A342989 Triangle read by rows: T(n,k) is the number of nonseparable rooted toroidal maps with n edges and k faces, n >= 2, k = 1..n-1.

1, 4, 4, 10, 39, 10, 20, 190, 190, 20, 35, 651, 1568, 651, 35, 56, 1792, 8344, 8344, 1792, 56, 84, 4242, 33580, 64667, 33580, 4242, 84, 120, 8988, 111100, 361884, 361884, 111100, 8988, 120, 165, 17490, 317680, 1607125, 2713561, 1607125, 317680, 17490, 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,   39,     10;
   20,  190,    190,     20;
   35,  651,   1568,    651,     35;
   56, 1792,   8344,   8344,   1792,     56;
   84, 4242,  33580,  64667,  33580,   4242,   84;
  120, 8988, 111100, 361884, 361884, 111100, 8988, 120;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 2..1276 (first 50 rows)
• 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 II.

Crossrefs

Columns 1..4 are A000292, A006408, A006409, A006410.

Row sums are A343089.

Cf. A082680 (planar case), A269921 (rooted toroidal maps), A343090, A343092.

Programs

PARI

MQ(n,g,x=1)={ \\ after Quadric in A269921.
  my(Q=matrix(n+1,g+1)); Q[1,1]=x;
  for(n=1, n, for(g=0, min(n\2,g),
     my(t = (1+x)*(2*n-1)/3 * Q[n, 1+g]
       + if(g && n>1, (2*n-3)*(2*n-2)*(2*n-1)/12 * Q[n-1, g])
       + sum(k = 1, n-1, sum(i = 0, g, (2*k-1) * (2*(n-k)-1) * Q[k, 1+i] * Q[n-k, 1+g-i]))/2);
     Q[1+n, 1+g] = t * 6/(n+1); ));
  Q
}
F(n,m,y,z)={my(Q=MQ(n,m,z)); sum(n=0, n, x^n*Ser(Q[1+n,]/z, y)) + O(x*x^n)}
H(n,g=1)={my(p=F(n,g,'y,'z), v=Vec(polcoef(subst(p, x, serreverse(x*p^2)), g, 'y))); vector(#v, n, Vecrev(v[n], n))}
{ my(T=H(10)); for(n=1, #T, print(T[n])) }

Formula

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