Triangular array

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]
• Lozanić'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

• Weisstein, Eric W. "Number Triangle". MathWorld.