Lobb number

In combinatorial mathematics, the Lobb number L_m,n counts the number of ways that n + m open parentheses and n − m close parentheses can be arranged to form the start of a valid sequence of balanced parentheses.

Lobb numbers form a natural generalization of the Catalan numbers, which count the number of complete strings of balanced parentheses of a given length. Thus, the nth Catalan number equals the Lobb number L_0,n. They are named after Andrew Lobb, who used them to give a simple inductive proof of the formula for the n^th Catalan number.

The Lobb numbers are parameterized by 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 the formula

$L_{m,n}={\frac {2m+1}{m+n+1}}{\binom {2n}{m+n}}\qquad {\text{ for }}n\geq m\geq 0.$

As well as counting sequences of parentheses, the Lobb numbers also count the number of ways in which n + m copies of the value +1 and n − m copies of the value −1 may be arranged into a sequence such that all of the partial sums of the sequence are non-negative.