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- W4298271772 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." @default.
- W4298271772 created "2022-10-01" @default.
- W4298271772 creator A5072571608 @default.
- W4298271772 date "2014-06-16" @default.
- W4298271772 modified "2023-09-29" @default.
- W4298271772 title "Inversions and Longest Increasing Subsequence for $k$-Card-Minimum Random Permutations" @default.
- W4298271772 doi "https://doi.org/10.48550/arxiv.1406.3911" @default.
- W4298271772 hasPublicationYear "2014" @default.
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