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
Σ
_{n≥0}*C*(*2n*,*n*)*x*= (1-4^{n}*x*)^{-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*c*^{λ}_{μν}is its definition <*s*_{λ},*s*_{μ}*s*_{ν}>.**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(*h*) =_{i}*h*_{i+1}for all*i*≥1. For all partitions λ such that λ_{n}≥*n*(where*n*is the number of parts of λ) express φ(*s*_{λ}) as a linear combination of Schur functions.

**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 φ(*s*_{λ}) = s_{λ1+1, λ2+1, ...}. Find a nice product formula for the generating function Σ_{m≥0}*f*(*m*)*x*.^{m}

- 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 exercise.**Hint 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 product*s*._{n-k}p_{k}

**Also hand in:**(A3) Let*n*≥3. Let*A*denote the set of all pairs (_{n}*i*,*T*), where*i*is an element of [*n*] and*T*is a 2-element subset of [*n*] not containing*i*. Thus #*A*=_{n}*n*(*n*-1)(*n*-2)/2. The symmetric group*S*acts on_{n}*A*in the obvious way by acting on each element separately. Find the decomposition of the character of this action into irreducible characters._{n}

- Due Thursday, April 12. Hand in the
following problem.

(A4) Let*S*denote the set of all sequences_{n}*a*,_{1}*a*, ...,_{2}*a*of_{2n}*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*a*=1 and_{i}*a*=-1. Find a simple product formula for the generating function Σ_{i+1}_{α∈Sn}*q*^{w(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.