arXiv subjects

Ross G. Pinsky

Publications and source records attributed to Ross G. Pinsky.

At least 19 recordsLinked to original sources

arXiv2026

A new look at some aspects of one-dimensional random sequential adsorption and its continuum limit

Fix a positive integer $k\ge2$, and for $n\ge k$, consider a row of $n$ molecules. From among the $n-k+1$ nearest-neighbor $k$-tuples of molecules, select one uniformly at random and bond the $k$ molecules. Now, from all the remaining nearest-neighbor $k$-tuples, again select one uniformly at random and bond the $k$ molecules. Continue like this until there are no nearest-neighbor $k$-tuples left. Let $M^{(n)}_k$ denote the expected value of the number of bonded molecules. An explicit integral formula for $m_k:=\lim_{n\to\infty}\frac{M^{(n)}_k}n$ is known, and an explicit formula for $m_\infty:=\lim_{k\to\infty}m_k$ is known. The constant $m_\infty$, known as the R\'enyi parking constant, arises as the limiting packing density for a continuous analog of the above discrete packing problems. These are all models of what is called random sequential adsorption (RSA). The first part of this paper studies the gaps of sizes $0,1,\cdots, k-1$ that arise between bonded $k$-tuples and shows that after scaling the $k$-grid, when $k\to\infty$ the empirical distribution of expected gaps in the discrete problem on the lattice converges weakly to an appropriate gap distribution that is known to hold for the above noted continuous analog. The second part of the this paper considers two different models of the discrete bonding problem when both $k_1$-bonding and $k_2$-bonding occur, with $2\le k_1<k_2$. Explicit formulas are obtained for the analogs of $m_k$, and the asymptotic behavior of these analogs is studied both when $k_2\to\infty$ with $k_1$ fixed, and when $k_1,k_2\to\infty$ at certain ratios.

math.PR↗

arXiv2026

A secretary for Messrs. Luce and Mallows

We analyze the secretary problem in the case that the $n$ ranked items arrive not in uniformly random order but rather according to a certain type of Luce distribution or according to a Mallows distribution on the set $S_n$ of permutations of $[n]$. The secretary problem for the Mallows distributions with parameter $q\in(0,1)$ was analyzed in a previous paper; in this paper the case $q>1$ is also analyzed. The Luce distribution with the class $\{q_j\}_{j=1}^\infty$ of weights is related in a certain sense to the Mallows distribution with parameter $q$, but is more difficult to analyze. It turns out that for every $n$ and every strategy, the probabilities for the secretary problem when the smallest number is considered of highest rank for the Luce distribution with this class of weights coincides with those for the corresponding Mallows distribution. We analyze the asymptotic optimal strategy and corresponding limiting probability for the above cases, as well as for the Luce distributions with other classes of weights, such as the Sukhatme weights.

math.PR↗

arXiv2025

Inversions in Random Permutations Under the Ewens Sampling Distribution With and Without a Prescribed Number of Fixed Points

In the first part of the paper, we study the inversion statistic of random permutations under the family $(\mathbb{P}_\theta^{(n)})_{\theta \ge 0}$ of Ewens sampling distributions on $S_n$. We obtain a rather simple exact formula for the expected number of inversions under $\mathbb{P}_\theta^{(n)}$. In particular, we show that this expected number of inversions is decreasing in the tilting parameter $\theta$ for any $n$ and that it is convex in $\theta$ for $n \not \in \{3,4\}$ only. Furthermore, we derive an exact formula for the probability that a specific pair of indices $(i,j) \in \{1,\dots,n\}^2$ is inverted and show that this probability is decreasing in $\theta$ if and only if $|j-i| \ge 2$ holds. We also exhibit the asymptotic behavior of these quantities as $n \to \infty$ and $\theta \to \infty$. In the second part of our paper, we analyze the inversion statistic of random permutations under~$(\mathbb{P}_\theta^{(n)})_{\theta > 0}$ conditioned on having a prescribed number of fixed points. Again, we obtain exact formulas for the expected number of inversions and for the probability that a specific pair of indices is inverted. Since, as expected, the resulting formulas are rather complicated, we focus on the asymptotic behavior of these quantities as $n \to \infty$, $\theta \to \infty$ and $\theta \to 0$.

