Associated Bell number

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

Basic Recurrence

• $$B_{0 \geq m} = 1$$
• $$B_{n \geq m} = 0 \text{ for } m > n > 0$$
• $$B_{n \geq k} = B_{n \geq k-1} - \sum_{i=1}^{\lfloor \frac{n}{k-1} \rfloor} \frac{n!}{(k-1)!i!(n-(k-1)i)!} B_{n-(k-1)i \geq k}$$

Generating Function

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

Relation to Bell and restricted Bell numbers

Where $B_n$ is a Bell number and $B_{n \leq m}$ is a restricted Bell number

• $$B_n = \sum_{i=0}^{n}\binom{n}{i} \cdot B_{i\leq m} \cdot B_{n-i\geq m+1}$$
• $$B_{n \geq k} = B_{n} - \sum_{i=1}^{n} \binom{n}{i} B_{i \leq k-1} B_{n-i \geq k}$$
• $$\sum_{i=0}^{n} \binom{n}{i} B_{i \geq 2} = B_{n}$$
• $$B_n = B_{n \geq 1}$$

Articles

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