**ERIC Number:**EJ862218

**Record Type:**Journal

**Publication Date:**2009-Mar

**Pages:**9

**Abstractor:**As Provided

**Reference Count:**0

**ISBN:**N/A

**ISSN:**ISSN-0746-8342

Lobb's Generalization of Catalan's Parenthesization Problem

Koshy, Thomas

College Mathematics Journal, v40 n2 p99-107 Mar 2009

A. Lobb discovered an interesting generalization of Catalan's parenthesization problem, namely: Find the number L(n, m) of arrangements of n + m positive ones and n - m negative ones such that every partial sum is nonnegative, where 0 = m = n. This article uses Lobb's formula, L(n, m) = (2m + 1)/(n + m + 1) C(2n, n + m), where C is the usual binomial coefficient, to prove that L(n, m) is odd for all m if and only if either n = 0 or n is a Mersenne number. It follows that L(n, m) and the Catalan number C[subscript n] have the same parity. We also show that L(n, m) = C(2n, n - m) - C(2n, n - m - 1), so every Lobb number can be read from Pascal's triangle. In addition to other interesting combinatorial identities, we establish that every Catalan number C[subscript 2n] is the sum of n + 1 squares.

Descriptors: Geometric Concepts, Generalization, Problem Solving, Mathematics Instruction, College Mathematics, Numbers, Mathematical Formulas

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**Publication Type:**Journal Articles; Reports - Descriptive

**Education Level:**Higher Education

**Audience:**N/A

**Language:**English

**Sponsor:**N/A

**Authoring Institution:**N/A