Lobb numbers

The Lobb numbers^[1]^[2], named after Andrew Lobb, are a natural generalization of the Catalan numbers, originally introduced^[3] to give a simple inductive proof of the formula for the n^th Catalan number.

The Lobb numbers depend on the input of two non-negative integers m and n with n ≥ m ≥ 0. The (m, n)^th Lobb number L_m,n is given in terms of binomial coefficients by

$L_{m,n} = \frac{2m%2B1}{m%2Bn%2B1}{2n\choose m%2Bn} \qquad\text{ for }n \ge m \ge 0.$

For m = 0, L_0,n coincides with the n^th Catalan number. The Lobb numbers solve the combinatorial problem that asks for the number of ways in which n + m values of +1 and n − m values of −1 can be arranged so that no partial sum is negative (when m = 0 this is a restatement of Catalan's problem). Equivalently, L_m,n is the number of ways that n + m opening parentheses and n − m closing parentheses can be arranged so that the resulting string is the prefix of (or, when m = 0, the entirety of) a valid string of balanced parentheses.

Notes

^ Koshy, Thomas (March 2009). "Lobb's generalization of Catalan's parenthesization problem". The College Mathematics Journal 40 (2): 99–107. doi:10.4169/193113409X469532.
^ Koshy, Thomas (2008). Catalan Numbers with Applications. Oxford University Press. ISBN 978-0195334548.
^ Lobb, Andrew (March 1999). "Deriving the nth Catalan number". Mathematical Gazette 83 (8): 109–110.

Salmassi, Mohammad (August 6–8 2009). "A generating function for Lobb's double sequence". The Annual Meeting of the The Mathematical Association of America MathFest. Portland, Oregon, USA.