Related papers: GL(n,q) and Increasing Subsequences in Nonuniform …
The rate of convergence of the distribution of the length of the longest increasing subsequence, toward the maximal eigenvalue of certain matrix ensembles, is investigated. For finite-alphabet uniform and nonuniform i.i.d. sources, a rate…
We study the fluctuations, in the large deviations regime, of the longest increasing subsequence of a random i.i.d. sample on the unit square. In particular, our results yield the precise upper and lower exponential tails for the length of…
We study the length of the longest increasing and longest decreasing subsequences of random permutations drawn from the Mallows measure. Under this measure, the probability of a permutation pi in S_n is proportional to q^{inv(pi)} where q…
It has been conjectured by W. Chen that the distribution of the length of the longest increasing subsequence in a uniformly random permutation is log-concave. We propose a stronger version of this conjecture which involves the Kronecker…
The Mallows measure on the symmetric group $S_n$ is the probability measure such that each permutation has probability proportional to $q$ raised to the power of the number of inversions, where $q$ is a positive parameter and the number of…
This paper is interested in independent sets (or equivalently, cliques) in uniform random cographs. We also study their permutation analogs, namely, increasing subsequences in uniform random separable permutations. First, we prove that,…
This paper is concerned with certain connections between the ensemble of n x n unitary matrices -- specifically the characteristic function of the random variable tr(U) -- and combinatorics -- specifically Ulam's problem concerning the…
We study the longest increasing subsequence problem for random permutations avoiding the pattern $312$ and another pattern $\tau$ under the uniform probability distribution. We determine the exact and asymptotic formulas for the average…
The paper discusses progress in understanding statistical properties of complex eigenvalues (and corresponding eigenvectors) of weakly non-unitary and non-Hermitian random matrices. Ensembles of this type emerge in various physical…
We survey the theory of increasing and decreasing subsequences of permutations. Enumeration problems in this area are closely related to the RSK algorithm. The asymptotic behavior of the expected value of the length is(w) of the longest…
Let $P_{n,k}$ be the number of permutations $\pi$ on [n]={1, 2,..., n} such that the length of the longest increasing subsequences of $\pi$ equals k, and let $M_{2n, k}$ be the number of matchings on [2n] with crossing number k. Define…
The authors consider the length, $l_N$, of the length of the longest increasing subsequence of a random permutation of $N$ numbers. The main result in this paper is a proof that the distribution function for $l_N$, suitably centered and…
We introduce a new permutation statistic, namely, the number of cycles of length $q$ consisting of consecutive integers, and consider the distribution of this statistic among the permutations of $\{1,2,...,n\}$. We determine explicit…
When n is odd, consider the finite general linear and unitary groups of rank n, extended by the inverse transpose automorphism. There are elements in the extended groups which square to a regular unipotent element, and we evaluate the…
Our interest is in the cumulative probabilities Pr(L(t) \le l) for the maximum length of increasing subsequences in Poissonized ensembles of random permutations, random fixed point free involutions and reversed random fixed point free…
We investigate linearity of amalgams of subgroups of algebraic groups along intersections with algebraic subgroups. In the process, we establish linearity of certain "doubles" of linear groups, and obtain new examples of finitely generated…
Using sequences of finite length with positive integer elements and the inversion statistic on such sequences, a collection of binomial and multinomial identities are extended to their $q$-analog form via combinatorial proofs. Using the…
We study the entropy $S$ of longest increasing subsequences (LIS), i.e., the logarithm of the number of distinct LIS. We consider two ensembles of sequences, namely random permutations of integers and sequences drawn i.i.d.\ from a limited…
We investigate permutations and involutions that avoid a pattern of length three and have a {\em unique} longest increasing subsequence.
It is proved that the limiting distribution of the length of the longest weakly increasing subsequence in an inhomogeneous random word is related to the distribution function for the eigenvalues of a certain direct sum of Gaussian unitary…