Some recurrence relations in finite topologies
Zapiski Nauchnykh Seminarov POMI, Modules and algebraic groups, Tome 114 (1982), pp. 174-179
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In a number of papers (see, e.g., RZhMat, 1977, 11B586) there is given for the number $T_0(n)$ of labeled topologies on $n$ points satisfying the $T_0$ separation axiom the formula
$$
T_0(n)=\sum\dfrac{n!}{p_1!\dots p_k!}V(p_1,\dots,p_k),
$$
where the summation extends over all ordered sets $(p_1,\dots,p_k)$ of natural numbers such that $p_1+\dots+p_k=n$. In the present paper there is found a relation for calculating, when $n\geqslant2$, the sum of all terms in this formula for which $p_2=1$ in terms of the values $V(q_1,\dots,q_t)$ with $q_1+\dots+q_t\leqslant n-2$. This permits the determination (with the aid of a computer) of the new value
$$
T_0(12)=414\,864\,951\,055\,853\,499.
$$
@article{ZNSL_1982_114_a15,
author = {V. I. Rodionov},
title = {Some recurrence relations in finite topologies},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {174--179},
publisher = {mathdoc},
volume = {114},
year = {1982},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_1982_114_a15/}
}
V. I. Rodionov. Some recurrence relations in finite topologies. Zapiski Nauchnykh Seminarov POMI, Modules and algebraic groups, Tome 114 (1982), pp. 174-179. http://geodesic.mathdoc.fr/item/ZNSL_1982_114_a15/