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In mathematics, the Bell triangle is a triangle of numbers analogous to Pascal's triangle, whose values count partitions of a set in which a given element is the largest singleton. It is named for its close connection to the Bell numbers , [ 1 ] which may be found on both sides of the triangle, and which are in turn named after Eric Temple Bell .
The triangular array whose right-hand diagonal sequence consists of Bell numbers. The Bell numbers can easily be calculated by creating the so-called Bell triangle, also called Aitken's array or the Peirce triangle after Alexander Aitken and Charles Sanders Peirce. [6] Start with the number one. Put this on a row by itself. (, =)
This number is known as the nth Bell number. ... This relation is specified by mapping n and k coordinates onto the Sierpiński triangle. More directly, let two sets ...
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number , other examples being square numbers and cube numbers . The n th triangular number is the number of dots in the triangular arrangement with n dots on each side, and is equal to the sum of the n natural ...
Eulerian number; Floyd's triangle; 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 number; Triangular pyramidal number; The (incomplete) Bell polynomials from a triangular array of polynomials (see also Polynomial ...
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]
The total number of these faces is 1 + 14 + 36 + 24 = 75, an ordered Bell number, corresponding to the summation formula above for =. [ 17 ] By expanding each Stirling number in this formula into a sum of binomial coefficients , the formula for the ordered Bell numbers may be expanded out into a double summation.
That is, let s be the lower triangular matrix of Stirling numbers of the first kind, whose matrix elements = (,). The inverse of this ... is the n-th Bell number