Median-Extremes Alternation

TL;DR

This paper introduces the Median-Extremes Alternation (MEA) permutation, generated by an elementary alternating process, revealing its rigid combinatorial structure, including its alternating nature, descent set, inversion number, and recursive properties.

Contribution

The paper defines a new permutation family with a simple process and proves its structural properties, including its alternating pattern and exact inversion count.

Findings

pi_n is always an alternating permutation

The descent set depends on the parity of n

Inversion number is exactly floor((n-1)^2/4)

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.