Solution Found!
a. The sequence of Catalan numbers is defined recursively by C(0) = 1 C(1) = 1 C(n) = n
Chapter 3, Problem 38(choose chapter or problem)
a. The sequence of Catalan numbers is defined recursively by C(0) = 1 C(1) = 1 C(n) = n k=1 C(k 1)C(n k) for n 2 Compute the values of C(2), C(3), and C(4) using this recurrence relation. b. Frank and Jody are both candidates for president of the County Council. The number of votes cast equals 2n, where n votes are cast for Frank and n for Jody. Votes are counted sequentially. The ballot problem asks: In how many ways can the votes be counted so that Jodys total is never ahead of Franks total? The answer, as it turns out, is C(n), the nth Catalan number. For example, if n = 5, one possible counting sequence that meets this requirement is FFJJFJFFJJ Using n = 3, write down all the satisfactory counting sequences and compare the result to the Catalan number C(3).
Questions & Answers
QUESTION:
a. The sequence of Catalan numbers is defined recursively by C(0) = 1 C(1) = 1 C(n) = n k=1 C(k 1)C(n k) for n 2 Compute the values of C(2), C(3), and C(4) using this recurrence relation. b. Frank and Jody are both candidates for president of the County Council. The number of votes cast equals 2n, where n votes are cast for Frank and n for Jody. Votes are counted sequentially. The ballot problem asks: In how many ways can the votes be counted so that Jodys total is never ahead of Franks total? The answer, as it turns out, is C(n), the nth Catalan number. For example, if n = 5, one possible counting sequence that meets this requirement is FFJJFJFFJJ Using n = 3, write down all the satisfactory counting sequences and compare the result to the Catalan number C(3).
ANSWER:Step 1 of 3
The Catalan numbers are sequences of natural numbers that are sequences of natural numbers that occur in counting problems. It has a recursive definition.