Covering n-Permutations with (n+1)-Permutations

TL;DR

This paper investigates the minimal size of subsets of (n+1)-permutations needed to cover all n-permutations, analyzing bounds, probabilistic thresholds, and phase transitions in coverage.

Contribution

It provides new bounds and probabilistic analysis for covering n-permutations with (n+1)-permutations, including phase transition behavior.

Findings

Derived upper bounds on the minimal covering set size

Identified probabilistic thresholds for coverage transition

Analyzed fine-magnification of coverage probability near phase transition

Abstract

Let S_n be the set of all permutations on [n]:={1,2,....,n}. We denote by kappa_n the smallest cardinality of a subset A of S_{n+1} that "covers" S_n, in the sense that each pi in S_n may be found as an order-isomorphic subsequence of some pi' in A. What are general upper bounds on kappa_n? If we randomly select nu_n elements of S_{n+1}, when does the probability that they cover S_n transition from 0 to 1? Can we provide a fine-magnification analysis that provides the "probability of coverage" when nu_n is around the level given by the phase transition? In this paper we answer these questions and raise others.