Skip to main content

Combinatorics and applications

Combinatorics and applications
Problem Set 3
1. Let an be the number of ways to tile a 1 × n board with 1 × 1 squares, 1 × 2 rectangles, and 1 × 3 rectangles.
(a) Find a recurrence for an.
(b) Find a simple expression for the generating function Pn≥0 anz n as a quotient of two polynomials.
2. Consider the recurrence an = 3an−1 − 4an−3, with initial conditions a0 = 1, a1 = 2, a2 = 6.
Find a simple expression for the generating function P n≥0 anz n.
3. Let an = 5 · 3n − 2 n+1 P
for n ≥ 0. Find a simple expression for the generating function n≥0 anz n as a quotient of two polynomials.
4. Find a generating function A(z) such that the coefficient of z 100 is the number of ways to give change of a dollar (that is, 100 cents) using cents, nickels (5-cent coins), dimes (10-cent coins), and quarters (25 cent coins).
5. Find a simple expression for the generating function P n≥1 nzn as a quotient of two polynomials.

Comments

Post a Comment

Popular posts from this blog

Tracy, one of your friends, is in the early stages of conducting a feasibility analysis for a small business consulting company.

Tracy, one of your friends, is in the early stages of conducting a feasibility analysis for a small business consulting company. Understanding the target audience and the marketing mix is a mandatory step for achieving success in her new venture. Branding is equally important. In this assignment, you will act as a consultant to Tracy to find the best answers to her problems related to understanding the market and planning and executing the correct marketing strategies. Tasks: Respond to the following questions: How would you advise Tracy to approach the following topics? Market segmentation Selection of a target market Development of a positioning strategy You are familiar with the concept of branding and know that because her company is new, she will have to build a brand from scratch. What needs to be done in this case? Submission Details: By the due date assigned,post your responses to this Discussion Area. Through the end of module, respond to at least two of your classmates' p...
Post a Comment
Read more

Case 7.1, “His Team Gets the Best Assignments” (pp. 151–152)

Case 7.1, “His Team Gets the Best Assignments” (pp. 151–152) There have been many stories throughout history of leaders who have been corrupted by their power. The very definition of leadership as “a process by which an individual influences the group to achieve a common goal” (Northouse, 2016, p. 6) implies the power to exert influence. Leaders must exert power in order to lead, but they must also strive to understand the complexities of power. Research by French and Raven (1959) and by others (Burns, 1976; Neck & Manz, 1994) provides insight on the types, or bases, of power available to a leader, the process of choosing among those bases, and the motivations for leaders’ choices (Northouse, 2016, p. 379). Recent researchers (Barbuto & Warneke, 2014; Vevere, 2014) also explore the effectiveness of using individual bases of power in particular situations and the effects certain bases of powers have on leaders and those they lead. Armed with this information, leaders can avoid u...
Post a Comment
Read more

Scenario: You have been promoted to senior level manager of a company with low employee morale and decreased productivity

Scenario: You have been promoted to senior level manager of a company with low employee morale and decreased productivity. You have been tasked with developing a strategic plan for the workforce to present to the CEO and other members of the leadership team.  Chosen Organization :Kaiser Permanente Hospital Create a 10- to 12-slide PowerPoint® presentation including the following:  Explain rationale for plan Discuss SWOT of organization Summarize one year plan of action Flowchart of one year plan Give an overview of Budget for plan Evaluate the potential benefit of the plan Write conclusion Proper citation of animation or graphics is required if any is used Click Here For More Details on How to Work on this Paper...... Need a Professional Writer to Work on this Paper?  Click Here and Get this Essay Done ………
Post a Comment
Read more