# HW2, Q.3(a)

Hi all,

I’m a little confused by what the sequences $S_0,S_1,\dots,S_k$ look like. I’m thinking these are cycles (ie. permutations) with elements from $[n]$. And so then we’d be looking for the number of increasing and decreasing sequences of subsets. Any thoughts? Thanks!

### About Anastasia Chavez

I am currently a 3rd year graduate student at UC Berkeley in mathematics. My research interests are in ennumerative combinatorics. I am originally from Santa Rosa, Ca and have been a Bay Area transplant since 2003. My husband, two children, dog and 5 chickens now reside in Berkeley. Besides counting, I enjoy hiking, yoga, playing games of the imagination with my daughters, stand-up comedy and day dreaming about backpacking adventures with my husband. I plan to make those dreams, among others, a reality!

1. I think they are simply a sequence of subsets each of which increases or decreases by one element comparing to the previous subset.

2. Anastasia Chavez

Ah, so we consider subsets of \$\latex [n]\$ as usual (ie. subsets of \$\latex [3]\$ are \$\latex\{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\}\$). Thanks!

3. plus the empty set, i do believe!

4. (assuming that |{}-{1}|=1, etc. why not?)

5. Hi everyone! For n=2 and k=3 is the following sequence a “valid sequence”?:
$S_0=\emptyset, \ S_1=\{1\}, S_2=\{1,2\}, S_3=\{2\}$
If so… Does n have to be even on part (b)?

6. Sorry, I mean, does k have to be even?
By the way…
Is there a way to edit a comment?

7. yes & yes!

8. & (last question) i don’t know!

9. i mean if k is odd there are zero sequences with this property

10. Thanks!