A longest increasing subsequence problem (LIS) is a well-known combinatorial problem with applications mainly in bioinformatics, where it is used in various projects on DNA sequences. Recently, a number of generalisations of this problem were proposed. One of them is to find an LIS among all fixed-size windows of the input sequence (LISW). We propose an algorithm for the LISW problem based on c...

ABSTRACT. We study the shape of the Young diagram λ associated via the Robinson–Schensted– Knuth algorithm to a random permutation in Sn such that the length of the longest decreasing subsequence is not bigger than a fixed number d; in other words we study the restriction of the Plancherel measure to Young diagrams with at most d rows. We prove that in the limit n → ∞ the rows of λ behave like ...

Given a sequence π1π2 . . . πn, a longest increasing subsequence (LIS) in a window π〈l, r〉= πlπl+1 . . . πr is a longest subsequence σ = πi1πi2 . . . πiT such that l ≤ i1 < i2 < · · · < iT ≤ r and πi1 < πi2 < · · · < πiT . We consider the Lisw problem, which is to find the longest increasing subsequences in a sliding window of fixed-size w over a sequence. Formally, it is to find a LIS for ever...

We survey three recent breakthroughs in algebraic combinatorics. The first is the proof by Knutson and Tao, and later Derksen and Weyman, of the saturation conjecture for Littlewood-Richardson coefficients. The second is the proof of the n! and (n + 1)n−1 conjectures by Haiman. The final breakthrough is the determination by Baik, Deift, and Johansson of the limiting behavior of the length of th...

Let u(d, n) denote the number of permuations in the symmetric group Sn with no increasing subsequence of length greater than d. u(d, n) may alternatively be interpreted as the number of closed Z-lattice walks which begin and end at the origin and take n positive steps followed by n negative steps while remaining confined to the Weyl chamber W = {(t1, t2, . . . , td) ∈ R : t1 ≥ t2 ≥ · · · ≥ td}....

Karlin and Altschul in their statistical analysis for multiple highscoring segments in molecular sequences introduced a distribution function which gives the probability there are at least r distinct and consistently ordered segment pairs all with score at least x. For long sequences this distribution can be expressed in terms of the distribution of the length of the longest increasing subseque...

The Mallows measure on the symmetric group Sn 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 inversions of π is equal to the number of pairs i < j such that πi > πj . We prove a weak law of large numbers for the length of the longest increasing subsequence ...