Triangular array

Not to be confused with Triangular matrix.

The triangular array whose right-hand diagonal sequence consists of Bell numbers

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:

• Bell numbers (the "Bell triangle", "Aitken's array", or the "Peirce triangle")
• Bell polynomials
• Boustrophedon transform
• Eulerian number
• Floyd's triangle
• Hosoya's triangle
• Lah number
• Lozanić's triangle
• Narayana number
• Pascal's triangle
• Rencontres numbers
• Romberg's method
• Stirling numbers of the first kind
• Stirling numbers of the second kind

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.^[1]

Practical use

Apart from the representation of triangular matrices, triangular arrays are used in several algorithms. One example is the CKY parsing algorithm for context-free grammars, an example of dynamic programming.

See also

• Triangular number is the number of entries in such an array.

References

1. ^ Barry, P. (2006), "On integer-sequence-based constructions of generalized Pascal triangles", J. Integer Sequences 9 (06.2.4): 1–34, http://www.emis.ams.org/journals/JIS/VOL9/Barry/barry91.pdf .

External links

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