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. 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[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]
- 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]
- 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]
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.[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]
Applications
The Boustrophedon transform uses a triangular array to transform one integer sequence into another.[13]
See also
- Triangular number, the number of entries in such an array up to some particular row
References
- MR 0624091.
- S2CID 18248485.
- MR 1363707.
- ISBN 9780534082444.
- The Fibonacci Quarterly, 14 (2): 173–178.
- .
- MR 2792161.
- ISBN 9780801869464.
- Bibcode:2006JIntS...9...24B.
- MR 2890929.
- ISBN 9780792313090.
- S2CID 29898282.
- S2CID 15637402.