Add integer partition algorithm using dynamic programming

[AliAlimohammadi]

Description

Adds an implementation of the integer partition algorithm using dynamic programming.

The partition function calculates the number of ways to express a positive integer as a sum of positive integers (order doesn't matter). For example, the number 5 can be partitioned in 7 ways:

• 5
• 4 + 1
• 3 + 2
• 3 + 1 + 1
• 2 + 2 + 1
• 2 + 1 + 1 + 1
• 1 + 1 + 1 + 1 + 1

Algorithm Overview

The implementation uses dynamic programming with a 2D memoization table where:

• memo[n][k] represents the number of partitions of n into at least k parts
• Base case: memo[i][0] = 1 (one way to partition into 0 parts)
• Recurrence relation: memo[n][k] = memo[n][k-1] + memo[n-k-1][k]

This approach is based on the mathematical principle that:

The number of partitions of n into at least k parts equals the number of partitions into exactly k parts plus the number of partitions into at least k-1 parts.

Changes Made

• Added src/dynamic_programming/integer_partition.rs
• Added module declaration in src/dynamic_programming/mod.rs
• Added public export in src/dynamic_programming/mod.rs

Testing

All tests pass:

cargo test integer_partition

Test cases include:

• ✅ partition(5) → 7
• ✅ partition(7) → 15
• ✅ partition(100) → 190,569,292
• ✅ partition(1000) → 24,061,467,864,032,622,473,692,149,727,991
• ✅ Invalid inputs (0, negative) → panic with assertion error

References

• Partition (number theory) - Wikipedia
• Partition function - Wikipedia

Checklist

• Code follows repository style guidelines
• All tests pass
• Documentation includes examples
• Function is publicly exported
• No compiler warnings

[codecov-commenter]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 95.77%. Comparing base (b77770e) to head (a509064).

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #976      +/-   ##
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+ Coverage   95.75%   95.77%   +0.01%     
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  Files         350      351       +1     
  Lines       22892    22934      +42     
==========================================
+ Hits        21920    21964      +44     
+ Misses        972      970       -2     

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[AliAlimohammadi]

@siriak, this is ready to be merged. Clippy error is a CI issue. No problem on other machines.