Bell numbers, named after Eric Temple Bell who was not the first to study them.

\begin{align} B_0 & = 1 \\ B_1 & = 1 \\ B_2 & = 2 \\ B_3 & = 5 \\ B_4 & = 15 \\ B_5 & = 52 \\ B_6 & = 203 \\ & \,\,\,\vdots \end{align}

The Bell number $B_n$ is the number of partitions of a set of $n$ members.

It is also the $n$th moment of the Poisson distribution with expected value $1.$

In 1964 Gian-Carlo Rota wrote about these and called them exponential numbers.¹ At some time after that they came to be conventionally called Bell numbers.

In that paper Rota never mentions the connection with the Poisson distribution, although he did in later writings. He defines a linear functional $L$ on the space of polynomials in $u$ by saying $L((u)_n) = 1$ for all $n,$ where $(u)_n$ is the $n$th-degree falling factorial $u(u-1)(u-2)\cdots(u-n+1)$ and states a lemma: $$ L(u p(u-1)) = L(p(u)) $$ for every polynomial $p(u).$ Those who have studied empirical Bayes methods in statistics will recognize that as a form of the Robbins lemma: $$ \text{If } X\sim\operatorname{Poisson}(\lambda) \text{ then } \operatorname E(Xf(X-1)) = \lambda\operatorname E(f(X)). $$ (The function $f$ need not be a polynomial.) This lemma was discovered by Herbert Robbins in the 1950s as a result of analyzing data on car insurance.

(1) Rota, Gian-Carlo (1964). "The number of partitions of a set". American Mathematical Monthly. 71 (5): 498–504.

doi:10.2307/2312585.

JSTOR 2312585.

MR 0161805.

Bell numbers, named after Eric Temple Bell who was not the first to study them.

\begin{align} B_0 & = 1 \\ B_1 & = 1 \\ B_2 & = 2 \\ B_3 & = 5 \\ B_4 & = 15 \\ B_5 & = 52 \\ B_6 & = 203 \\ & \,\,\,\vdots \end{align}

The Bell number $B_n$ is the number of partitions of a set of $n$ members.

It is also the $n$th moment of the Poisson distribution with expected value $1.$

In 1964 Gian-Carlo Rota wrote about these and called them exponential numbers.¹ At some time after that they came to be conventionally called Bell numbers.

In that paper Rota never mentions the connection with the Poisson distribution, although he did in later writings. He defines a linear functional $L$ on the space of polynomials in $u$ by saying $L((u)_n) = 1$ for all $n,$ where $(u)_n$ is the $n$th-degree falling factorial $u(u-1)(u-2)\cdots(u-n+1)$ and states a lemma: $$ L(u p(u-1)) = L(p(u)) $$ for every polynomial $p(u).$ Those who have studied empirical Bayes methods in statistics will recognize that as a form of the Robbins lemma: \begin{align} & \text{If } X\sim\operatorname{Poisson}(\lambda) \\[6pt] & \text{then } \operatorname E(Xf(X-1)) = \lambda\operatorname E(f(X)). \end{align} (The function $f$ need not be a polynomial.) This lemma was discovered by Herbert Robbins in the 1950s as a result of analyzing data on car insurance.

(1) Rota, Gian-Carlo (1964). "The number of partitions of a set". American Mathematical Monthly. 71 (5): 498–504.

doi:10.2307/2312585.

JSTOR 2312585.

MR 0161805.

Bell numbers, named after Eric Temple Bell who was not the first to study them.

\begin{align} B_0 & = 1 \\ B_1 & = 1 \\ B_2 & = 2 \\ B_3 & = 5 \\ B_4 & = 15 \\ B_5 & = 52 \\ B_6 & = 203 \\ & \,\,\,\vdots \end{align}

The Bell number $B_n$ is the number of partitions of a set of $n$ members.

It is also the $n$th moment of the Poisson distribution with expected value $1.$

In 1964 Gian-Carlo Rota wrote about these and called them exponential numbers.¹ At some time after that they came to be conventionally called Bell numbers.

In that paper Rota never mentions the connection with the Poisson distribution, although he did in later writings. He defines a linear functional $L$ on the space of polynomials in $u$ by saying $L((u)_n) = 1$ for all $n,$ where $(u)_n$ is the $n$th-degree falling factorial $u(u-1)(u-2)\cdots(u-n+1)$ and states a lemma: $$ L(u p(u-1)) = L(p(u)) $$ for every polynomial $p(u).$ Those who have studied empirical Bayes methods in statistics will recognize that as a form of the Robbins lemma: $$ \text{If } X\sim\operatorname{Poisson}(\lambda) \text{ then } \operatorname E(Xf(X-1)) = \lambda\operatorname E(f(X)). $$ This lemma was discovered by Herbert Robbins in the 1950s as a result of analyzing data on car insurance.

(1) Rota, Gian-Carlo (1964). "The number of partitions of a set". American Mathematical Monthly. 71 (5): 498–504.

doi:10.2307/2312585.

JSTOR 2312585.

MR 0161805.

Bell numbers, named after Eric Temple Bell who was not the first to study them.

\begin{align} B_0 & = 1 \\ B_1 & = 1 \\ B_2 & = 2 \\ B_3 & = 5 \\ B_4 & = 15 \\ B_5 & = 52 \\ B_6 & = 203 \\ & \,\,\,\vdots \end{align}

The Bell number $B_n$ is the number of partitions of a set of $n$ members.

It is also the $n$th moment of the Poisson distribution with expected value $1.$

In 1964 Gian-Carlo Rota wrote about these and called them exponential numbers.¹ At some time after that they came to be conventionally called Bell numbers.

In that paper Rota never mentions the connection with the Poisson distribution, although he did in later writings. He defines a linear functional $L$ on the space of polynomials in $u$ by saying $L((u)_n) = 1$ for all $n,$ where $(u)_n$ is the $n$th-degree falling factorial $u(u-1)(u-2)\cdots(u-n+1)$ and states a lemma: $$ L(u p(u-1)) = L(p(u)) $$ for every polynomial $p(u).$ Those who have studied empirical Bayes methods in statistics will recognize that as a form of the Robbins lemma: $$ \text{If } X\sim\operatorname{Poisson}(\lambda) \text{ then } \operatorname E(Xf(X-1)) = \lambda\operatorname E(f(X)). $$ (The function $f$ need not be a polynomial.) This lemma was discovered by Herbert Robbins in the 1950s as a result of analyzing data on car insurance.

(1) Rota, Gian-Carlo (1964). "The number of partitions of a set". American Mathematical Monthly. 71 (5): 498–504.

doi:10.2307/2312585.

JSTOR 2312585.

MR 0161805.