Mathematics 501
Homework (due Nov 1)
A. Hulpke
43) Show that there are exactly Cn Young diagrams that “fit in the shape” (n − 1, n −
2, . . . , 1).
44) Using the Hook formula, determine the number of tableaux of the shape 71 51 32 1.
45) a) Show that if a diagram has a hook of length a + b it must have a hook of length
a or of length b. (Hint: Consider a zig-zag path between the ends of the hook that goes
around the bottom edge of the diagram)
b) Show that if a diagram has a hook of length ab it must have hooks of length a and
length b.
46) Using the RSK correspondence, determine the tableaux corresponding to the permutation (1, 7)(3, 8, 6, 4, 5). Determine a permutation corresponding to the following pair
1 4 8
1 3 8
2 6
2 4
3 7
5 7
5
and 6
.
of tableaux:
47) Let s(n) be the number of permutations on n points for which π 2 = 1.
a) Show that s(n) = s(n − 1) + (n − 1)s(n − 2)
b) Show that the exponential generating function for s(n) is
tn
2
∑ s(n) n! = et+t /2
n
(Hint: Consider the differential equation f 0 (t) = (1 + t) f (t).)
48) Let f (x) = Q[x] be a monic (i.e the coefficient of the largest power of x occurring is
1) polynomial. We define the discriminant of f as
D( f ) = ∆2 ( f ) = ∏(αi − α j )
i6= j
where the {αi } are the roots of f , including multiplicities.
a) Show that D( f ) = 0 if and only f has multiple roots.
b) Show that if f (x) ∈ Z[x] then D( f ) ∈ Z.
c) Show that if f (x) ∈ Z[x] has no multiple roots that there are only finitely many primes
p such that f has multiple roots modulo p.