Most suitable combinatorics algorithm

Question

There are 6 seats and 4 persons, for which seats must be allocated according to some optimality criterion. For example:

Allocation 1:
_ _ _ _ _ _
1 2   3 4 

Allocation 2:
_ _ _ _ _ _
1 3   2   4 

...

Question 1: Which combinatorics problem is it?

Question 2: What is the name of most suitable algorithm for searching through all possible combinations?

Answer 1

1.) k-permutation of n without repetion: http://www.statlect.com/comdis1.htm

2.) It depends on which you are searching for. For example, I offer you genetic algorithms which can find the best candidate based on a special heuristic if you can order a "goodness degree" to the possible solutions.

Answer 2

To answer Question 1, note that for the sequence of 4 persons there are 4! possibilites. Additionally, the 6-4=2 non-occupied seats must be positioned between the people, for which there are 4+1=5 slots (before every person and behind the last person), resulting in 5+2-1 choose 2 possibilities, where choose denotes the binomial coeffiecient, by interpretation as stars and bars problem. In total, there are

4!(6 choose 2)

possibilities, or parameterized

m!(m+1+nm-1 choose nm) = m!(n choose nm)

where m is the number of people and n is the number of seats; using the identity

n choose k = n!/(k!(nk)!)

this can be simplified to

n!/(nm)!

which is indeed the n -permutation of m objects as defined here .

Concerning Question 2, this really depends on the optimality criterion and whether an exact, approximate or heuristic algorithm is desired.