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.
Highlights
A random n-permutation may be generated with a deck of n cards D = {1, ..., n} as follows
We find that I and L still obey the same n2 and n scalings, but decreased and increased, respectively, by constant factors
→ ∞ the scaling rates o(n), I scales as n2/kn are and altered
Summary
One is given a choice between k ≥ 2 (uniformly) random cards to remove at each step one can bias the resulting permutation by an appropriate selection rule to achieve a particular objective. We refer to the resulting procedure for constructing our random permutation as the k-card-minimum procedure. Ct,1, ..., Ct,k are the k random card choices from the remaining set Dt, and the minimum of these, Ct, is selected for removal. With k = 1, the kCM procedure reduces to the original procedure in which a single random card is drawn at each step, and the final permutation σ is uniform.
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