Although research findings have emphasized the importance of students being engaged in dialogue and co-construction of science knowledge, implementing the teaching required to promote such engagement is challenging for teachers. Co-construction is crucial not only for students' understanding of science content but also for their familiarity with some of the attributes associated with the nature of science (NOS). Inquiry-based science classroom discourses could facilitate NOS familiarization processes by promoting creativity and collaboration in upper primary school. This article presents findings from a case study of how one teacher created and navigated opportunities for students’ co-construction of science stories in a grade 6 class (age 12) working on the topic “Substances around us”. To examine the classroom dialogue and the teacher’s enablement of students’ co-construction in class, observation data were gathered over seven weeks and analysed using principles indicative of classroom dialogue. Findings show that while opportunities for co-construction were created in science class through for example open-ended questions, the teacher’s use of IRE, complemented with boosted communication for evaluation of student answers, often hampered dialogue and co-construction of novel science stories regardless of how the teaching was organized.
We study the enumeration of inversion sequences that avoid the pattern 021 and another pattern of length four. We determine the generating trees for all possible pattern pairs and compute the corresponding generating functions. We introduce the concept of dregular generating trees and conjecture that for any 021-avoiding pattern τ , the generating tree T ({021, τ }) is d-regular for some integer d.
Abstract In this study, we first obtain a new identity for generalized fractional integrals which contains some parameters. Then by this equality, we establish some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals. Moreover, we show that the results proved in the main section reduce to several Simpson-, trapezoid- and midpoint-type inequalities for various values of parameters.
Given a tree embedded in a disk, we define two lattices - the oriented flip graph of noncrossing arcs and the lattice of noncrossing tree partitions. When the interior vertices of the tree have degree 3, the oriented flip graph is equivalent to the oriented exchange graph of a type A cluster algebra. Our main result is an isomorphism between the shard intersection order of the oriented flip graph and the lattice of noncrossing tree partitions. As a consequence, we deduce a simple characterization of c-matrices of type A cluster algebras.
We prove that the product of a monomial and a Demazure atom is a positive sum of Demazure atoms combinatorially. This result proves one particular case in a conjecture which provides an approach to a combinatorial proof of Schubert positivity property.
Recently, Kenyon and Wilson introduced Dyck tilings, which are certain tilings of the region between two Dyck paths. The enumeration of Dyck tilings is related with hook formulas for forests and the combinatorics of Hermite polynomials. The first goal of this work is to give an alternative point of view on Dyck tilings by making use of the weak order and the Bruhat order on permutations. Then we introduce two natural generalizations: $k$-Dyck tilings and symmetric Dyck tilings. We are led to consider Stirling permutations, and define an analogue of the Bruhat order on them. We show that certain families of $k$-Dyck tilings are in bijection with intervals in this order. We enumerate symmetric Dyck tilings and show that certain families of symmetric Dyck tilings are in bijection with intervals in the weak order on signed permutations.
We describe a new uniform generation tree for permutations with the specific property that, for most permutations, all of their descendants in the generation tree have the same number of fixed points. Our tree is optimal for the number of permutations having this property. We then use this tree to describe a new random generation algorithm for derangements, using an expected n+O(1) calls to a random number generator. Another application is a combinatorial algorithm for exact sampling from the Poisson distribution with parameter 1.
A general lattice theoretic construction of Reading constructs Hopf subalgebras of the Malvenuto-Reutenauer Hopf algebra (MR) of permutations. The products and coproducts of these Hopf subalgebras are defined extrinsically in terms of the embedding in MR. The goal of this paper is to find an intrinsic combinatorial description of a particular one of these Hopf subalgebras. This Hopf algebra has a natural basis given by permutations that we call Pell permutations. The Pell permutations are in bijection with combinatorial objects that we call sashes, that is, tilings of a 1 by n rectangle with three types of tiles: black 1 by 1 squares, white 1 by 1 squares, and white 1 by 2 rectangles. The bijection induces a Hopf algebra structure on sashes. We describe the product and coproduct in terms of sashes, and the natural partial order on sashes. We also describe the dual coproduct and dual product of the dual Hopf algebra of sashes.
