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The exponential Bell polynomial encodes the information related to the ways a set can be partitioned. For example, if we consider a set {A, B, C}, it can be partitioned into two non-empty, non-overlapping subsets, which are also referred to as parts or blocks, in 3 different ways:
The Bell numbers are named after Eric Temple Bell, who wrote about them in 1938, following up a 1934 paper in which he studied the Bell polynomials. [27] [28] Bell did not claim to have discovered these numbers; in his 1938 paper, he wrote that the Bell numbers "have been frequently investigated" and "have been rediscovered many times". Bell ...
Touchard polynomials. The Touchard polynomials, studied by Jacques Touchard (1939), also called the exponential polynomials or Bell polynomials, comprise a polynomial sequence of binomial type defined by. where is a Stirling number of the second kind, i.e., the number of partitions of a set of size n into k disjoint non-empty subsets. [1][2][3][4]
Permutations, n! {\displaystyle n!} ! n n! {\displaystyle {\frac {!n} {n!}}} In combinatorial mathematics, a derangement is a permutation of the elements of a set in which no element appears in its original position. In other words, a derangement is a permutation that has no fixed points.
In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of n objects into k non-empty subsets and is denoted by or . [1] Stirling numbers of the second kind occur in the field of mathematics called combinatorics and the study of partitions.
Triangular array. The triangular array whose right-hand diagonal sequence consists of Bell numbers. 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 i th row contains only i elements.
A common property of all three kinds is that they describe coefficients relating three different sequences of polynomials that frequently arise in combinatorics. Moreover, all three can be defined as the number of partitions of n elements into k non-empty subsets, where each subset is endowed with a certain kind of order (no order, cyclical, or ...
The figure shows the 13 weak orderings on three elements. Starting from , the ordered Bell numbers are. 1, 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 102247563, ... (sequence A000670 in the OEIS). When the elements to be ordered are unlabeled (only the number of elements in each tied set matters, not their identities) what remains is a ...
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