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Counting alternating permutations with restricted prefix and suffix

Ran Pan, Jeffrey Remmel

TL;DR

This work develops a systematic framework for counting alternating permutations with restricted prefixes and suffixes by leveraging Hasse diagrams and generating functions. It derives explicit formulas and exponential generating functions for prefixes of lengths 3 and 4, including patterns 231, 132, 1324, 1423, and 3412, and studies joint prefix-suffix restrictions via inclusion-exclusion. Core results include closed-form EGFs such as $f^{231}(x)=(x-1)(\sec x+\tan x)$, $f^{132}(x)=(2-x)(\sec x+\tan x)$, and various length-4 cases, along with asymptotic comparisons among patterns. The methodology generalizes to longer prefixes/suffixes and sets of patterns, providing a foundation for broader enumerative investigations of constrained alternating permutations and related statistics.

Abstract

In this paper, we use Hasse diagrams and generating functions to count alternating permutations with restricted prefix and suffix of lengths 3 and 4. In other words, for an alternating permutation $σ=σ_1σ_2σ_3\cdotsσ_{n}\in S_{n}$, we restrict length-3 prefixes $σ_1σ_2σ_3$ to follow certain patterns, such as $231$ and $132$, or follow certain restrictions such as $σ_2 \geq \max\{σ_1,σ_3\}+2$, similarly for prefixes of length 4. We also study the enumeration of alternating permutations with restrictions on both prefix and suffix.

Counting alternating permutations with restricted prefix and suffix

TL;DR

This work develops a systematic framework for counting alternating permutations with restricted prefixes and suffixes by leveraging Hasse diagrams and generating functions. It derives explicit formulas and exponential generating functions for prefixes of lengths 3 and 4, including patterns 231, 132, 1324, 1423, and 3412, and studies joint prefix-suffix restrictions via inclusion-exclusion. Core results include closed-form EGFs such as , , and various length-4 cases, along with asymptotic comparisons among patterns. The methodology generalizes to longer prefixes/suffixes and sets of patterns, providing a foundation for broader enumerative investigations of constrained alternating permutations and related statistics.

Abstract

In this paper, we use Hasse diagrams and generating functions to count alternating permutations with restricted prefix and suffix of lengths 3 and 4. In other words, for an alternating permutation , we restrict length-3 prefixes to follow certain patterns, such as and , or follow certain restrictions such as , similarly for prefixes of length 4. We also study the enumeration of alternating permutations with restrictions on both prefix and suffix.

Paper Structure

This paper contains 18 sections, 1 theorem, 57 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Prefix of length $3$
  3. Pattern $231$ as prefix
  4. Pattern $132$ as prefix
  5. $\sigma_2>\max\{\sigma_1,\sigma_3\}+1$ in prefix
  6. Prefix of length $4$
  7. Pattern $1324$ as prefix
  8. Pattern $1423$ as prefix
  9. Pattern $3412$ as prefix
  10. Brief comparison
  11. Prefix and Suffix of length $4$
  12. Pattern $1324$ as prefix and $1324$ as suffix
  13. Pattern $1423$ as prefix and $1423$ as suffix
  14. Pattern $2413$ as prefix and $2413$ as suffix
  15. Pattern $3412$ as prefix and $1324$ as suffix
  16. ...and 3 more sections

Key Result

Theorem 1

Suppose $P=p_1p_2\cdots p_{k}\in A_{k}$, $Q=q_1q_2\cdots q_{k}\in A_{k}$. Let $f^{P}$ denote the exponential generating function of $|A^P(n)|$ for $n\geq k$. If $p_{k}=q_{k}$, then $f^{P}=f^{Q}.$

Figures (16)

  • Figure 1: $A_5$ and $A_6$
  • Figure 2: alternating permutations with pattern $231$ as prefix (even-length case)
  • Figure 3: formula for $A^{231}(8)$
  • Figure 4: alternating permutations with pattern $132$ as prefix (even-length case)
  • Figure 5: formula for $A^{132}(8)$
  • ...and 11 more figures

Theorems & Definitions (1)

  • Theorem 1