math.PR↗

arXiv2025

A family of non-uniform distributions on the set of parking functions generated by random permutations

We introduce a rather natural family of non-uniform distributions on $PF_n$, $n\in\mathbb{N}$, the set of parking functions of length $n$. One of the motivations for this comes from a similar situation in the context of integer partitions. For a permutation $\sigma\in S_n$ and for $j\in[n]$, let $I_{n, 0$, the above map along with the distribution $P_n^{(q)}\times P_n$ induces an exchangeable distribution $\mathcal{P}_n^{(q)}$ on $PF_n$. We study the asymptotic behavior of two fundamental statistics of parking functions under the family of distributions $\mathcal{P}_n^{(q)}$.

math.PR↗

arXiv2025

The inversion statistic in derangements and in other permutations with a prescribed number of fixed points

We study how the inversion statistic is influenced by fixed points in a permutation. %The expected number of inversions in a uniformly random permutation in $S_n$ is $\frac{n(n-1)}4$. For each $n\in\mathbb{N}$, and each $k\in\{0,1,\cdots, n\}$, let $P_n^{(k)}$ denote the uniform probability measure on the set of permutations in $S_n$ with exactly $k$ fixed points. We obtain an exact formula for the expected number of inversions under the measure $P_n^{(k)}$ as well as for $P_n^{(k)}(\sigma^{-1}_i<\sigma^{-1}_j)$, for $1\le i<j\le n$, the $P_n^{(k)}$-probability that the number $i$ precedes the number $j$. In particular, up to a super-exponentially small correction as $n\to\infty$, the expected number of inversions in a random derangement $(k=0)$ is $\frac16n+\frac1{12}$ more than the value $\frac{n(n-1)}4$ that one obtains for a uniformly random general permutation in $S_n$. On the other hand, up to a super-exponentially small correction, for $k\ge2$, the expected number of inversions in a random permutation with $k$ fixed points is $\frac{k-1}6n+\frac{k^2-k-1}{12}$ less than $\frac{n(n-1)}4$. In the borderline case, $k=1$, up to a super-exponentially small correction, the expected number of inversions in a random permutation with one fixed point is $\frac1{12}$ more than $\frac{n(n-1)}4$. The proofs make strategic and perhaps novel use of the Chinese restaurant construction for a uniformly random permutation.

math.PR↗

arXiv2024

Longest subsequence for certain repeated up/down patterns in random permutations avoiding a pattern of length three

Let $S_n$ denote the set of permutations of $[n]$ and let $\sigma=\sigma_1\cdots\sigma_n\in S_n$. For a subsequence $\{\sigma_{i_j}\}_{j=1}^k$ of $\{\sigma_i\}_{i=1}^n$ of length $k\ge2$, construct the ``up/down'' sequence $V_1\cdots V_{k-1}$ defined by $$ V_j=\begin{cases} U,\ \text{if}\ \sigma_{i_j+1}-\sigma_{i_j}>0;\\ D,\ \text{if}\ \sigma_{i_j+1}-\sigma_{i_j}<0.\end{cases} $$ Consider now a fixed up/down pattern: $V_1\cdots V_l$, where $l\in\mathbb{N}$ and $V_j\in\{U, D\},\ j\in[l]$. Given a permutation $\sigma\in S_n$, consider the length of the longest subsequence of $\sigma$ that repeats this pattern. For example, consider $l=3$ and $V_1V_2V_3=UUD$. Then for the permutation $342617985\in S_9$, the length of the longest subsequence that repeats the pattern $UUD$ is 7; it is obtained by 3461798 and 3461785. The above framework includes two well-known cases. The pattern $U$ is the celebrated case of the longest increasing subsequence. The pattern $UD$ (or $DU$) is the case of the longest alternating subsequence. These have been studied both under the uniform distribution on $S_n$ as well as under the uniform distribution on those permutations in $S_n$ which avoid a particular pattern of length three. In this paper, we consider the patterns $UUD$ and $UUUD$ under the uniform distribution on those permutations in $S_n$ which avoid the pattern $132$. We prove that the expected value of the longest increasing subsequence following the pattern $UUD$ is asymptotic to $\frac37n$ and the expected value of the longest increasing subsequence following the pattern $UUUD$ is asymptotic to $\frac4{11}n$. (For $UD$ (alternating subsequences) it is known to be $\frac12n$.) This leads directly to appropriate corresponding results for permutations avoiding any particular pattern of length three.

