Count In The Finite Topology

In this article we study the counting problem of topology of pairwise non-homeomorphism on n-ary finite set. First, for two special cases: 1-level and 2-level, we study these two counting problem of topological structure. Then, we discuss the general case for rn-level, and we have these results:1.For 1-level topological structure, the number of topology of pairwise non-homeomorphism is: F1 (n) = P(ri), and P(n) is a partition of integern.We have P(n) .2.For 2-level topological structure, a upper bound of the counting result of topology of pairwise non-homeomorphism is found:(pxq眫 [⁺⁼.Rp,q(k) is the coefficient of ^P<sup>kyk in polynomial:N(Gp,q) zgEGpqGp,q is a permutation group consisted of the row-column exchanging of p x q compoment of matrix, which is a 0-1 matrix denoting a order structure; N(Gp,q) is the cardinal of group Gp,q; A1(g) is number of i-circle in a permutation g of Gp,q.On the base of that, we have a counting formule for 2-level topological structure:F2(n)>7>3 (Allpxq ?Allpx(q....1) ?All(p.1)xq + A11(P....1)x(^....1))]^pqAllpxq is number of coloring style of p x q squares, and we knowAllpxq =1N(Gp,q)3.It is more and more diffcult to enumerate all elements of Gp,q by increaing of integer i-i. So we discuss deeply about the permutation group Gp,q, and give two new forms of Rp,q(k)抯 generated polynomial and Allpxq, which are more detailed and more operable:F /31[ I1>3>3 a3bt . + ^f) LCM(.,i)=r) js1r1IandAllpxq ?7 (a^<sup>tLCM(.,,) (s)^t)]forp.(s)(⁾(⁾a3. 1% 202 ... pop=which is number of elements of the set consisted of permutation belonging to the same conjugate class ic4) 2⁴) . . . p0 of row exchanging group Si., and the same conjugate class 1??? qL4 of column exchanging group Sq. P(p), P(q) are the partition of p, q, and LCM(i,j) is lowest common multiple of i, j.4.In the end of this article, we discuss the general case for rn-level.About case of non-skip-levels for partition of P, we have a upper bound:1P(pi)1H p^! s⁼1IP, A(g) are defined in the article.About case of counting of rn-level (rn > 3) which has skip-levels for partition of P, we don抰 get a satisfying result in this article. But we get a upper bound formula of counting result for 3-level:F3(n) +n抏9v拁...