Why is there uncertainty in combinatorial proofs?

[s.hamid.ef]

There's something I can not understand about proofs in combinatorics. Whenever I solve a counting problem, there's a non-negligible amount of uncertainty about the solution which I really don't feel when I solve problems in other fields, say in analysis or abstract algebra. It happens too often that someone sees my solution and tells me I've counted more or fewer than the correct answer. And I've observed this happens even to more experienced students and even teachers. But every time we come to a general agreement after refining the solution.
What's wrong with me? Or does it have anything to do with how it's presented? I've never seen an axiomatic treatment of this field, like say, abstract algebra. Of course all the books I've seen start with the two counting principles, but they seem like too informal to use in rigorous proofs.


Thanks in advance.

 

[mathman]

You should supply some sort example of what you are concerned about.

 

[s.hamid.ef]

Hi Mathman,
I'm not concerned about any particular examples, and now I'm realizing it's not as common an issue as I thought. I guess I need to gain more experience in the field before I can compare it to other fields.
Anyways, what's the most rigorous treatment of enumerative combinatorics you ( and others!) know?

 

[mathman]

Hi Mathman,
I'm not concerned about any particular examples, and now I'm realizing it's not as common an issue as I thought. I guess I need to gain more experience in the field before I can compare it to other fields.
Anyways, what's the most rigorous treatment of enumerative combinatorics you ( and others!) know?

I have no answer for your question.

 

[Stephen Tashi]

There's something I can not understand about proofs in combinatorics. Whenever I solve a counting problem, there's a non-negligible amount of uncertainty about the solution which I really don't feel when I solve problems in other fields, say in analysis or abstract algebra.

I share your feeling. I also note that in threads about complicated combinatorial problems we often see fairly skilled people make "small" mistakes in the answers they propose and get corrrected by others. Often it is a technicality about how the English statement of the problem is to be translated into precise requirements. In the field of combinatorics, what we see on math forums are usually the solutions to problems, not formal proofs. I haven't read enough formal proofs of combinatorial results to form an opinion about the formal proofs.

However, there is a similar uncertainty when solving problems that involve doing long symbolic manipulations by hand, or problems that involve considering a large number of different cases. Do you enjoy combinatorial problems? I've never been interested in the kind that just count the number of ways. I do have an interest in algorithms that actually generate a list of all the ways.