Inversions and Longest Increasing Subsequence for $k$-Card-Minimum Random Permutations

TL;DR

This paper studies how a biased permutation process, where at each step the smallest of k randomly chosen remaining cards is removed, affects the permutation's order, analyzing measures like inversions and longest increasing subsequence.

Contribution

It introduces a model of biased permutations based on k-card minimum selection and provides probabilistic analysis of their order properties, including laws of large numbers and limit theorems.

Findings

Weak law of large numbers for inversions

Central limit theorem for inversions

Scaling behavior of the longest increasing subsequence

Abstract

A random $n$-permutation may be generated by sequentially removing random cards $C_1,...,C_n$ from an $n$-card deck $D = \{1,...,n\}$. The permutation $\sigma$ is simply the sequence of cards in the order they are removed. This permutation is itself uniformly random, as long as each random card $C_t$ is drawn uniformly from the remaining set at time $t$. We consider, here, a variant of this simple procedure in which one is given a choice between $k$ random cards from the remaining set at each step, and selects the lowest numbered of these for removal. This induces a bias towards selecting lower numbered of the remaining cards at each step, and therefore leads to a final permutation which is more "ordered" than in the uniform case (i.e. closer to the identity permutation id $=(1,2,3,...,n)$). We quantify this effect in terms of two natural measures of order: The number of inversions $I$ and the length of the longest increasing subsequence $L$. For inversions, we establish a weak law of large numbers and central limit theorem, both for fixed and growing $k$. For the longest increasing subsequence, we establish the rate of scaling, in general, and existence of a weak law in the case of growing $k$. We also show that the minimum strategy, of selecting the minimum of the $k$ given choices at each step, is optimal for minimizing the number of inversions in the space of all online $k$-card selection rules.