Bell polynomials

In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are used in the study of set partitions. They are related to Stirling and Bell numbers. They also occur in many applications, such as in the Faà di Bruno's formula.

The partial or incomplete exponential Bell polynomials are a triangular array of polynomials given by

where the sum is taken over all sequences j₁, j₂, j₃, ..., j_n−k+1 of non-negative integers such that these two conditions are satisfied:

The sum

is called the nth complete exponential Bell polynomial.

Likewise, the partial ordinary Bell polynomial, in contrast to the usual exponential Bell polynomial defined above, is given by

where the sum runs over all sequences j₁, j₂, j₃, ..., j_n−k+1 of non-negative integers such that

The ordinary Bell polynomials can be expressed in the terms of exponential Bell polynomials:

In general, Bell polynomial refers to the exponential Bell polynomial, unless otherwise explicitly stated.

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 is also referred to as parts or blocks, in 3 different ways:

Thus, we can encode the information regarding these partitions as

Here, the subscripts of B_3,2 tells us that we are considering the partitioning of set with 3 elements into 2 blocks. The subscript of each x_i indicates the presence of block with i elements (or block of size i) in a given partition. So here, x₂ indicates the presence of a block with two elements. Similarly, x₁ indicates the presence of a block with a single element. The exponent of x_i^j indicates that there are j such blocks of size i in a single partition. Here, since both x₁ and x₂ has exponent 1, it indicates that there is only one such block in a given partition. The coefficient of the monomial indicates how many such partitions there are. For our case, there are 3 partitions of a set with 3 elements into 2 blocks, where in each partition the elements are divided into two blocks of sizes 1 and 2.

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