Triangular array

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Numbers arranged in a triangle

title: "Triangular array" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["triangles-of-numbers"] description: "Numbers arranged in a triangle" topic_path: "general/triangles-of-numbers" source: "https://en.wikipedia.org/wiki/Triangular_array" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Numbers arranged in a triangle ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/a/ab/BellNumberAnimated.gif" caption="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. That is, the ith row contains only i elements.

Examples

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{{citation | last = Shallit | first = Jeffrey | authorlink = Jeffrey Shallit | editor1-first = Verner E. Jr. | editor1-last = Hoggatt | editor2-first = Marjorie | editor2-last = Bicknell-Johnson | contribution = A triangle for the Bell numbers | location = Santa Clara, Calif. | mr = 624091 | pages = 69–71 | publisher = Fibonacci Association | title = A collection of manuscripts related to the Fibonacci sequence | url = https://www.fq.math.ca/collection.html | contribution-url = http://www.fq.math.ca/Books/Collection/shallit.pdf | year = 1980}}.
  • Catalan's triangle, which counts strings of matched parentheses{{citation | title = Harmonic numbers, Catalan's triangle and mesh patterns | last1 = Kitaev | first1 = Sergey | author1-link = Sergey Kitaev | last2 = Liese | first2 = Jeffrey | journal = Discrete Mathematics | year = 2013 | volume = 313 | issue = 14 | pages = 1515–1531 | doi = 10.1016/j.disc.2013.03.017 | mr = 3047390 | arxiv = 1209.6423 | s2cid = 18248485 | url = https://personal.strath.ac.uk/sergey.kitaev/Papers/mesh1.pdf
  • Euler's triangle, which counts permutations with a given number of ascents{{citation | title = Permutations and combination locks | last1 = Velleman | first1 = Daniel J. | last2 = Call | first2 = Gregory S. | journal = Mathematics Magazine | year = 1995 | volume = 68 | issue = 4 | pages = 243–253 | doi = 10.1080/0025570X.1995.11996328 | mr = 1363707 | jstor = 2690567
  • Floyd's triangle, whose entries are all of the integers in order{{citation | title = Programming by design: a first course in structured programming | pages=211–212 | first1=Philip L. | last1=Miller | first2 = Lee W. | last2 = Miller | first3 = Purvis M. | last3=Jackson | publisher = Wadsworth Pub. Co. | year = 1987 | isbn = 978-0-534-08244-4
  • Hosoya's triangle, based on the Fibonacci numbers{{citation | title = Fibonacci triangle | last = Hosoya | first = Haruo | author-link = Haruo Hosoya | journal = The Fibonacci Quarterly | volume = 14 | issue = 2 | pages = 173–178 | year = 1976| doi = 10.1080/00150517.1976.12430575 }}.
  • Lozanić's triangle, used in the mathematics of chemical compounds{{citation | title = Die Isomerie-Arten bei den Homologen der Paraffin-Reihe | trans-title = The isomery species of the homologues of the paraffin series | last = Losanitsch | first = Sima M. | author-link = Sima Lozanić | journal = Chem. Ber. | lang = de | volume = 30 | issue = 2 | year = 1897 | pages = 1917–1926 | doi = 10.1002/cber.189703002144 | url = https://zenodo.org/record/1425862
  • Narayana triangle, counting strings of balanced parentheses with a given number of distinct nestings{{citation | title = On a generalization of the Narayana triangle | last = Barry | first = Paul | journal = Journal of Integer Sequences | issue = 4 | volume = 14 | article-number = 11.4.5 | mr = 2792161 | url = https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry4/barry142.pdf | year = 2011}}.
  • Pascal's triangle, whose entries are the binomial coefficients{{citation | title = Pascal's Arithmetical Triangle: The Story of a Mathematical Idea | first = A. W. F. | last = Edwards | author-link = A. W. F. Edwards | publisher=JHU Press | year=2002 | isbn = 978-0-8018-6946-4}}.

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.{{citation | title = On integer-sequence-based constructions of generalized Pascal triangles | last = Barry | first = Paul | journal = Journal of Integer Sequences | volume = 9 | issue = 2 | article-number = 6.2.4 | url = http://www.emis.de/journals/JIS/VOL9/Barry/barry91.pdf | year = 2006 | bibcode = 2006JIntS...9...24B

Generalizations

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.{{citation | last1 = Rota Bulò | first1 = Samuel | last2 = Hancock | first2 = Edwin R. | last3 = Aziz | first3 = Furqan | last4 = Pelillo | first4 = Marcello | doi = 10.1016/j.laa.2011.08.017 | issue = 5 | journal = Linear Algebra and Its Applications | mr = 2890929 | pages = 1436–1441 | title = Efficient computation of Ihara coefficients using the Bell polynomial recursion | volume = 436 | year = 2012| doi-access = free

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.

Applications

Romberg's method can be used to estimate the value of a definite integral by completing the values in a triangle of numbers.

The Boustrophedon transform uses a triangular array to transform one integer sequence into another.{{citation | last1 = Millar | first1 = Jessica | last2 = Sloane | first2 = N. J. A. | last3 = Young | first3 = Neal E. | arxiv = math.CO/0205218 | issue = 1 | journal = Journal of Combinatorial Theory | pages = 44–54 | series = Series A | title = A new operation on sequences: the Boustrouphedon transform | volume = 76 | year = 1996 | doi=10.1006/jcta.1996.0087| s2cid = 15637402

In general, a triangular array is used to store any table indexed by two natural numbers where j ≤ i.

Indexing

Storing a triangular array in a computer requires a mapping from the two-dimensional coordinates (i, j) to a linear memory address. If two triangular arrays of equal size are to be stored (such as in LU decomposition), they can be combined into a standard rectangular array. If there is only one array, or it must be easily appended to, the array may be stored where row i begins at the ith triangular number Ti. Just like a rectangular array, one multiplication is required to find the start of the row, but this multiplication is of two variables (i*(i+1)/2), so some optimizations such as using a sequence of shifts and adds are not available.

References

References

  1. (1991). "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.
  2. Thacher Jr., Henry C.. (July 1964). "Remark on Algorithm 60: Romberg integration". Communications of the ACM.

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