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The classical recurrence relation generalizes: the Catalan number of a Coxeter diagram is equal to the sum of the Catalan numbers of all its maximal proper sub-diagrams. [19] The Catalan numbers are a solution of a version of the Hausdorff moment problem. [20]
Analogously to Pascal's triangle, these numbers may be calculated using the recurrence relation [2] = + (). As base cases, p 1 ( 1 ) = 1 {\displaystyle p_{1}(1)=1} , and any value on the right hand side of the recurrence that would be outside the triangle can be taken as zero.
Catalan's trapezoids are a countable set of number trapezoids which generalize Catalan’s triangle. Catalan's trapezoid of order m = 1, 2, 3, ... is a number trapezoid whose entries (,) give the number of strings consisting of n X-s and k Y-s such that in every initial segment of the string the number of Y-s does not exceed the number of X-s by m or more. [6]
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.
In mathematics, a recurrence relation is an equation according to which the th term of a sequence of numbers is equal to some combination of the previous terms. Often, only previous terms of the sequence appear in the equation, for a parameter that is independent of ; this number is called the order of the relation.
An alternative method for deriving the same generating function uses the recurrence relation for the Bell numbers in terms of binomial coefficients to show that the exponential generating function satisfies the differential equation ′ = (). The function itself can be found by solving this equation. [11] [12] [13]
Knowing that some Catalan number counts the number of triangulations of a polygon is part of the excitement of what mathematics is all about. Knowing which Catalan number it is, through the mnemonic that "the n-th Catalan number counts the case of n triangles" is intended to be the frosting on the cake. I'd like to put in the count of triangles ...
The r-Stirling number of the second kind {} counts the number of partitions of a set of n objects into k non-empty disjoint subsets, such that the first r elements are in distinct subsets. [15] These numbers satisfy the recurrence relation