"The Catalan numbers are named after the French-Belgian mathematician Eugène Charles Catalan."
A sequence of natural numbers that count various combinatorial objects such as balanced parentheses, binary trees, and noncrossing partitions.
Combinatorics: The branch of mathematics that deals with counting and arranging objects.
Permutations and Combinations: The ways in which objects can be arranged or grouped.
Factorials: The product of all positive integers from 1 to a given number.
Binomial Coefficients: The coefficients of the terms in the expansion of (a + b)^n.
Pascal's Triangle: A triangular arrangement of numbers that shows the coefficients of the binomial expansion.
Recurrence Relations: A sequence where each term is defined in terms of previous terms in the sequence.
Generating Functions: A power series used to encode a sequence of numbers.
Catalan Numbers: A sequence of numbers that arises in many combinatorial problems, including counting the number of ways to arrange objects in a particular pattern, and counting the number of different ways that a sequence of parentheses can be arranged.
Catalan Triangle: A triangular arrangement of Catalan numbers that shows the number of ways to arrange objects in different patterns.
Dyck Paths: A path that starts at (0,0) and ends at (n,n) that only moves up or right and never crosses the diagonal line y = x.
Trees: A connected acyclic graph.
Parentheses Expressions: A sequence of parentheses that can be nested such that each opening parenthesis has a corresponding closing parenthesis.
Non-Crossing Partitions: A partition of a set where the subsets are non-intersecting.
Continued Fractions: A way of writing a number as a fraction of integers.
Riordan Arrays: A way of defining sequences of numbers using generating functions as rows and columns of an array.
Ordinary Catalan Numbers: The number of different ways a sequence of n pairs of parentheses can be properly nested.
Motzkin Numbers: The number of different ways a sequence of n pairs of parentheses and up to n dots can be arranged, where each dot must be paired with a left or right parenthesis.
Schröder Numbers: The number of different ways n+1 elements can be arranged in a binary tree structure, with each node having either 0, 1, or 2 children.
Eulerian Numbers of the Second Kind: The number of permutations of length n with k descents, where a descent is defined as an index i such that ai > ai+1.
Narayana Numbers: The number of different ways a convex polygon can be triangulated with non-intersecting diagonals that all pass through a fixed point on the boundary.
Riordan Numbers: A generalization of previous Catalan number types, where the total number of objects comes from two different generating functions.
Mahonian Numbers: The number of permutations of length n with k inversions of adjacent entries, where an inversion is defined as a pair of entries (i,j) such that i < j but ai > aj.
Poset Numbers: The number of acyclic orientations of a given poset with n elements, where an acyclic orientation is chosen in a directed graph such that there is no directed cycle.
Fuss-Catalan Numbers: The number of rooted trees with n+1 vertices, where each vertex has a label from 1 to n+1, such that the number of branches attached to each vertex follows a particular pattern.
Signed Catalan Numbers: The number of permutations of length 2n that have an even number of inversions, where an inversion is defined as a pair of entries (i,j) such that i < j but ai > aj.
"The nth Catalan number can be expressed directly in terms of the central binomial coefficients."
"Cn = (1/(n+1)) * (2n choose n)"
"Cn = ((2n)!)/((n+1)!(n!))"
"Cn = ∏(n+k)/k for n ≥ 0."
"The first Catalan numbers for n = 0, 1, 2, 3, ... are 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ..."
"The Catalan numbers correspond to sequence A000108 in the OEIS."
"The Catalan numbers are a part of combinatorial mathematics."
"The Catalan numbers occur in various counting problems, often involving recursively defined objects."
"The nth Catalan number represents the count of some combinatorial objects in various counting problems."
"The use of central binomial coefficients allows for a direct expression of the nth Catalan numbers."
"The Catalan numbers can be calculated using the factorial function in their formula."
"The second Catalan number is 1."
"The nth Catalan number can be expressed in terms of combinations or binomial coefficients."
"The Catalan numbers are often involved in counting problems that deal with recursively defined objects."
"The sixth Catalan number is 42."
"As n increases, the Catalan numbers grow in magnitude."
"The nth Catalan number can be calculated using a product notation equivalent to ∏(n+k)/k for n ≥ 0."
"The Catalan numbers have various applications in real-life counting problems, including those involving recursive objects."
"The Catalan numbers can be represented as a ratio of factorials in their formula."