Gessel's theorem

Gessel’s theorem is a result in combinatorics that provides a way to count certain structured arrangements called permutations, especially those avoiding specific patterns. It connects these permutations to algebraic objects called generating functions, which encode counting information efficiently. Essentially, the theorem helps mathematicians find formulas for counting permutations with particular restrictions, revealing deep relationships between combinatorial structures and algebraic functions. This insight simplifies understanding complex counting problems and advances the study of permutation patterns and their properties.