But if you believe that you're a lost cause in any case :P It's still reasonable to ask for further justification, though. I'd attempt to give some, but I've been awake for like 30 hours and it would be even more handwavey than the above. The short version: If you do enough enumerative combinatorics, you start to see that nice formulas for enumerating structures arise from one of a few situations. Some really big ones are:.
So my answer boils down to: It's intractable 'cause it is. Not a particularly satisfying reason, but sometimes that's the way combinatorics works. Sorry if you read that whole post -- I meant for it to be shorter and have more content, but it ended up like most tales told by idiots.
But hopefully you learned something, or at least had fun with it? The best result I know is found in the article The number of finite topologies , by D.
Kleitman and B. I'll expand a little on Harrison's answer. There are several techniques in algebraic combinatorics that allow for exact enumeration of unordered structures; they include. Writing down the generating function in terms of elementary functions. Important ways to combine structures include sum, product, composition, and derivative, all of which have known combinatorial meanings.
But, on the surface, there appears to be no way to encode the structure of a topology in this way. Writing down a functional equation the generating function satisfies. This is most useful with tree-like structures, since the recursive nature of their definition is reflected in certain functional equations. But topologies don't seem to be a tree-like structure in any obvious way. Exhibiting the sequence as the evaluation of a sequence of determinants. This is an incredibly useful technique; it can count tableaux and perfect matchings and all sorts of crazy things, but problems to which this technique applies have a certain feel to them and this one doesn't even come close to fitting the bill.
Using some variant of Burnside's lemma or Polya's theorem. But there seems to be no useful way to interpret finite topologies in terms of a group action. Since finite topologies are an algebraic structure of sorts - a set closed under two operations - you might like to think about a correspondingly difficult algebraic problem, which is the enumeration of finite groups. Sloane's encyclopedia gives some references. I think that if something moves in this question, then Sloane's site will be one of the first to be updated I am not sure in how far this has to do the Union-Closed Sets Conjecture - the latter is just an extremal-combinatorics style assertion.
I don't see how to use it for an enumeration. This question is closely related to the Union-Closed Sets Conjecture which is an open question proposed in Sign up to join this community.
The best answers are voted up and rise to the top. Number of valid topologies on a finite set of n elements Ask Question. Asked 11 years, 11 months ago. Bus technology is mainly suited for small networks like LAN, etc. In this topology, the bus acts as the backbone of the network, which joins every computer and peripherals in the network.
Both ends of the shared channel have line terminators. The data is sent only in one direction and as soon as it reaches the end, the terminator removes the data from the communication line to prevent signal bounce and data flow disruption. In a bus topology, each computer communicates to another computer on the network independently.
Every computer can share the network's total bus capabilities. The devices share the responsibility for the flow of data from one point to the other in the network. Ring topology is a topology in which each computer is connected to exactly two other computers to form the ring.
The message passing is unidirectional and circular in nature. This network topology is deterministic in nature, i. All the nodes are connected in a closed-loop. This topology mainly works on a token-based system and the token travels in a loop in one specific direction.
In a ring topology, if a token is free then the node can capture the token and attach the data and destination address to the token, and then leaves the token for communication. When this token reaches the destination node, the data is removed by the receiver and the token is made free to carry the next data.
Star topology is a computer network topology in which all the nodes are connected to a centralized hub. The hub or switch acts as a middleware between the nodes. Any node requesting for service or providing service, first contact the hub for communication.
The central device hub or switch has point to point communication link the dedicated link between the devices which can not be accessed by some other computer with the devices. The central device then broadcast or unicast the message based on the central device used. The hub broadcasts the message, while the switch unicasts the messages by maintaining a switch table. Broadcasting increases unnecessary data traffic in the network.
In a star topology, hub and switch act as a server, and the other connected devices act as clients. Also, it is understood that the empty set is in all these topologies. I organized these topologies by the number of sets in each topology. The lowest row consists of just the trivial topology.
Neither one is finer than the other. As each path moves from the bottom to the top of the picture, we move from coarser to finer topologies.
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