Lobb's Generalization of Catalan's Parenthesization Problem

Autor/in	Koshy, Thomas
Titel	Lobb's Generalization of Catalan's Parenthesization Problem
Quelle	In: College Mathematics Journal, 40 (2009) 2, S.99-107 (9 Seiten)Infoseite zur Zeitschrift PDF als Volltext Verfügbarkeit
Sprache	englisch
Dokumenttyp	gedruckt; online; Zeitschriftenaufsatz
ISSN	0746-8342
Schlagwörter	Geometric Concepts; Generalization; Problem Solving; Mathematics Instruction; College Mathematics; Numbers; Mathematical Formulas + Suchen Sie Ihr Suchwort? Elementare Geometrie; Problemlösen; Mathematics lessons; Mathematikunterricht; Zahlenraum; Mathematische Formel
Abstract	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. (As Provided).
Anmerkungen	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
Erfasst von	ERIC (Education Resources Information Center), Washington, DC
Update	2017/4/10