Some statistics on permutations avoiding generalized patterns

In the last decade a huge amount of articles has been published studying pattern avoidance on permutations. From the point of view of enumeration, typically one tries to count permutations avoiding certain patterns according to their lengths. Here we tackle the problem of refining this enumeration by considering the statistics "first/last entry". We give complete results for every generalized patterns of type $(1,2)$ or $(2,1)$ as well as for some cases of permutations avoiding a pair of generalized patterns of the above types.Comment: 5 figure

On the inverse image of pattern classes under bubble sort

Let B be the operation of re-ordering a sequence by one pass of bubble sort. We completely answer the question of when the inverse image of a principal pattern class under B is a pattern class.Comment: 11 page

Slicings of parallelogram polyominoes: Catalan, schröder, baxter, and other sequences

We provide a new succession rule (i.e. generating tree) associated with Schröder numbers, that interpolates between the known succession rules for Catalan and Baxter numbers. We define Schröder and Baxter generalizations of parallelogram polyominoes, called slicings, which grow according to these succession rules. In passing, we also exhibit Schröder subclasses of Baxter classes, namely a Schröder subset of triples of non-intersecting lattice paths, a new Schröder subset of Baxter permutations, and a new Schröder subset of mosaic floorplans. Finally, we define two families of subclasses of Baxter slicings: the m-skinny slicings and the m-rowrestricted slicings, for m ∈ N. Using functional equations and the kernel method, their generating functions are computed in some special cases, and we conjecture that they are algebraic for any m

Graph Reconstruction via Distance Oracles

We study the problem of reconstructing a hidden graph given access to a distance oracle. We design randomized algorithms for the following problems: reconstruction of a degree bounded graph with query complexity $\tilde{O}(n^{3/2})$; reconstruction of a degree bounded outerplanar graph with query complexity $\tilde{O}(n)$; and near-optimal approximate reconstruction of a general graph

Sorption of Ni by birnessite: equilibrium controls on Ni in seawater

Synthetic hexagonal birnessite (Hx-birnessite) is a close analogue to natural poorly crystalline phyllomanganate phases found in soils and marine ferromanganese deposits. These phases are often highly enriched in trace metals such as Ni and Co. We measured the sorption of Ni(II) onto synthetic hexagonal birnessite (Hx-birnessite) from pH 1 to 7. EXAFS spectra show that, at pH 3.7, Ni is adsorbed to the Hx-birnessite surface above vacancy sites on {001} as a tridentate corner-sharing complex. We developed a surface complexation model for Ni adsorption based on the equilibria<br/><br/>3(?Mn2O? 2/3) + Ni+ 2 = (?Mn2O)3Ni0 <br/><br/><br/><br/>3(?Mn2O? 2/3) + Ni+ 2 + H2O = (?Mn2O)3Ni(OH)? + H+ <br/><br/><br/>Using this surface complexation model, we predict the concentration of Ni in seawater in equilibrium with Ni-bearing birnessite found in hydrogenetic FeMn crusts and nodules. Our predicted results are in good agreement with observed Ni concentrations in seawater and suggest that the concentration of dissolved Ni in seawater is buffered by sorption to birnessite or a related MnO2 phase. However, in addition to the surface complex, Ni also sorbs by structural incorporation into the vacancy site. In our synthetic samples at pH 7, EXAFS shows 10% of Ni is structurally incorporated into Hx-birnessite. In natural birnessites found in marine ferromanganese crusts and nodules, EXAFS shows that all of the sorbed Ni is structurally incorporated. Structural incorporation suggests that Ni sorption may be irreversible

Competitive Group Testing and Learning Hidden Vertex Covers with Minimum Adaptivity

Abstract. Suppose that we are given a set of n elements d of which are “defective”. A group test can check for any subset, called a pool, whether it contains a defective. It is well known that d defectives can be found by using O(d log n) pools. This nearly optimal number of pools can be achieved in 2 stages, where tests within a stage are done in parallel. But then d must be known in advance. Here we explore group testing strategies that use a nearly optimal number of pools and a few stages although d is not known to the searcher. One easily sees that O(log d) stages are sufficient for a strategy with O(d log n) pools. Here we prove a lower bound of Ω(log d / log log d) stages and a more general pools vs. stages tradeoff. As opposed to this, we devise a randomized strategy that finds d defectives using O(d log(n/d)) pools in 3 stages, with any desired probability 1 − ɛ. Open questions concern the optimal constant factors and practical implications. A related problem motivated by, e.g., biological network analysis is to learn hidden vertex covers of a small siz

Pattern Matching for Separable Permutations

International audienceGiven a permutation π (called the text) of size n and another permutation σ (called the pattern) of size k, the NP-complete permutation pattern matching problem asks whether σ occurs in π as an order-isomorphic subsequence. In this paper, we focus on separable permutations (those permutations that avoid both 2413 and 3142, or, equivalently, that admit a separating tree). The main contributions presented in this paper are as follows.We simplify the algorithm of Ibarra (Finding pattern matchings for permutations, Information Processing Letters 61 (1997), no. 6) to detect an occurrence of a separable permutation in a permutation and show how to reduce the space complexity from O(n3k) to O(n3logk) .In case both the text and the pattern are separable permutations, we give a more practicable O(n2k) time and O(nk) space algorithm. Furthermore, we show how to use this approach to decide in O(nk3ℓ2) time whether a separable permutation of size n is a disjoint union of two given permutations of size k and ℓ .Given a permutation of size n and a separable permutation of size k, we propose an O(n6k) time and O(n4log k) space algorithm to compute the largest common separable permutation that occurs in the two input permutations. This improves upon the existing O(n8) time algorithm by Rossin and Bouvel (The longest common pattern problem for two permutations, Pure Mathematics and Applications 17 (2006)).Finally, we give a O(n6k) time and space algorithm to detect an occurrence of a bivincular separable permutation in a permutation. (Bivincular patterns generalize classical permutations by requiring that positions and values involved in an occurrence may be forced to be adjacent)