math.CO↗

arXiv2024

The distribution on permutations induced by a random parking function

A parking function on $[n]$ creates a permutation in $S_n$ via the order in which the $n$ cars appear in the $n$ parking spaces. Placing the uniform probability measure on the set of parking functions on $[n]$ induces a probability measure on $S_n$. We initiate a study of some properties of this distribution. Let $P_n^{\text{park}}$ denote this distribution on $S_n$ and let $P_n$ denote the uniform distribution on $S_n$. In particular, we obtain an explicit formula for $P_n^{\text{park}}(σ)$ for all $σ\in S_n$. Then we show that for all but an asymptotically $P_n$-negligible set of permutations, one has $P_n^{\text{park}}(σ)\in\left(\frac{(2-ε)^n}{(n+1)^{n-1}},\frac{(2+ε)^n}{(n+1)^{n-1}}\right)$. However, this accounts for only an exponentially small part of the $P_n^{\text{park}}$-probability. We also obtain an explicit formula for $P_n^{\text{park}}(σ^{-1}_{n-j+1}=i_1,σ^{-1}_{n-j+2}=i_2,\cdots, σ^{-1}_n=i_j)$, the probability that the last $j$ cars park in positions $i_1,\cdots, i_j$ respectively, and show that the $j$-dimensional random vector $(n+1-σ^{-1}_{n-j+l}, n+1-σ^{-1}_{n-j+2},\cdots, n+1-σ^{-1}_{n})$ under $P_n^{\text{park}}$ converges in distribution to a random vector $(\sum_{r=1}^jX_r,\sum_{r=2}^j X_r,\cdots, X_{j-1}+X_j,X_j)$, where $\{X_r\}_{r=1}^j$ are IID with the Borel distribution. We then show that in fact for $j_n=o(n^\frac16)$, the final $j_n$ cars will park in increasing order with probability approaching 1 as $n\to\infty$. We also obtain an explicit formula for the expected value of the left-to-right maximum statistic $X_n^{\text{LR-max}}$, which counts the total number of left-to-right maxima in a permutation, and show that $E_n^{\text{park}}X_n^{\text{LR-max}}$ grows approximately on the order $n^\frac12$.

math.PR↗

arXiv2023

Mean and variance of the longest alternating subsequence in a random separable permutation

A permutation is \it separable \rm if it can be obtained from the singleton permutation by iterating direct sums and skew sums. Equivalently, it is separable if and only it avoids the patterns 2413 and 3142. Under the uniform probability on separable permutations of $[n]$, let the random variable $A_n$ denote the length of the longest alternating subsequence. Also, let $A_n^{+,-}$ denote the length of the longest alternating subsequence that begins with an ascent and ends with a descent, and define $A_n^{-,+}, A_n^{+,+}, A_n^{-,-}$ similarly. By symmetry, the first two and the last two of these latter four random variables are equi-distributed. We prove that the expected value of any of these five random variables behaves asymptotically as $(2-\sqrt2)n\approx0.5858\thinspace n$. We also prove that the variance of any of the four random variables $A_n^{\pm,\pm}$ behaves asymptotically as $\frac{16-11\sqrt2}2n\approx0.2218\thinspace n$.

math.PR↗

arXiv2023

Large deviations for the longest alternating and the longest increasing subsequence in a random permutation avoiding a pattern of length three

