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
- Catalan's triangle, which counts strings of parentheses in which no close parenthesis is unmatched
- Euler's triangle, which counts permutations with a given number of ascents
- Floyd's triangle, whose entries are all of the integers in order
- Hosoya's triangle, based on the Fibonacci numbers
- Lozanić's triangle, used in the mathematics of chemical compounds
- Narayana triangle, counting strings of balanced parentheses with a given number of distinct nestings
- Pascal's triangle, whose entries are the binomial coefficients
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.
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.
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.
- Triangular number, the number of entries in such an array up to some particular row
- Shallit, Jeffrey (1980), "A triangle for the Bell numbers", A collection of manuscripts related to the Fibonacci sequence (PDF), Santa Clara, Calif.: Fibonacci Association, pp. 69–71, MR 624091.
- Kitaev, Sergey; Liese, Jeffrey (2013), "Harmonic numbers, Catalan's triangle and mesh patterns", Discrete Mathematics, 313 (14): 1515–1531, MR 3047390, doi:10.1016/j.disc.2013.03.017.
- Velleman, Daniel J.; Call, Gregory S. (1995), "Permutations and combination locks", Mathematics Magazine, 68 (4): 243–253, MR 1363707, doi:10.2307/2690567.
- Miller, Philip L.; Miller, Lee W.; Jackson, Purvis M. (1987), Programming by design: a first course in structured programming, Wadsworth Pub. Co., pp. 211–212, ISBN 9780534082444.
- Hosoya, Haruo (1976), "Fibonacci triangle", The Fibonacci Quarterly, 14 (2): 173–178.
- Losanitsch, S. M. (1897), "Die Isomerie-Arten bei den Homologen der Paraffin-Reihe", Chem. Ber., 30: 1917–1926, doi:10.1002/cber.189703002144.
- Barry, Paul (2011), "On a generalization of the Narayana triangle", Journal of Integer Sequences, 14 (4): Article 11.4.5, 22, MR 2792161.
- Edwards, A. W. F. (2002), Pascal's Arithmetical Triangle: The Story of a Mathematical Idea, JHU Press, ISBN 9780801869464.
- Barry, P. (2006), "On integer-sequence-based constructions of generalized Pascal triangles" (PDF), Journal of Integer Sequences, 9 (06.2.4): 1–34.
- Rota Bulò, Samuel; Hancock, Edwin R.; Aziz, Furqan; Pelillo, Marcello (2012), "Efficient computation of Ihara coefficients using the Bell polynomial recursion", Linear Algebra and its Applications, 436 (5): 1436–1441, MR 2890929, doi:10.1016/j.laa.2011.08.017.
- Fielder, Daniel C.; Alford, Cecil O. (1991), "Pascal's triangle: Top gun or just one of the gang?", in Bergum, Gerald E.; Philippou, Andreas N.; Horadam, A. F., Applications of Fibonacci Numbers (Proceedings of The Fourth International Conference on Fibonacci Numbers and Their Applications, Wake Forest University, N.C., U.S.A., July 30–August 3, 1990), Springer, pp. 77–90, ISBN 9780792313090.
- Indurkhya, Nitin; Damerau, Fred J., eds. (2010), Handbook of Natural Language Processing, Second Edition, CRC Press, p. 65, ISBN 9781420085938.
- Thacher, Jr., Henry C. (July 1964), "Remark on Algorithm 60: Romberg integration", Communications of the ACM, 7 (7): 420–421, doi:10.1145/364520.364542.
- Millar, Jessica; Sloane, N. J. A.; Young, Neal E. (1996), "A new operation on sequences: the Boustrouphedon transform", Journal of Combinatorial Theory, Series A, 76 (1): 44–54, arXiv: , doi:10.1006/jcta.1996.0087.