Triangular Array

In mathematics and computing, a **triangular array** of numbers, polynomials, or the like, is a doubly indexed sequence in which each row is only as long as the row's own index.

Notable particular examples include these:

- The Bell triangle, whose numbers count the partitions of a set in which a given element is the largest singleton
^{[1]} - Catalan's triangle, which counts strings of parentheses in which no close parenthesis is unmatched
^{[2]} - Euler's triangle, which counts permutations with a given number of ascents
^{[3]} - Floyd's triangle, whose entries are all of the integers in order
^{[4]} - Hosoya's triangle, based on the Fibonacci numbers
^{[5]} - Lozani?'s triangle, used in the mathematics of chemical compounds
^{[6]} - Narayana triangle, counting strings of balanced parentheses with a given number of distinct nestings
^{[7]} - Pascal's triangle, whose entries are the binomial coefficients
^{[8]}

Triangular arrays of integers in which each row is symmetric and begins and ends with 1 are sometimes called **generalized Pascal triangles**; examples include Pascal's triangle, the Narayana numbers, and the triangle of Eulerian numbers.^{[9]}

Triangular arrays may list mathematical values other than numbers; for instance the Bell polynomials form a triangular array in which each array entry is a polynomial.^{[10]}

Arrays in which the length of each row grows as a linear function of the row number (rather than being equal to the row number) have also been considered.^{[11]}

Apart from the representation of triangular matrices, triangular arrays are used in several algorithms. One example is the CYK algorithm for parsing context-free grammars, an example of dynamic programming.^{[12]}

Romberg's method can be used to estimate the value of a definite integral by completing the values in a triangle of numbers.^{[13]}

The Boustrophedon transform uses a triangular array to transform one integer sequence into another.^{[14]}

- Triangular number, the number of entries in such an array up to some particular row

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