We calculate the large deviations for the length of the longest alternating subsequence and for the length of the longest increasing subsequence in a uniformly random permutation that avoids a pattern of length three. We treat all six patterns in the case of alternating subsequences. In the case of increasing subsequences, we treat two of the three patterns for which a classical large deviations result is possible. The same rate function appears in all six cases for alternating subsequences. This rate function is in fact the rate function for the large deviations of the sum of IID symmetric Bernoulli random variables. The same rate function appears in the two cases we treat for increasing subsequences. This rate function is twice the rate function for alternating subsequences.

math.PR↗

arXiv2023

The secretary problem with items arriving according to a random permutation avoiding a pattern of length three

In the classical secretary problem, $n$ ranked items arrive one by one, and each item's rank relative to its predecessors is noted. The observer must select or reject each item as it arrives, with the object of selecting the item of highest rank. For $M_n\in\{0,1,\cdots, n-1\}$, let $\mathcal{S}(n,M_n)$ denote the strategy whereby the observer rejects the first $M_n$ items, and then selects the first later-arriving item whose rank is higher than that of any of the first $M_n$ items (if such an item exists). If the ranked items arrive in a uniformly random order, it is well-known that the limiting optimal probability of success is $\frac1e$, which occurs if $M_n\sim\frac ne$. It has been shown that when the ranked items arrive according to certain non-uniform distributions on the set of permutations, $\frac1e$ serves as a lower bound for the optimal probability. There is a fundamental reason for this phenomenon. We consider certain distributions for which that reason does not apply. We begin by noting a cooked-up class of distributions for which $\mathcal{S}(n,M)$ yields the lowest possible probability of success -- namely $\frac1n$, for all $M$. We then consider the uniform distribution over all permutations avoiding a particular pattern of length three. In the case of the pattern 231 or 132, for any choice of $M_n$, the strategy $\mathcal{S}(n,M_n)$ yields the very same probability of success; namely $\frac{n+1}{2(2n-1)}$, which gives a limiting probability of $\frac14$. For the pattern 213, the optimal strategy is obtained for $M\in\{0,1\}$, also yielding a limiting probability of $\frac14$. For the pattern 123, the optimal strategy is obtained for $M=1$, yielding a limiting probability of $\frac34$. For the other two patterns, 312 and 321, an optimal strategy will yield a limiting probability of at least $\frac7{16}$.

math.PR↗

arXiv2023

Comparison of Brownian jump and Brownian bridge resetting in search for Gaussian target on the line and in space

For $d\ge1$ and $r>0$, let $X^{(d;r)}(\cdot)$ be a $d$-dimensional Brownian motion with diffusion coefficient $D$, equipped with an exponential clock with rate $r$. When the clock rings, the process jumps to the origin and begins anew. For a parameter $T>0$, let $X^{\text{bb},d;T}(\cdot)$ be the process that performs a $d$-dimensional Brownian bridge with diffusion coefficient $D$ and bridge interval $T$, and then at time $T$ starts anew from the origin, and let $X^{d;T}$ be the process that performs a $d$-dimensional Brownian motion with diffusion coefficient $D$ up until time $T$, at which time it jumps to the origin and begins anew. Denote expectations by $E_0^{d;r},E_0^{\text{bb},d;T}$ and $E_0^{d;T}$. These Markov processes with resetting search for a random target $a\in\mathbb{R}^d$ with centered Gaussian distribution of variance $σ^2$, denoted by $μ_{σ^2}^{\text{Gauss},d}$. Fix $ε_0>0$. Let $τ_a$ be the hitting time of $a$, for $d=1$, and the hitting time of the $ε_0$-ball around $a$, for $d\ge2$. The expected time to locate the target for each of the processes is $\int_{\mathbb{R}^d}\big(E_0^*τ_a\big)μ_{σ^2}^{\text{Gauss},d}(da)$, where $E_0^*$ stands for $E_0^{d;r}, E_0^{\text{bb},d;T}$ or $E_0^{d;T}$. For $d=1$ and $d=3$, we calculate the infimum of each of the above expressions over $r>0$ or $T>0$ as appropriate, in order to compare the relative efficiencies of the three search processes. In terms of the parameters $D$ and $σ$, in the 1-dimensional case these infima scale as $\frac{σ^2}D$, which is a natural scaling, but in the 3-dimensional case, they scale anomalously as $\frac{σ^3}D$. We also show that in the 2-dimensional case, the infimum over $r>0$ for the first of the three search processes scales as $\frac{σ^2}D$ as in the 1-dimensional case.

