Triangular array

Not to be confused with Triangular matrix.

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.

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

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 matched parentheses^[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]

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

Romberg's method can be used to estimate the value of a definite integral by completing the values in a triangle of numbers.^[12]

The Boustrophedon transform uses a triangular array to transform one integer sequence into another.^[13]

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 T_i. 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.

See also

• Triangular number, the number of entries in such an array up to some particular row