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The number on the left hand side of a given row is the Bell number for that row. ( B i ← x i , 1 {\displaystyle B_{i}\leftarrow x_{i,1}} ) Here are the first five rows of the triangle constructed by these rules:
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]
Construction of the Bell triangle. The Bell numbers may also be computed using the Bell triangle in which the first value in each row is copied from the end of the previous row, and subsequent values are computed by adding two numbers, the number to the left and the number to the above left of the position. The Bell numbers are repeated along ...
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.
Since the Stirling number {} counts set partitions of an n-element set into k parts, the sum = = {} over all values of k is the total number of partitions of a set with n members. This number is known as the nth Bell number.
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series.Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series.
I tried to edit with a tabular display providing n on the first row and B n on the second row while keeping the numbers from 0 to 19, but @Deacon Vorbis: undid the change as the table was too wide. I'd see three options: truncate the sequence to the first 10 or so Bell numbers to reduce the table width
The codimension of a face gives the number of equivalence classes in the corresponding weak ordering. [16] In this geometric representation the partial cube of moves on weak orderings is the graph describing the covering relation of the face lattice of the permutohedron.