math.PR↗

arXiv2023

The secretary problem with non-uniform arrivals via a left-to-right-minimum exponentially tilted distribution

We solve the secretary problem in the case that the ranked items arrive in a statistically biased order rather than in uniformly random order. The bias is given by the left-to-right-minimum exponentially tilted distribution with parameter $q\in(0,\infty)$. That is, for $σ\in S_n$, $P_n(σ)$ is proportional to $q^{\text{LR}^{-}_n(σ)}$, where the left-to-right minimum statistic $\text{LR}^-_n$ is defined by $$ \text{LR}^{-}_n(σ)=|\{j\in[n]: σ_j=\min\{σ_i:1\le i\le j\}\}|,\ σ\in S_n. $$ For $q\in(0,1)$, higher ranked items tend to arrive earlier than in the case of the uniform distribution, and for $q\in(1,\infty)$, they tend to arrive later. In the classical problem, the asymptotically optimal strategy is to reject the first $M_n^*$ items, where $M_n^*\sim\frac ne$, and then to select the first item ranked higher than any of the first $M_n^*$ items (if such an item exists). This yields $e^{-1}$ as the limiting probability of success. With the above bias on arrivals, we calculate the asymptotic behavior of the optimal strategy $M_n^*$ and the corresponding limiting probability of success, for all regimes of $\{q_n\}_{n=1}^\infty$. In particular, if the leading order asymptotic behavior of $\{q_n\}_{n=1}^\infty$ is at least $\frac1{\log n}$, and if also its order is no more than $o(n)$, then the limiting probability of success when using an asymptotically optimal strategy is $e^{-1}$; otherwise, this limiting probability of success is greater than $e^{-1}$. Also, the limiting fraction of numbers, $\lim_{n\to\infty}\frac{M^*_n}n$, that are summarily rejected by an asymptotically optimal strategy lies in $(0,1)$ if and only if $\lim_{n\to\infty}q_n\in(0,\infty)$.

math.PR↗

arXiv2023

Clustering of consecutive numbers in permutations avoiding a pattern of length three or avoiding a finite number of simple patterns

For $η\in S_3$, let $S_n^{\text{av}(η)}$ denote the set of permutations in $S_n$ that avoid the pattern $η$, and let $E_n^{\text{av}(η)}$ denote the expectation with respect to the uniform probability measure on $S_n^{\text{av}(η)}$. For $n\ge k\ge2$ and $τ\in S_k^{\text{av}(η)}$, let $N_n^{(k)}(σ)$ denote the number of occurrences of $k$ consecutive numbers appearing in $k$ consecutive positions in $σ\in S_n^{\text{av}(η)}$, and let $N_n^{(k;τ)}(σ)$ denote the number of such occurrences for which the order of the appearance of the $k$ numbers is the pattern $τ$. We obtain explicit formulas for $E_n^{\text{av}(η)}N_n^{(k;τ)}$ and $E_n^{\text{av}(η)}N_n^{(k)}$, for all $2\le k\le n$, all $η\in S_3$ and all $τ\in S_k^{\text{av}(η)}$. These exact formulas then yield asymptotic formulas as $n\to\infty$ with $k$ fixed, and as $n\to\infty$ with $k=k_n\to\infty$. We also obtain analogous results for $S_n^{\text{av}(η_1,\cdots,η_r)}$, the subset of $S_n$ consisting of permutations avoiding the patterns $\{τ_i\}_{i=1}^r$, where $τ_i\in S_{m_i}$, in the case that $\{τ_i\}_{i=1}^n$ are all simple permutations. A particular case of this is the set of separable permutations, which corresponds to $r=2$, $τ_1=2413,τ_2=3142$.

