Combinatorics -> Bounded Compositions

Distribute sums under lower and upper bounds using stars and bars, generating functions, or inclusion-exclusion.

• Topic slug: combinatorics/bounded-compositions
• Tutorial page: Open tutorial
• Ladder page: Open ladder
• Repo problems currently tagged here: 1
• Repo companion pages: 0
• Curated external problems: 6

Microtopics

• stars-and-bars
• lower-bounds
• upper-bounds
• bounded-sums
• compositions
• generating-functions
• PIE

Learning Sources

Source	Type
cp-algorithms stars and bars	Reference
OI Wiki combinations	Reference
OI Wiki PIE	Reference

Practice Sources

Source	Type
CSES Distributing Apples	Practice
AtCoder DP M	Practice

Curated External Problems

Core

Problem	Source	Difficulty	Context	Style	Prerequisites	Tags	Why it fits
Coins	AtCoder	Easy	Coin-Change	Brute Force; Counting	Nested Loops; Case Analysis	Bounded Composition; Coin Sums	A tiny bounded-composition problem where all constraints are small enough to enumerate.
Distributing Apples	CSES	Easy-Medium	Stars-And-Bars, Bounded Counting	Math; Combinatorics	Combinations	Nonnegative Parts; Combinations	The cleanest unconstrained composition baseline.
M - Candies	AtCoder	Medium	Prefix DP	-	-	Upper Bounds; Children And Candies; DP	A canonical bounded-distribution DP.
D - Between Two Arrays	AtCoder	Hard	Prefix DP	-	-	Nondecreasing Arrays; Range Bounds; Mod 998244353	Counting bounded monotone sequences with DP.

Practice

Problem	Source	Difficulty	Context	Style	Prerequisites	Tags	Why it fits
How Many Ways?	AtCoder	Medium	Triple-Counting	Counting; Brute Force	Case Analysis; Sum Constraints	Triples; Sum Constraint	A small but useful constrained-composition style problem over ordered triples.
Tak and Cards	AtCoder	Medium	Subset Sum	DP; Counting	Cardinality Constraints	Sum Constraints; Card Selection	A good bridge from composition thinking to constrained subset counting.

Repo Problems

Code	Title	Fit	Difficulty	Pattern	Note	Solution
VOSFENCE	Xay hang rao	primary	hard	bounded compositions; run decomposition; gap distribution	Note	Code

Regeneration

python3 scripts/generate_problem_catalog.py