MTH 786-Q (Spring 2018): PROBLEM ASSIGNMENTS
Problems should be solved primarily on your own. Some
"reasonable" collaboration is permitted, but you shouldn't just obtain
the solution from another source. Do not hand in a
solution that you did not obtain on your own or by collaboration with
another student in the course!
- Due Tuesday, January 23. Problems 1, 4, 7, 8.
- Due Thursday, February 1. Problems 10, 11, 15, 20, 24. For
Problem 20(b) you will need some knowledge of enumerative
combinatorics that can be found in EC1, Chapter 1.
- Due Tuesday, February 13. Problems 21, 28, 31, 49, 54. For #54
you may assume the (easy) result
(1-4x)-1/2, where C(2n,n)
denotes a binomial coefficient (since I don't know how to write the
usual binomial coefficient notation in html). See EC1, Exercise 8(a).
- Due Thursday, February 22. Problems 18, 57, 61, 78(a)
(for a(m,n) only). For #78(a), all you need to know
about cλμν is its
Bonus. Problem 55.
Also hand in the following computational problem: let A
be the 3×3 matrix of all 1's. (a) Find the plane partition
with at most three rows and columns associated with A as
discussed in class. (b) Find the pair (P,Q) of
tableaux obtained by applying the dual RSK algorithm to A.
- Due Tuesday, March 6. Problems 30(a), 53, 66, 67. Also do
the following two (easy, I hope) problems.
(A1) Let λ be a partition, identified with its
diagram. Show that the number of ways to add a horizontal
strip of size m to λ followed by adding a
vertical strip of size n is equal to the number of
ways of first adding a vertical strip of size n to
λ followed by adding a horizontal strip of
size m. (Minus 5 points for not finding a
sufficiently elegant solution.)
(A2) Let φ be the homomorphism from Λ to Λ
defined by φ(hi)
= hi+1 for
all i≥1. For all partitions λ such that
λn≥n (where n
is the number of parts of λ) express
φ(sλ) as a linear combination of
Hint for #30(a). Use
the proof of Theorem 7.15.1.
- March 22: no class
- Due Tuesday, March 27. Problems 29(a), 46, 63, 84(a), 90(a).
Hint for #46. Use RSK.
Hint for #90(a). At some
point use Exercise 7.82(a).
Bonus. Let φ be the
homomorphism from (A2) of the previous problem set (now
corrected). Let f(m) denote the number of
partitions λ of m for which
λ2+1, .... Find a nice product
formula for the generating function
- Due Tuesday, April 10 (note postponement). Problems
73(a,d,e,f), 94(a), 98,
108. For #73(f), you need only give the first (easy) proof. The
difficulty level of #108 is [2+]. Note Exercise 7.42
errata here. Remark
for #73(f). You can use part
(b). Another way is to use another Chapter 7
for #98. In equation (7.20) on page 299, set the
first t y-variables equal to one and the rest equal to
zero. Differentiate with respect to t, set t=1,
and consider only terms of degree n. Also consider the
Also hand in: (A3)
Let An denote the set of all pairs
(i,T), where i is an element of
[n] and T is a 2-element subset of [n]
not containing i. Thus #An =
n(n-1)(n-2)/2. The symmetric
group Sn acts on An in the
way by acting on each element separately. Find the decomposition of
the character of this action into irreducible characters.
- Due Thursday, April 12. Hand in the
(A4) Let Sn denote the set of all
sequences a1, a2,
..., a2n of n 1's and n
-1's such that every partial sum is nonnegative. Given such a
sequence α, define w(α) to be the sum of all
indices i such that ai=1 and
ai+1=-1. Find a simple product formula for
the generating function
qw(α). (Show that this result is
equivalent to a result involving Schur functions.)
- Thursday, April 19. IN-CLASS
QUIZ. The quiz will be open book (EC2 only) and class
material (returned homework, personal class notes). No outside material
such as other books, internet connection, etc. No phones or computers.