3.(6.3) Let an be the number of ordered partitions of n. For example, the ordered partitions...
Show that the number of partitions of n in which each part appears either 0, 2, 5, or 7 times is the same as the number of partitions of n in which each part is either 2 mod 4, or 5 mod 10.
Problem 2: Recall that pi(n) denotes the number of integer partitions of n with exactly k parts. Show that Pk(n)an- m11 n20 Problem 2: Recall that pi(n) denotes the number of integer partitions of n with exactly k parts. Show that Pk(n)an- m11 n20
Exercise 5.6 (10 pts). Let an be the number of partitions of In] in which each block has odd size. Find the exponential generating function associated to an
QUESTION 3 Let S = {(6, 0, 3),(0,5,5),(0,1,0)} be an ordered basis of R3. Let v be a vector in R3, v=(4,7,-1) You calculate V in the basis of S. And get: (a1, a1, a3) What is the value of a3?
Problem 12.29. A basic example of a simple graph with chromatic number n is the complete graph on n vertices, that is x(Kn) n. This implies that any graph with Kn as a subgraph must have chromatic number at least n. It's a common misconception to think that, conversely, graphs with high chromatic number must contain a large complete sub- graph. In this problem we exhibit a simple example countering this misconception, namely a graph with chromatic number four that...
From 6.3-6. For n E N, let W < W<< W2nt be the order statistics of (2n 1) independent draws from Unifl-1 2ra-+1 (1) Find the PDF of W and W2n+1 (2) By symmetry or otherwise, compute EW+
5. Partitions For each n e Z, let T={(x, y) + R n<I- g < n+1}. Is T = {T, n € Z} a partition of R?? Justify your answer using the definition.
12 ) - 2. Let p(n) denote the number ofdstinct prime divisors ofn. For example, p( p(24)-2 and p(60) 3. Let q(n)an, where a is fixed and show that qn) is multiplicative, but not completely multiplicative. 12 ) - 2. Let p(n) denote the number ofdstinct prime divisors ofn. For example, p( p(24)-2 and p(60) 3. Let q(n)an, where a is fixed and show that qn) is multiplicative, but not completely multiplicative.
Example 1.9: 1.23 "The median of an ordered set is an element such that the number of elements less than the median is within one of the number that are greater, assuming no ties. a. Write an algorithm to find the median of three distinct integers a, b, and c. b. Describe D. the set of inputs for your algorithm. in light of the discussion in Sec- tion 1.4.3 following Example 1.9. c. How many comparisons does your algorithm do...
Let S = {(-6, 0, 3),(0, -7, -7),(0,2,0)} be an ordered basis of R3. Let v be a vector in R3, v=(4,7,-1) You calculate V in the basis of S. And get: (a1, a1, a3) What is the value of a3?