Show that the number of triangulations of a regular n-gon is the same as the number...

Show that the number of triangulations of a regular n-gon is the same as the number of Catalan paths from (0,0) to (n−2, n−2).

A Catalan path is defined as the following:

we want to count the number of distinct paths from the point (0,0) to the point (n, n) subject to the following rules:

•We must stay inside the box [0, n]×[0, n].

•We move one step at a time, either moving one unit East or one unit North.

•We cannot visit the same point twice.

•The path must always stay at or below the line y = x.

Expert Solution

Colby Messinger answered 2 years ago

Use the pumping lemma to show that the following languages are not regular. A a. A1={0^n...

Use the pumping lemma to show that the following languages are not regular. A a. A1={0^n 1^n 2^n | n≥0} b. A2 = {www | w ∈ {a,b}∗} A c. A3 ={a^2^n | n≥0} (Here, a^2^n means a string of 2^n a’s.) A ={a3n |n > 0 }

For all integers n > 2, show that the number of integer partitions of n in...

For all integers n > 2, show that the number of integer partitions of n in which each part is greater than one is given by p(n)-p(n-1), where p(n) is the number of integer partitions of n.

show that the power set of N and R have the same cardinality

Two vessels contain the same number N of molecules of the same perfect gas. Initially the...

Two vessels contain the same number N of molecules of the same perfect gas. Initially the two vessels are isolated from each other, the gases being at the same temperature T but at different pressures P1 and P2. The partition separating the two gases is removed. Find the change of entropy of the system when equilibrium has been re-established, in terms of initial pressures P1 and P2. Show that this entropy change is non-negative.

Show that for any square-free integer n > 1, √ n is an irrational number

Let n ≥ 3. Show that if G is a graph with the same chromatic polynomial...

Let n ≥ 3. Show that if G is a graph with the same chromatic polynomial as Cn, then G is isomorphic to Cn. (Hint: Use induction. What kind of graph must G − e be for any edge e? Why?)

Show that if S is bounded above and below, then there exists a number N >...

Show that if S is bounded above and below, then there exists a number N > 0 for which - N < or equal to x < or equal to N if x is in S

Show that two m×n matrices are equivalent if and only if they have the same invariant...

Show that two m×n matrices are equivalent if and only if they have the same invariant factors, i.e. (by Problem 4), if and only if they have the same Smith normal form.

1. Given an array of integers a dimension n. If the array contains the same number...

1. Given an array of integers a dimension n. If the array contains the same number of even and odd elements get (a1 + an) (a2 + an-1) ... 2. Given an array of integers dimension n. All array elements with even numbers preceding the first element to the maximum, multiplied by the maximum. 3. Given an array of dimension n. Insert after each zero element of the element in the middle (or the amount of secondary elements for even...

By constructing a suitable bijection, show that the number of subsets of an n-set of odd...

By constructing a suitable bijection, show that the number of subsets of an n-set of odd size is equal to the number of subsets of an n-set of even size.

Question