Permutation Sum-Fun

Player 1 finds an integer m by taking a permutation {b(j)} of the
first n positive integers,
then forms m:
m = sum{k=1 to n} k * b(k).
Player 2 then, in some time limit, tries to find ANY permutation of
the first n positive integers which also sums to m.
Players take turns making up puzzles and trying to solve other
player's puzzles.
( It is advantageous for the proposer to come up with m's with
relatively few solutions, but are not too easy {easy, such as b(k) = k
or = (n+1-k), which each are the only solutions to their sum m, but
are easy to solve}.)
2 things:
1) One can play a solitaire version of this game, where the
permutation (of 1 through 10, for example) is picked by shuffling
cards, and the player tries to come up with a solution which is
a) a derrangement of the cards' permutation;
b) is not the inverse perm. of the card-perm..
2) How can we determine the number of permutations (for a given n)
which have a given sum m?
thanks,
Leroy
Quet