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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.
Mathematical Association of America. 1529 Eighteenth Street NW, Washington, DC 20036. Tel: 800-741-9415; Tel: 202-387-5200; Fax: 202-387-1208; e-mail: maahq@maa.org; Web site: http://www.maa.org/pubs/cmj.html
Publication Type: Journal Articles; Reports - Descriptive
Education Level: Higher Education
Audience: N/A
Language: English
Sponsor: N/A
Authoring Institution: N/A