Given a string S = a1a2a3 · · · an, the longest increasing subsequence (LIS) problem is to find a subsequence of S such that the subsequence is increasing and its length is maximal. In this paper, we propose and solve two variants of the LIS problem. The first one is the minimal height LIS where the height means the difference between the greatest and smallest elements. We propose an algorithm ...

MOTIVATION The popular BLAST algorithm is based on a local similarity search strategy, so its high-scoring segment pairs (HSPs) do not have global alignment information. When scientists use BLAST to search for a target protein or DNA sequence in a huge database like the human genome map, the existence of repeated fragments, homologues or pseudogenes in the genome often makes the BLAST result fi...

the set of all non-increasing non-negative integer sequences $pi=(d_1‎, ‎d_2,ldots,d_n)$ is denoted by $ns_n$‎. ‎a sequence $piin ns_{n}$ is said to be graphic if it is the degree sequence of a simple graph $g$ on $n$ vertices‎, ‎and such a graph $g$ is called a realization of $pi$‎. ‎the set of all graphic sequences in $ns_{n}$ is denoted by $gs_{n}$‎. ‎the complete product split graph on $l‎ ...

We prove a number of results related to problem Po-Shen Loh, which is equivalent in Ramsey theory. Let $a=(a_1,a_2,a_3)$ and $b=(b_1,b_2,b_3)$ be two triples integers. Define $a$ 2-less than $b$ if $a_i<b_i$ for at least values $i$, define sequence $a^1,\dots,a^m$ 2-increasing $a^r$ $a^s$ whenever $r<s$. Loh asks how long can all the take $\{1,2,\dots,n\}$, gives $\log_*$ improvement over trivi...

It was recently discovered by Baik, Deift and Johansson [4] that the asymptotic distribution of the length of the longest increasing subsequence in a permutation chosen uniformly at random from Sn, properly centred and normalised, is the same as the asymptotic distribution of the largest eigenvalue of an n × n GUE random matrix, properly centred and normalised, as n → ∞. This distribution had e...

In a famous paper 8] Hammersley investigated the length L n of the longest increasing subsequence of a random n-permutation. Implicit in that paper is a certain one-dimensional continuous-space interacting particle process. By studying a hydrodynamical limit for Hammersley's process we show by fairly \soft" arguments that limn ?1=2 EL n = 2. This is a known result, but previous proofs (Vershik-...

We compute the limit distribution for the (centered and scaled) length of the longest increasing subsequence of random colored permutations. The limit distribution function is a power of that for usual random permutations computed recently by Baik, Deift, and Johansson (math.CO/9810105). In the two–colored case our method provides a different proof of a similar result by Tracy and Widom about t...

Connections between longest increasing subsequences in random permutations and eigenvalues of random matrices with complex entries have been intensely studied. This note applies properties of random elements of the finite general linear group to obtain results about the longest increasing subsequence in non-uniform random permutations.