Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Median-Extremes Alternation

David Carr

Abstract

We define a deterministic family of permutations generated by an alternating center-edge extraction process on the ordered set [n] = {1,2,...,n}. Starting from the ordered list (1,2,...,n), one repeatedly removes the median element or elements of the current list, then removes its extreme elements, alternating these two operations until the list is exhausted. The resulting output is a permutation pi_n in S_n, which we call the Median-Extremes Alternation (MEA) permutation. Although the construction is elementary, the resulting permutations exhibit unexpectedly rigid combinatorial structure. We prove that pi_n is always an alternating permutation, with parity-dependent alternating type. As a consequence, its descent set is completely determined by the parity of n. We also prove an exact formula for the inversion number, inv(pi_n) = floor((n-1)^2/4), which immediately yields a characterization of the sign of pi_n. In addition, we give an exact recursive description of the family and a recursive formula for the inverse permutation.

Median-Extremes Alternation

Abstract

We define a deterministic family of permutations generated by an alternating center-edge extraction process on the ordered set [n] = {1,2,...,n}. Starting from the ordered list (1,2,...,n), one repeatedly removes the median element or elements of the current list, then removes its extreme elements, alternating these two operations until the list is exhausted. The resulting output is a permutation pi_n in S_n, which we call the Median-Extremes Alternation (MEA) permutation. Although the construction is elementary, the resulting permutations exhibit unexpectedly rigid combinatorial structure. We prove that pi_n is always an alternating permutation, with parity-dependent alternating type. As a consequence, its descent set is completely determined by the parity of n. We also prove an exact formula for the inversion number, inv(pi_n) = floor((n-1)^2/4), which immediately yields a characterization of the sign of pi_n. In addition, we give an exact recursive description of the family and a recursive formula for the inverse permutation.
Paper Structure (7 sections, 6 theorems, 60 equations)

This paper contains 7 sections, 6 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Definition of the MEA permutation
  3. Exact recursive structure
  4. Alternating structure and descent set
  5. Inversion number and sign
  6. Recursive description of the inverse permutation
  7. Conclusion

Key Result

Proposition 3.1

Let $\pi_n$ denote the MEA permutation of $[n]$. If $n=2m+1$ is odd with $m\ge 1$, then where $T_{2m+1}$ is obtained from $\pi_{2m-2}$ by replacing each entry $r$ with If $n=2m$ is even with $m\ge 2$, then where $T_{2m}$ is obtained from $\pi_{2m-4}$ by replacing each entry $r$ with Here $\|$ denotes concatenation.

Theorems & Definitions (14)

  • Remark
  • Proposition 3.1
  • proof
  • Remark
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • proof
  • Theorem 5.1
  • proof
  • ...and 4 more