20. List the odd permutations of 1, 2, 3, 4.
 
R 21. Let P be a permutation i, j, h, ... k of 1, 2, 3, ... , n.

(a) Show that if i and j are interchanged, P changes from odd to even or from even to odd.
 
(b) Show that if any two adjacent terms in P are interchanged, P changes from odd to even or from even to odd.
 
(c) Show that the interchange of any two terms in P can be considered to be the result of an odd number of interchanges of adjacent terms.
 
(d) Show that if any two terms in the permutation P are interchanged, P changes from odd to even or from even to odd.
 
(e) Given thatshow that half of the permutations of 1, 2, ... , n are even and half are odd, that is, that there are even permutations and the same number of odd ones.
 
R 22. (a) Let P be a permutation a, b, c, d of the numbers 1, 2, 3, 4. Let d = 4 and let Q be the associated permutation a, b, c of 1, 2, 3. Show that P and Q are either both even or both odd.
 
(b) Let R be a permutation i, j, ..., h, n of the numbers 1, 2, ..., n-1, n in which the last term of R is n. Let S be the associated permutation i, j, ..., h of 1, 2, ..., n-1 obtained by dropping the last term of R. Show that R and S are either both even or both odd.
 
23. How many triples of positive integers r, s, and t are there with r < s < t and:
 
(a) r + s + t = 52?
 
(b) r + s + t = 352?

Tuesday, May 12, 1998