This quiz is taken from the book "Statistical Mechanics:
Rigorous Results"
by D. Ruelle *Addison-Wesley 1969).

Let *A _{ij}* be a

where:

- the sum is over the
*2*subsets of {1,..,n}^{n} - S*{1,..,k} denotes the intersection of the subsets S and {1,..,k}; the coefficient in front of the double product is therefore +1 or -1 depending whether the cardinality of this set is even or odd, respectively;
- the product over i runs on the integers in S
- the product over j runs over the integers in the complement S' of S
- if S is empty or equal to {1,..,n} then the products are 1.

For exaample, consider the case *n*=2:

- P(0,2) = 1 + A
_{12}+ A_{21}+ 1 - P(1,2) = 1 - A
_{12}+ A_{21}- 1 - P(2,2) = 1 - A
_{12}- A_{21}+ 1

Show that P(k,n) is greater or equal to 0 for every *n*
and *k* (between 0 and *n*).

Marco Corvi - 2006