is a collection of combinatorial objects.

=> and are combinatorically isomorphic

if there exist a size-preserving bijection between and

Q1

Prove that number of Dyck Paths equals to number of triangulations of a n-polygon

8.jpg

7.jpg

  1. Label all the points

  2. if is an edge, , then we draw an edge

  3. Finish by drawing an

OGF

Q2

Find the value of

take

HW

Find the value of

Q4

Q5

Prove

q5.jpg

m is an even number

formula.jpg

Q6

Prove (Euler's Identity)

HW

Prove

Thm

The OGF's for partitions

using parts in Misplaced &;

partitions with largest part ;

all partitions,are

Def

Misplaced &

a run is a maximal increasing string in the word

84135726 has 4 runs: 8, 4, 1357, 26

The Eulerian number

is the number of permutations of containing runs.

Thm (Worpitzky's Identity)

: number of functions from to

boxes, there can be nothing in a box, or some balls in a box. If there are some balls in a box, then list the balls from small to large, like:

separates, runs, so add separates, there are ways

so the answer is

Thm