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Repeat step three until there is a new row with one more number than the previous row (do step 3 until = +) The number on the left hand side of a given row is the Bell number for that row. (,) Here are the first five rows of the triangle constructed by these rules:
The remaining positions in each row are filled by a rule very similar to that for Pascal's triangle: they are the sum of the two values to the left and upper left of the position. Thus, after the initial placement of the number 1 in the top row, it is the last position in its row and is copied to the leftmost position in the next row.
The number of alternative assignments for a given number of workers, taking into account the choices of how many stages to use and how to assign workers to each stage, is an ordered Bell number. [29] As another example, in the computer simulation of origami , the ordered Bell numbers give the number of orderings in which the creases of a crease ...
The total number of partitions of an n-element set is the Bell number B n. The first several Bell numbers are B 0 = 1, B 1 = 1, B 2 = 2, B 3 = 5, B 4 = 15, B 5 = 52, and B 6 = 203 (sequence A000110 in the OEIS). Bell numbers satisfy the recursion + = = and have the exponential generating function
This number is known as the nth Bell number. Analogously, ... For instance, the number 25 in column k = 3 and row n = 5 is given by 25 = 7 + (3×6), ...
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.
For n ≥ 0 and 0 ≤ k ≤ n, the rencontres number D n, k is the number of permutations of { 1, ..., n } that have exactly k fixed points. For example, if seven presents are given to seven different people, but only two are destined to get the right present, there are D 7, 2 = 924 ways this could happen.
There are known formulae to evaluate the prime-counting function (the number of primes smaller than a given value) faster than computing the primes. This has been used to compute that there are 1,925,320,391,606,803,968,923 primes (roughly 2 × 10 21 ) smaller than 10 23 .