We investigate combinatorial enumeration problems related to subsequences of strings; in contrast to substrings, subsequences need not be contiguous. For a finite alphabet Σ, the following three problems are solved. (1) Number of distinct subsequences: Given a sequence s ∈ Σ and a nonnegative integer k ≤ n, how many distinct subsequences of length k does s contain? A previous result by Chase st...

Inspired by the results of Stanley and Widom concerning the limiting distribution of the lengths of longest alternating subsequences in random permutations, and results of Deutsch, Hildebrand and Wilf on the limiting distribution of the longest increasing subsequence for pattern-restricted permutations, we find the limiting distribution of the longest alternating subsequence for pattern-restric...

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 ...

We present a number of results relating partial Cauchy-Littlewood sums, integrals over the compact classical groups, and increasing subsequences of permutations. These include: integral formulae for the distribution of the longest increasing subsequence of a random involution with constrained number of fixed points; new formulae for partial Cauchy-Littlewood sums, as well as new proofs of old f...

In this paper, we present algorithms and lower bounds for the Longest Increasing Subsequence (LIS) and Longest Common Subsequence (LCS) problems in the data streaming model. For the problem of deciding whether the LIS of a given stream of integers drawn from {1, . . . ,m} has length at least k, we discuss a one-pass streaming algorithm using O(k log m) space, with update time either O(log k) or...

A rollercoaster is a sequence of real numbers for which every maximal contiguous subsequence, that is increasing or decreasing, has length at least three. By translating this sequence to a set of points in the plane, a rollercoaster can be defined as a polygonal path for which every maximal sub-path, with positiveor negative-slope edges, has at least three points. Given a sequence of distinct r...

The longest increasing subsequence of a random walk with mean zero and finite variance is known to be n1/2+o(1). We show that this is not universal for symmetric random walks. In particular, the symmetric fat-tailed random walk has a longest increasing subsequence that is asymptotically at least n0.690 and at most n0.815. An exponent strictly greater than 1/2 is also shown for the symmetric sta...

integrating natural resources management in the secondary school curriculum in kenya has received a lot of talk without adequate practical activities. the farmers of the future programme under the world agroforestry centre, formally the international centre for research in agroforestry (icraf), initiated a practical approach to integration of natural resources management in the secondary school...