342980, A342989, A343092.

Programs

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*(1+'z), 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])) }

Formula

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