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+# The Bell numbers
+
+```agda
+module elementary-number-theory.bell-numbers where
+```
+
+Imports
+
+```agda
+open import elementary-number-theory.binomial-coefficients
+open import elementary-number-theory.multiplication-natural-numbers
+open import elementary-number-theory.natural-numbers
+open import elementary-number-theory.strict-inequality-natural-numbers
+open import elementary-number-theory.strong-induction-natural-numbers
+open import elementary-number-theory.sums-of-natural-numbers
+```
+
+
+
+## Idea
+
+The {{#concept "Bell numbers" Agda=bell-number-ℕ}} count the number of ways to partition a set of size $n$. The Bell numbers can be defined recursively by $B_0 := 1$ and
+
+$$
+ B_{n+1} := \sum_{k=0}^{n} \binom{n}{k}B_k.
+$$
+
+The Bell numbers are listed as sequence A000110 in the [OEIS](literature.oeis.md) {{#cite OEIS}}
+
+## Definitions
+
+### The Bell numbers
+
+```agda
+bell-number-ℕ : ℕ → ℕ
+bell-number-ℕ =
+ strong-rec-ℕ 1
+ ( λ n B →
+ bounded-sum-ℕ
+ ( succ-ℕ n)
+ ( λ k k