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Counting occurrences of patterns in permutations

Andrew R Conway, Anthony J Guttmann

Abstract

We develop a new, powerful method for counting elements in a multiset. As a first application, we use this algorithm to study the number of occurrences of patterns in a permutation. For patterns of length 3 there are two Wilf classes, and the general behaviour of these is reasonably well-known. We slightly extend some of the known results in that case, and exhaustively study the case of patterns of length 4, about which there is little previous knowledge. For such patterns, there are seven Wilf classes, and based on extensive enumerations and careful series analysis, we have conjectured the asymptotic behaviour for all classes.

Counting occurrences of patterns in permutations

Abstract

We develop a new, powerful method for counting elements in a multiset. As a first application, we use this algorithm to study the number of occurrences of patterns in a permutation. For patterns of length 3 there are two Wilf classes, and the general behaviour of these is reasonably well-known. We slightly extend some of the known results in that case, and exhaustively study the case of patterns of length 4, about which there is little previous knowledge. For such patterns, there are seven Wilf classes, and based on extensive enumerations and careful series analysis, we have conjectured the asymptotic behaviour for all classes.
Paper Structure (20 sections, 59 equations, 22 figures, 1 table)

This paper contains 20 sections, 59 equations, 22 figures, 1 table.

Table of Contents

  1. Introduction
  2. Generating the number of occurrences of a given pattern
  3. Set construction algorithm
  4. Set representation
  5. Multiset representation
  6. Cardinality algorithm of a MBDD
  7. Software availability
  8. Counting occurrences of 132, 231, 312, 213
  9. Counting occurrences of 123 and 321
  10. Length 4 patterns
  11. Class I
  12. Class II
  13. Class III
  14. Class IV
  15. Class V
  16. ...and 5 more sections

Figures (22)

  • Figure 1: Histogram of normalised 132 occurrences.
  • Figure 2: Histogram of normalised 123 occurrences.
  • Figure 3: Three different representations of a BDD of two variables.
  • Figure 4: Three different representations of a MBDD of two variables, similar to figure \ref{['fig:BDD_example']}.
  • Figure 5: Ratio of $\psi_1(n)/\psi_0(n)$ plotted against $1/n$ for 1234 patterns.
  • ...and 17 more figures