MAT 246             University of California, Davis                 Winter 2011

Homework 1
due Jan ???, 2011 , in class

Haven't quite figured when is the most sensible due date for this. Let's say Jan 12 for now, subject to postponement (or adding of exercises). In fact, it's fine w/ me if you wait till Jan 14 to turn it in.

I'm going to list several to start on and think about. I'll pick out some subset to write up nicely and hand in. Without a reader, I'm not sure how many problems will get graded. But you should do many hw problems to learn the material, regardless of whether they are collected!
And if you've taken the time to write up more than what's required to hand in, feel free to turn that in too--especially if I may be lazy and use that as the posted solution (I'll certainly factor those extra efforts into your grade).

    Make sure your homework paper is legible, stapled, and has your name clearly showing on each page.    Also, for alphabetizing purposes, in the upper LeftHand corner of your paper please put the first letter of your last name, large and circled.  

No late homeworks will be accepted!

Collaboration with classmates is encouraged, so long as you write up your solutions in your own words. Please note on the front page of your HW who you have collaborated with.    

Read EC II, Sections 7.1, 7.2 , 7.3 , 7.4, 7.5, 7.6, 7.7
(that list may well shorten as I fine tune the HW)

1.     7.2c (p 450) OPTIONAL
2.     7.3 (p 450)
3.     7.4 (p 450)     Please WRITE this one up to hand in.
4.     7.6 (p 450)     Please (definitely) WRITE this one up to hand in.
5.     7.8 (p 451)

6.     7.12 (p 451)     (This one may get POSTPONED, depending on when in lecture we get to Kostka numbers. Or possibly the whole set will be postponed till after I've had a chance to hold an official office hour. But the rest of the problems should be do-able after Friday's lecture.)

7.     Show that dominance order ◁ on Par(n) satisfies
λ ◁ μ     iff     μT ◁ λT ,
where T denotes the transpose of the partition (denoted λ' in Stanley).
(WARNING-- had a typo here (had lambda and mu reversed in the 2nd statement. now fixed. updated jan6.) You do not need to write this one up, but it's worth doing to get a feel for dominance order.)

8.     Show that dominance order ◁ on Par(n) is the transitive closure of the relation λ → μ if μ = Rijλ for some i < j, where
Rij1, ..., λi, ..., λj, ... , λl) = (λ1, ..., λi +1, ..., λj-1, ... , λl).