math.PR↗

arXiv2022

Large time probability of failure in diffusive search with resetting in arbitrary dimension--a functional analytic approach

We consider a stochastic search model with resetting for an unknown stationary target $a\in\mathbb{R}^d,\ d\ge1$, with known distribution $μ$. The searcher begins at the origin and performs Brownian motion with diffusion coefficient $D$. The searcher is also armed with an exponential clock with rate $r>0$, so that if it has failed to locate the target by the time the clock rings, then its position is reset to the origin and it continues its search anew from there. In dimension one, the target is considered located when the process hits the point $a$, while in dimensions two and higher, one chooses an $ε_0>0$ and the target is considered located when the process hits the $ε_0$-ball centered at $a$. Denote the position of the searcher at time $t$ by $X(t)$, let $τ_a$ denote the time that a target at $a$ is located, and let $P^{d;(r,0)}_0$ denote probabilities for the process starting from 0. Taking a functional analytic point of view, and using the generator of the Markovian search process and its adjoint, we obtain precise estimates, uniformly in $a$, on the asymptotic behavior of $P^{d;(r,0)}_0(τ_a>t)$ for large time, and then use this to obtain large time estimates on $\int_{\mathbb{R}^d}P^{d;(r,0)}_0(τ_a>t)dμ(a)$, the probability that the searcher has failed up to time $t$ to locate the random target, distributed according to $μ$.

math.PR↗

arXiv2022

Clustering of consecutive numbers in permutations avoiding a pattern and in separable permutations

Let $S_n$ denote the set of permutations of $[n]:=\{1,\cdots, n\}$, and denote a permutation $σ\in S_n$ by $σ=σ_1σ_2\cdots σ_n$. For $l\ge2$ an integer, let $A^{(n)}_{l;k}\subset S_n$ denote the event that the set of $l$ consecutive numbers $\{k, k+1,\cdots, k+l-1\}$ appears in a set of consecutive positions: $\{k,k+1,\cdots, k+l-1\}=\{σ_a,σ_{a+1},\cdots, σ_{a+l-1}\}$, for some $a$. For $τ\in S_m$, let $S_n(τ)$ denote the set of $τ$-avoiding permutations in $S_n$, and let $P_n^{\text{av}(τ)}$ denote the uniform probability measure on $S_n(τ)$. Also, let $S_n^{\text{sep}}$ denote the set of separable permutations in $S_n$, and let $P_n^{\text{sep}}$ denote the uniform probability measure on $S_n^{\text{sep}}$. We investigate the quantities $P_n^{\text{av}(τ)}(A^{(n)}_{l;k})$ and $P_n^{\text{sep}}(A^{(n)}_{l;k})$ for fixed $n$, and the limiting behavior as $n\to\infty$. We also consider the asymptotic properties of this limiting behavior as $l\to\infty$.

math.CO↗

arXiv2022

Two measures of efficiency for the secretary problem with multiple items at each rank

