A342061 Triangle read by rows: T(n,k) is the number of sensed 2-connected (nonseparable) planar maps with n edges and k vertices, n >= 2, 2 <= k <= n.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 8, 3, 1, 1, 4, 16, 16, 4, 1, 1, 5, 38, 63, 38, 5, 1, 1, 7, 72, 218, 218, 72, 7, 1, 1, 8, 134, 622, 1075, 622, 134, 8, 1, 1, 10, 224, 1600, 4214, 4214, 1600, 224, 10, 1, 1, 12, 375, 3703, 14381, 22222, 14381, 3703, 375, 12, 1

Offset

2,8

Comments

The number of faces is n + 2 - k.

Links

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

Timothy R. Walsh, Efficient enumeration of sensed planar maps, Discrete Math. 293 (2005), no. 1-3, 263--289. MR2136069 (2006b:05062).

Formula

T(n,k) = T(n, n+2-k).

Example

Triangle begins:
  1;
  1, 1;
  1, 1,   1;
  1, 2,   2,   1;
  1, 3,   8,   3,    1;
  1, 4,  16,  16,    4,   1;
  1, 5,  38,  63,   38,   5,   1;
  1, 7,  72, 218,  218,  72,   7, 1;
  1, 8, 134, 622, 1075, 622, 134, 8, 1;
  ...

Prog

(PARI) \\ See section 4 of Walsh reference.
T(n)={
  my(B=matrix(n, n, i, j, if(i+j <= n+1, (2*i+j-2)!*(2*j+i-2)!/(i!*j!*(2*i-1)!*(2*j-1)!))));
  my(C(i, j)=((i+j-1)*(i+1)*(j+1)/(2*(2*i+j-1)*(2*j+i-1)))*B[(i+1)/2, (j+1)/2]);
  my(D(i, j)=((j+1)/2)*B[i/2, (j+1)/2]);
  my(E(i, j)=((i-1)*(j-1) + 2*(i+j)*(i+j-1))*B[i, j]);
  my(F(i, j)=if(!i, j==1, ((i+j)*(6*j+2*i-5)*j*(2*i+j-1)/(2*(2*i+1)*(2*j+i-2)))*B[i, j]) + if(j-1, binomial(i+2, 2)*B[i+1, j-1]));
  vector(n, n, vector(n, i, my(j=n+1-i); B[i, j]
    + (i+j)*if(i%2, if(j%2, C(i, j), D(j, i)), if(j%2, D(i, j)))
    + sumdiv(i+j, d, if(d>1, eulerphi(d)*( if(i%d==0, E(i/d, j/d) ) + if(i%d==1, F((i-1)/d, (j+1)/d)) + if(j%d==1, F((j-1)/d, (i+1)/d)) )))
   )/(2*n+2));
}
{ my(A=T(10)); for(n=1, #A, print(A[n])) }

Crossrefs

Column k=3 is A001399(n-3).

Row sums are A006402.

Cf. A082680 (rooted), A239893, A342059.

Sequence in context: A266378 A092113 A331485 * A045995 A360625 A157654

Adjacent sequences: A342058 A342059 A342060 * A342062 A342063 A342064

Keyword

nonn,tabl

Author

Andrew Howroyd, Mar 30 2021

Status

approved