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The book Enumerative Combinatorics: Volume 2 by combinatorialist Richard P. Stanley contains a set of exercises which describe 66 different interpretations of the Catalan numbers. Following are some examples, with illustrations of the cases C 3 = 5 and C 4 = 14 .
Richard Peter Stanley (born June 23, 1944) is an Emeritus Professor of Mathematics at the Massachusetts Institute of Technology, and an Arts and Sciences Distinguished Scholar at the University of Miami. From 2000 to 2010, he was the Norman Levinson Professor of Applied Mathematics. [1]
Substituting k = 1 into this formula gives the Catalan numbers and substituting k = 2 into this formula gives the Schröder–Hipparchus numbers. [7] In connection with the property of Schröder–Hipparchus numbers of counting faces of an associahedron, the number of vertices of the associahedron is given by the Catalan numbers.
Similarly, the Schröder numbers count the number of ways to divide a rectangle into + smaller rectangles using cuts through points given inside the rectangle in general position, each cut intersecting one of the points and dividing only a single rectangle in two (i.e., the number of structurally-different guillotine partitions).
3 out of 4638576 [1] or out of 580717, [2] if rotations and reflections are not counted as distinct, Hamiltonian cycles on a square grid graph 8х8. Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed.
The following algebraic identity involving the Catalan numbers = + (), is apparently due to Touchard (according to Richard P. Stanley, who mentions it in his panorama article "Exercises on Catalan and Related Numbers" giving an overwhelming plenitude of different definitions for the Catalan numbers).
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The growth rate (or Stanley–Wilf limit) of a permutation class is defined as , where a n denotes the number of permutations of length n in the class. Clearly not every positive real number can be a growth rate of a permutation class, regardless of whether it is defined by a single forbidden pattern or a set of forbidden patterns.