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Lobb Numbers
In combinatorics, 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 Dyck language, 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 ''n''th Catalan number equals the Lobb number ''L''0,''n''. They are named after Andrew Lobb, who used them to give a simple mathematical induction, 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_ = \frac\binom \qquad\textn \ge m \ge 0. An alternative expression for Lobb number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |