We show how to represent various families of Laguerre polynomials by the three-dimensional Riordan arrays, and use the fundamental theorem of Riordan arrays to obtain the corresponding exponential generating functions.
New upper bounds on the size of the torsion group of a $\mathbb{Q}$-acyclic simplicial complex are introduced which depend only on the vertex degree sequence of the complex and its dimension.
We provide lower bounds on the connectivity of the independence complexes of hypergraphs. Additionally, we compute the homotopy types of the independence complexes of $d$-uniform properly-connected triangulated hypergraphs.
We show that the binary coin set minimizes the number of coins needed to guarantee the ability to make change in any one transaction and its asymptotic uniform average cost is no worse than that of any completely greedy coin set.
Gateway math courses can be a major hurdle for students to get over in order to continue on with their majors, especially STEM majors. Previously, students could have taken up to four semesters of remedial math before being able to place into the first gateway math course needed for their major. With Math Support, students no longer have to take those remedial courses; the remediation is built in to the course in addition to the regular level content. These co-requisite courses allow students to start into their major courses much sooner than in the past, therefore allowing them to stay on track in their major and finish successfully.
 Developmental math courses have been replaced with co-requisite and accelerated learning models. All students can now progress through gateway courses within their first year, regardless of test scores. Gateway courses are offered with additional support to aid students in learning and understanding and fill in any gaps. Fairmont State was one the first colleges in WV to offer these services at full-scale. Attendees will be given an overview of Math Support at Fairmont State and how we have adapted our course to meet the changing needs of students as a result of the COVID pandemic. More data will be reported.
Milagros Sainz, Katja Upadyaya, Katariina Salmela-Aro
The present two studies with a 3-year longitudinal design examined the co-development of science, math, and language (e.g., Spanish/Finnish) interest among 1,317 Spanish and 804 Finnish secondary school students across their transition to post-compulsory secondary education, taking into account the role of gender, performance, and socioeconomic status (SES). The research questions were analyzed with parallel process latent growth curve (LGC) modeling. The results showed that Spanish students’ interest in each domain slightly decreased over time, whereas Finnish students experienced an overall high and relatively stable level of interest in all domains. Further, boys showed greater interest in math and science in both countries, whereas girls reported having a greater interest in languages. Moreover, Spanish and Finnish students with high academic achievement typically experienced high interest in different domains, however, some declines in their interest occurred later on.
We present some results on the freeness or non freeness of some tridendriform algebras. In particular, we give a combinatorial proof of the freeness of WQSym, an algebra based on packed words, result already known with an algebraic proof. Then, we prove the non-freeness of an another tridendriform algebra, PQSym, a conjecture remained open. The method of these proofs is generalizable, in particular it has been used to prove the freeness of the dendriform algebra FQSym and the quadrialgebra of 2-permutations.
A new triple product formulae for plane partitions with bounded size of parts is derived from a combinato- rial interpretation of biorthogonal polynomials in terms of lattice paths. Biorthogonal polynomials which generalize the little q-Laguerre polynomials are introduced to derive a new triple product formula which recovers the classical generating function in a triple product by MacMahon and generalizes the trace-type generating functions in double products by Stanley and Gansner.
We introduce a Hopf algebra structure of subword complexes, including both finite and infinite types. We present an explicit cancellation free formula for the antipode using acyclic orientations of certain graphs, and show that this Hopf algebra induces a natural non-trivial sub-Hopf algebra on c-clusters in the theory of cluster algebras.
The poset P of all permutations ordered by pattern containment is a fundamental object of study in the field of permutation patterns. This poset has a very rich and complex topology and an understanding of its Möbius function has proved particularly elusive, although results have been slowly emerging in the last few years. Using a variety of topological techniques we present a two term formula for the Mo ̈bius function of intervals in P. The first term in this formula is, up to sign, the number of so called normal occurrences of one permutation in another. Our definition of normal occurrences is similar to those that have appeared in several variations in the literature on the Möbius function of this and other posets, but simpler than most of them. The second term in the formula is (still) complicated, but we conjecture that it equals zero for a significant proportion of intervals. We present some cases where the second term vanishes and others where it is nonzero. Computing the Möbius function recursively from its definition has exponential complexity, whereas the computation of the first term in our formula is polynomial and the exponential part is isolated to the second term, which seems to often vanish. This is thus the first polynomial time formula for the Möbius function of what appears to be a large proportion of all intervals of P.
In this paper, we use the multivariate analytic techniques of Pemantle and Wilson to find asymptotic for- mulae for the coefficients of a broad class of multivariate generating functions with algebraic singularities. Flajolet and Odlyzko (1990) analyzed the coefficients of a class of univariate generating functions with algebraic singularities. These results have been extended to classes of multivariate generating functions by Gao and Richmond (1992) and Hwang (1996, 1998), in both cases by immediately reducing the multivariate case to the univariate case. Pemantle and Wilson (2013) outlined new multivariate analytic techniques and used them to analyze the coefficients of rational generating functions.
For integers $k \geq 2$ and $n \geq k+1$, we prove the following: If $n\cdot k$ is even, there is a connected $k$-regular graph on $n$ vertices. If $n\cdot k$ is odd, there is a connected nearly $k$-regular graph on $n$ vertices.
We study Hartnell's firefighter problem on infinite trees and characterise the branching number in terms of the firefighting game. Using our results about trees, we give a partial answer to a question of Martínez-Pedroza concerning firefighting on Cayley graphs.
In terms of the derivative operator and three hypergeometric series identities, several interesting summation formulas involving generalized harmonic numbers are established.
This short note proves that every incomplete $k$-list-critical graph has average degree at least $k-1 + \frac{k-3}{k^2-2k+2}$. This improves the best known bound for $k = 4,5,6$. The same bound holds for online $k$-list-critical graphs.
In this work we provide a decomposition theorem for the class of quaternary and non-binary signed-graphic matroids. This generalizes previous results for binary signed-graphic matroids and graphic matroids, and it provides the theoretical basis for a recognition algorithm.