Concepts6
Pólya Enumeration
Pólya Enumeration Theorem generalizes Burnside’s Lemma by turning counting under symmetry into a polynomial substitution problem.
Partition Function
The partition function p(n) counts the number of ways to write n as a sum of positive integers where order does not matter.
Stirling Numbers of First Kind
Stirling numbers of the first kind count permutations by their number of cycles and connect power polynomials to rising/falling factorials.
Binomial Theorem and Identities
The binomial theorem expands (x + y)^n into a sum of terms using binomial coefficients that count how many ways to choose k items from n.
Stars and Bars
Stars and Bars counts the ways to distribute n identical items into k distinct bins using combinations.
Polynomial Operations
Fast polynomial operations treat coefficients like numbers but use FFT/NTT to multiply in O(n \log n) time instead of O(n^2).