Lobb's Generalization of Catalan's Parenthesization Problem
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 Cn 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 C2n is the sum of n + 1 squares.
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Document Type: Research Article
Publication date: 2009-03-01
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