Define the <b>interval rank</b> $r_[i,j] : Gr_k(\mathbb C^n) →\mathbb{N}$ of a k-plane V as the dimension of the orthogonal projection $π _[i,j](V)$ of V to the $(j-i+1)$-dimensional subspace that uses the coordinates $i,i+1,\ldots,j$. By measuring all these ranks, we define the <b>interval rank stratification</b> of the Grassmannian $Gr_k(\mathbb C^n)$. It is finer than the Schubert and Richardson stratifications, and coarser than the positroid stratification studied by Lusztig, Postnikov, and others, so we call the closures of these strata <b>interval positroid varieties</b>. We connect Vakil's "geometric Littlewood-Richardson rule", in which he computed the homology classes of Richardson varieties (Schubert varieties intersected with opposite Schubert varieties), to Erdős-Ko-Rado shifting, and show that all of Vakil's varieties are interval positroid varieties. We build on his work in three ways: (1) we extend it to arbitrary interval positroid varieties, (2) we use it to compute in equivariant K-theory, not just homology, and (3) we simplify Vakil's (2+1)-dimensional "checker games" to 2-dimensional diagrams we call "IP pipe dreams". The ring Symm of symmetric functions and its basis of Schur functions is well-known to be very closely related to the ring $\bigoplus_a,b H_*(Gr_a(\mathbb{C}^{(a+b)})$ and its basis of Schubert classes. We extend the latter ring to equivariant K-theory (with respect to a circle action on each $\mathbb{C}^{(a+b)}$, and compute the structure constants of this two-parameter deformation of Symm using the interval positroid technology above.
Alexis Darrasse, Konstantinos Panagiotou, Olivier Roussel
et al.
This paper is devoted to the construction of Boltzmann samplers according to various distributions, and uses stochastic bias on the parameter of a Boltzmann sampler, to produce a sampler with a different distribution for the size of the output. As a significant application, we produce Boltzmann samplers for words defined by regular specifications containing shuffle operators and linear recursions. This sampler has linear complexity in the size of the output, where the complexity is measured in terms of real-arithmetic operations and evaluations of generating functions.
The study of Schubert varieties in G/B has led to numerous advances in algebraic combinatorics and algebraic geometry. These varieties are indexed by elements of the corresponding Weyl group, an affine Weyl group, or one of their parabolic quotients. Often times, the goal is to determine which of the algebraic and topological properties of the Schubert variety can be described in terms of the combinatorics of its corresponding Weyl group element. A celebrated example of this occurs when G/B is of type A, due to Lakshmibai and Sandhya. They showed that the smooth Schubert varieties are precisely those indexed by permutations that avoid the patterns 3412 and 4231. Our main result is a characterization of the rationally smooth Schubert varieties corresponding to affine permutations in terms of the patterns 4231 and 3412 and the twisted spiral permutations.
We consider a Markov chain Monte Carlo approach to the uniform sampling of meanders. Combinatorially, a meander $M = [A:B]$ is formed by two noncrossing perfect matchings, above $A$ and below $B$ the same endpoints, which form a single closed loop. We prove that meanders are connected under appropriate pairs of balanced local moves, one operating on $A$ and the other on $B$. We also prove that the subset of meanders with a fixed $B$ is connected under a suitable local move operating on an appropriately defined meandric triple in $A$. We provide diameter bounds under such moves, tight up to a (worst case) factor of two. The mixing times of the Markov chains remain open.
Haglund, Luoto, Mason, and van Willigenburg introduced a basis for quasisymmetric functions called the $\textit{quasisymmetric Schur function basis}$ which are generated combinatorially through fillings of composition diagrams in much the same way as Schur functions are generated through reverse column-strict tableaux. We introduce a new basis for quasisymmetric functions called the $\textit{row-strict quasisymmetric Schur function basis}$ which are generated combinatorially through fillings of composition diagrams in much the same way as Schur functions are generated through row-strict tableaux. We describe the relationship between this new basis and other known bases for quasisymmetric functions, as well as its relationship to Schur polynomials. We obtain a refinement of the omega transform operator as a result of these relationships.
It is becoming increasingly clear that the supercharacter theory of the finite group of unipotent upper-triangular matrices has a rich combinatorial structure built on set-partitions that is analogous to the partition combinatorics of the classical representation theory of the symmetric group. This paper begins by exploring a connection to the ring of symmetric functions in non-commuting variables that mirrors the symmetric group's relationship with the ring of symmetric functions. It then also investigates some of the representation theoretic structure constants arising from the restriction, tensor products and superinduction of supercharacters.
In the early 1990s, Garsia and Haiman conjectured that the dimension of the Garsia-Haiman module $R_{\mu}$ is $n!$, and they showed that the resolution of this conjecture implies the Macdonald Positivity Conjecture. Haiman proved these conjectures in 2001 using algebraic geometry, but the question remains to find an explicit basis for $R_{\mu}$ which would give a simple proof of the dimension. Using the theory of Orbit Harmonics developed by Garsia and Haiman, we present a "kicking basis" for $R_{\mu}$ when $\mu$ has two columns.
Here we show the existence of the o-minimal simplicial and singular (co)homology in o-minimal expansions of real closed fields and prove several comparison theorems for o-minimal (co)homology theories.
We study the average behavior of variants of the UNION-FIND algorithm to maintain partitions of a finite set under the random spanning tree model. By applying the method of moments we can characterize the limiting distribution of the total costs of the algorithms "Quick Find Weighted'' and "Quick Find Biased'' extending the analysis of Knuth and Schönhage, Yao, and Chassaing and Marchand.