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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.
8
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
2,2
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
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.
FORMULA
T(n,n-k) = T(n,k).
EXAMPLE
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;
...
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*(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])) }
CROSSREFS
Columns 1..4 are A000292, A006422, A006423, A006424.
Row sums are A343091.
Sequence in context: A342989 A377149 A161719 * A161433 A180498 A107856
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Apr 04 2021
STATUS
approved