Alapfogalmak
The authors develop methods to count pattern occurrences in permutations, extending known results for patterns of length 3 and exploring new insights for patterns of length 4.
Kivonat
The content explores the counting of pattern occurrences in permutations, focusing on patterns of different lengths. It discusses Wilf classes, generating functions, and asymptotic behaviors for various patterns. The authors present data and conjectures for different Wilf classes and provide insights into the distribution of pattern occurrences.
Statisztikák
For length-3 PAPs: sn(1234) ∼ 81√3·9n/16π·n^4.
For length-4 PAPs: sn(1324) ∼ C · µ^n · µ√n1 · n^g.
Generating function for r=1 in class I: Ψ1(x) = 1/2x^3(1 - 6x + 9x^2 - 2x^3).
Generating function for r=2 in class II: Ψ2(x) = 1/2x^5(1 - 8x + 20x^2 - 17x^3 + 7x^4 - 5x^5).
Coefficients for ψr(n) are provided for various Wilf classes up to r=6.