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In this paper, we provide generating functions and counting formulas for spanning trees and spanning forests in hypergraphs in two different ways: (1) We represent spanning trees and spanning forests in hypergraphs through Berezin-Grassmann integrals on Zeon algebra and hyper-Hafnians (orders and signs are not considered); (2) We establish a Hyper-Pfaffian-Cactus Spanning Forest Theorem through Berezin-Grassmann integrals on Grassmann algebra (orders and signs are considered), which generalizes the Hyper-Pfaffian-Cactus Theorem by Abdesselam (2004) [1] and Pfaffian matrix tree theorem by Masbaum and Vaintrob (2002) [15]. • We provide generating functions and counting formulas for spanning trees and spanning forests in hypergraphs through Berezin-Grassmann integrals on Zeon algebra and hyper- Hafnians when orders and signs are not considered. • We establish a Hyper-Pfaffian-Cactus Spanning Forest Theorem through Berezin-Grassmann integrals on Grassmann algebra (orders and signs are considered), which generalizes the Hyper-Pfaffian-Cactus Theorem by Abdesselam [Advances in Applied Mathematics (2004) Vol. 33: 51-70] and Pfaffian matrix tree theorem by Masbaum and Vaintrob [Internat. Math. Res. Notices (2002) Vol. 27: 1397-1426]. [ABSTRACT FROM AUTHOR]

In this paper, we study the desingularization problem in the first q -Weyl algebra. We give an order bound for desingularized operators, and thus derive an algorithm for computing desingularized operators in the first q -Weyl algebra. Moreover, an algorithm is presented for computing a generating set of the first q -Weyl closure of a given q -difference operator. As an application, we certify that several instances of the colored Jones polynomial are Laurent polynomial sequences by computing the corresponding desingularized operator. [ABSTRACT FROM AUTHOR]

This paper presents a geometric model of the Auslander–Reiten quiver of a type A quiver together with a stability function for which all indecomposable modules are stable. We also introduce a new Catalan object which we call a maximal almost rigid representation. We show that its endomorphism algebra is a tilted algebra of type A. We define a partial order on the set of maximal almost rigid representations and use our new geometric model to show that this partial order is a Cambrian lattice. [ABSTRACT FROM AUTHOR]

The Morris constant term identity is known to be equivalent to the famous Selberg integral. In this paper, we regard Morris type constant terms as polynomials of a certain parameter. Thus, we can take the leading coefficients and obtain new identities. These identities happen to be crucial in finding lower bounds for cardinalities of some restricted sumsets, and in calculating the volume of a Tesler polytope. [ABSTRACT FROM AUTHOR]

Abstract: The partially asymmetric exclusion process (PASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites. It is partially asymmetric in the sense that the probability of hopping left is q times the probability of hopping right. Additionally, particles may enter from the left with probability α and exit from the right with probability β. In this paper we prove a close connection between the PASEP and the combinatorics of permutation tableaux. (These tableaux come indirectly from the totally nonnegative part of the Grassmannian, via work of Postnikov, and were studied in a paper of Steingrímsson and the second author.) Namely, we prove that in the long time limit, the probability that the PASEP is in a particular configuration τ is essentially the generating function for permutation tableaux of shape enumerated according to three statistics. The proof of this result uses a result of Derrida, Evans, Hakim, and Pasquier on the matrix ansatz for the PASEP model. As an application, we prove some monotonicity results for the PASEP. We also derive some enumerative consequences for permutations enumerated according to various statistics such as weak excedence set, descent set, crossings, and occurrences of generalized patterns. [Copyright &y& Elsevier]

