Generation
# Stirling Numbers
There are two kinds of Stirling numbers:
## Stirling Numbers of the Second Kind: S(n,k)
**Definition:** S(n,k) counts the number of ways to partition a set of n elements into exactly k non-empty subsets.
**Recurrence relation:**
S(n,k) = k·S(n-1,k) + S(n-1,k-1)
with S(0,0) = 1 and S(n,0) = 0 for n > 0.
## Stirling Numbers of the First Kind: s(n,k)
**Definition:** The unsigned version counts the number of permutations of n elements with exactly k cycles.
**Recurrence relation:**
|s(n,k)| = (n-1)·|s(n-1,k)| + |s(n-1,k-1)|
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## Worked Example: S(4,2)
Find the number of ways to partition {1,2,3,4} into 2 non-empty subsets.
**Using the recurrence:**
- S(4,2) = 2·S(3,2) + S(3,1)
We need:
- S(3,2) = 2·S(2,2) + S(2,1) = 2·1 + 1 = 3
- S(3,1) = 1 (only one partition: all elements together)
Therefore:
- S(4,2) = 2·3 + 1 = **7**
**Verification by listing:**
The 7 partitions are:
1. {{1},{2,3,4}}
2. {{2},{1,3,4}}
3. {{3},{1,2,4}}
4. {{4},{1,2,3}}
5. {{1,2},{3,4}}
6. {{1,3},{2,4}}
7. {{1,4},{2,3}}
19 tokens in · 478 tokens out · 9961 ms · $0.0072
Trail
03:25:40 received prompt_chars=58 verifiers=['scripture_anchors', 'theology_doctrine']
03:25:40 red_gate decision=pass reason=no disqualifying input patterns detected
03:25:50 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=19 tokens_out=478 latency_ms=9960.9 cost_usd=0.0072
03:25:50 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
03:25:50 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
03:25:50 floor_gate decision=pass reason=output above protective minimum; no mismatches
03:25:50 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
03:25:50 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
03:25:50 signed hash_algo=sha256 hash=sha256:e1ca17b5be439a2115af67b433ea368d8c4ca953137ac9ea587d023f26c61e03