Restricted Bell number

$$B_{n \leq m}$$ is the number of partitions of an $n$ element set such that every block contains at most $m$ elements.

Cases

• General case: OEIS A229223
• Case $m=2$: OEIS A000085
• Case $m=3$: OEIS A001680
• Case $m=4$: OEIS A001681
• Case $m=5$: OEIS A110038
• Case $m=6$: OEIS A148092

Basic Recurrence

• $$B_{0 \leq m} = 1$$
• $$B_{n \leq m} = B_n \text{ for } m \geq n$$
• $$B_{n+1 \leq m}=\sum_{k=0}^{m-1} \binom{n}{k} B_{n-k \leq m}$$

Formulas

There is no known formula for the general case.

• $$B_{n \leq 2} = \sum_{j=0}^{\lfloor n/2 \rfloor} \binom{n}{2j} \frac{(2j)!}{2^j j!}$$
• $$B_{n \leq 3} = \sum_{i=0}^{\lfloor n/3 \rfloor} \sum_{j=0}^{\lfloor n/2 \rfloor} \binom{n}{3i} \frac{(3i)!}{6^i i!} \binom{n-3i}{2j} \binom{2j}{j} \frac{j!}{2^j}$$

Generating Functions

Exponential Generating Function

$$\sum_{n=0}^\infty \frac{B_{n \leq m}}{n!} x^n = \exp\left(\sum_{i=1}^{m} \frac{x^i}{i!}\right)$$

Relation to Bell and Associated Bell numbers

Where $B_n$ is a Bell number and $B_{n \geq m}$ is an Associated Bell number.

• $$B_n = \sum_{i=0}^{n}\binom{n}{i} \cdot B_{i \leq m} \cdot B_{n-i \geq m+1}$$
• $$B_{n \geq k} = B_{n} - \sum_{i=1}^{n} \binom{n}{i} B_{i \leq k-1} B_{n-i \geq k}$$
• $$B_n = B_{n \leq \infty}$$

Articles

Moll, Ramirez, Villamizar: Combinatorial and Arithmetical Properties of the Restricted and Associated Bell and Factorial Numbers