For $2\le k\in\mathbb{N}$, consider the following adaptation of the classical secretary problem. There are $k$ items at each of $n$ linearly ordered ranks. The $kn$ items are revealed, one item at a time, in a uniformly random order, to an observer whose objective is to select an item of highest rank. At each stage the observer only knows the relative ranks of the items that have arrived thus far, and must either select the current item, in which case the process terminates, or reject it and continue to the next item. For $M\in\{0,1,\cdots, kn-1\}$, let $\mathcal{S}(n,k;M)$ denote the strategy whereby one allows the first $M$ items to pass, and then selects the first later arriving item whose rank is \it either equal to or greater than\rm\ the highest rank of the first $M$ items (if such an item exists). Let $W_{\mathcal{S}(n,k;M)}$ denote the event that one selects an item of highest rank using strategy $\mathcal{S}(n,k;M)$ and let $P_{n,k}(W_{\mathcal{S}(n,k;M)})$ denote the corresponding probability. We obtain a formula for $P_{n,k}(W_{\mathcal{S}(n,k;M)})$, and for $\lim_{n\to\infty}P_{n,k}(W_{\mathcal{S}(n,k;M_n)})$, when $M_n\sim ckn$, with $c\in(0,1)$. In the classical secretary problem, the asymptotically optimal strategy yields a probability of success of $\frac1e\approx 0.368$. For $k=2$, the asymptotically optimal strategy yields yields a probability of success of about 0.701. For $k=3$, the optimal probability is above 0.85, for $k=7$, that probability exceeds 0.99, and for $k\ge12$, it is 1.000 to three decimal places. In the problem with multiple items at each rank, there is an additional measure of efficiency of a strategy besides the probability of selecting an item of highest rank; namely how quickly one selects an item of highest rank. We give a rather complete picture of this efficiency.

math.PR↗

arXiv2022

Exact formula and asymptotic behavior for the expected number of inversions in a random permutation avoiding a pattern of length three

For $τ\in S_3$, let $S_n(τ)$ denote the set of permutations in $S_n$ which avoid the pattern $τ$, and let $E_n^τ$ denote the expectation with respect to the uniformly random probability measure on $S_n(τ)$. Let $\mathcal{I}_n(σ)$ denote the number of inversions in $σ\in S_n$. We study $E_n^τ\mathcal{I}_n$ for $τ\in\{231,132,213,312\}\subset S_3$. We prove that $$ E_n^{231}\mathcal{I}_n=E_n^{312}\mathcal{I}_n=\frac12\frac{n!(n+1)!4^n}{(2n)!}-\frac12(3n+1), $$ and that $$ E_n^{132}\mathcal{I}_n=E_n^{213}\mathcal{I}_n=\frac12(n-1)n-E_n^{231}\mathcal{I}_n. $$ From the first equation it follows that $$ E_n^{231}\mathcal{I}_n=E_n^{312}\mathcal{I}_n\sim\frac{\sqrtπ}2n^\frac32. $$ We also show that the variance $\text{Var}_{P_n^τ}(\mathcal{I}_n)$ of $\mathcal{I}_n$ under $P_n^τ$ satisfies $$ \text{Var}_{P_n^τ}(\mathcal{I}_n)\sim (\frac56-\frac\pi4)n^3\approx 0.048n^3,\ \text{for}\ τ\in\{231,132,213,312\}. $$

math.PR↗

arXiv2021

The secretary problem with biased arrival order via a Mallows distribution

We solve the secretary problem in the case that the ranked items arrive in a statistically biased order rather than in uniformly random order. The bias is given by a Mallows distribution with parameter $q\in(0,1)$, so that higher ranked items tend to arrive later and lower ranked items tend to arrive sooner. In the classical problem, the asymptotically optimal strategy is to reject the first $M_n^*$ items, where $M_n^*\sim\frac ne$, and then to select the first item ranked higher than any of the first $M_n^*$ items (if such an item exists). This yields $\frac1e$ as the limiting probability of success. The Mallows distribution with parameter $q=1$ is the uniform distribution. For the regime $q_n=1-\frac cn$, with $c>0$, the case of weak bias, the optimal strategy occurs with $M_n^*\sim n\Big(\frac1c\log\big(1+\frac{e^c-1}e\big)\Big)$, with the limiting probability of success being $\frac1e$. For the regime $q_n=1-\frac c{n^α}$, with $c>0$ and $α\in(0,1)$, the case of moderate bias, the optimal strategy occurs with $n-M_n\sim\frac{n^α}c$, with the limiting probability of success being $\frac1e$. For fixed $q\in(0,1)$, the case of strong bias, the optimal strategy occurs with $M_n^*=n-L$ where $\frac{L-1}L \frac1e$.

math.PR↗