Abstract: The algebraic theory of variable-length codes was initiated by Schützenberger in the 1950s. Almost all the subsequently stated results in this theory are constructive and therefore lead to algorithms. However, there are some basic problems that are still open. For instance, we still do not know how to decide whether a finite code can be embedded in a finite maximal one. We answer this question under additional hypotheses. Precisely, let A be a finite alphabet with , let . Let , let be the number of factors in the prime factorization of n. We give an algorithm to decide whether there exists n, with , and a finite maximal code C which is also factorizing such that . We recall that a factorizing code is a finite maximal code which satisfies the factorization conjecture, proposed by Schützenberger. The above-mentioned statement is a consequence of another result proved in this paper. Namely, given a factorizing code C, it is known that the words in satisfy a property defined by using factorizations of cyclic groups. In this paper we give an algorithm to decide whether a set can be embedded in a set satisfying . Furthermore, we prove that, conversely, for each set satisfying , under additional hypotheses, there exists a factorizing code C such that and, as a consequence, is a code. In this case, C can be constructed starting with prefix/suffix codes and by using two types of operations on codes (composition and substitution). The additional required hypotheses concern the structure of the factorizations involved and are always satisfied when, for each , we have , with . In addition, we prove that there exist sets which satisfy and which are not codes. [Copyright &y& Elsevier]

Abstract: The main aim of this paper is the adaptation of the results of the papers [Y.A. Bahturin, S.K. Sehgal, M.V. Zaicev, Group gradings on associative algebras, J. Algebra 241 (2) (2001) 677–698; Y.A. Bahturin, S.K. Sehgal, M.V. Zaicev, Finite-dimensional graded simple algebras, preprint, and M.V. Zaĭtsev, S.K. Segal, Finite gradings of simple Artinian rings, Vestnik Moskov. Univ. Ser. I Mat. Mekh., vol. 3, 2001, pp. 21–24, 77 (in Russian); translation in Moscow Univ. Math. Bull. 56 (3) (2001) 21–24] about the structure of graded simple rings from the gradings by groups to the gradings by cancellative semigroups. [Copyright &y& Elsevier]

In a recent paper, I. Selesnick and C.S. Burrus developed a design method for maximally flat FIR low-pass digital filters with reduced group delay. Their approach leads to a system of polynomial equations depending on three integer design parameters K,L,M. In certain cases (their “Region I”), Selesnick and Burrus were able to derive solutions using only linear algebra; for the remaining cases (“Region II”), they proposed using Gro¨bner bases. This paper introduces a different method, based on multipolynomial resultants, for analyzing and solving the Selesnick–Burrus design equations. The results of calculations are presented, and some patterns concerning the number of solutions as a function of the design parameters are proved. [Copyright &y& Elsevier]

Abstract: The classic Cayley identity states that where is an matrix of indeterminates and is the corresponding matrix of partial derivatives. In this paper we present straightforward algebraic/combinatorial proofs of a variety of Cayley-type identities, both old and new. The most powerful of these proofs employ Grassmann algebra (=exterior algebra) and Grassmann–Berezin integration. Among the new identities proven here are a pair of “diagonal-parametrized” Cayley identities, a pair of “Laplacian-parametrized” Cayley identities, and the “product-parametrized” and “border-parametrized” rectangular Cayley identities. [Copyright &y& Elsevier]

Abstract: The colored Jones polynomial of the trefoil knot is a q-holonomic sequence of two variables with natural origin, namely quantum topology. The paper presents an explicit set of generators for the annihilator ideal of this q-holonomic sequence as a case study. On the one hand, our results are new and useful to quantum topology: this is the first example of a rank 2 Lie algebra computation concerning the colored Jones polynomial of a knot. On the other hand, this work illustrates the applicability and computational power of the employed computer algebra methods. [Copyright &y& Elsevier]

Abstract: The Harborth graph is widely regarded as the smallest known example of a 4-regular planar unit-distance graph. In this paper we sketch an exact proof of this property, and present some of the minimal polynomials of the coordinates of the vertices of its most prominent embedding, the calculations of which give an example for the heavy use of computer algebra in the area of geometric graph theory. Finally we use the particular algebraic structure of these polynomials to deduce some geometric properties of the Harborth graph. [Copyright &y& Elsevier]

Abstract: In this paper we will study the structure of algebras associated to two directed, layered graphs Γ. These are algebras associated with Hasse graphs of n-gons and the algebras related to pseudoroots of noncommutative polynomials. We will find the filtration preserving automorphism group of these algebras and then we will find the multiplicities of the irreducible representations of acting on the homogeneous components of and . [Copyright &y& Elsevier]

Frieze patterns form a nexus between algebra, combinatorics, and geometry. T -paths with respect to triangulations of surfaces have been used to obtain expansion formulae for cluster variables. This paper will introduce the concepts of weak friezes and T -paths with respect to dissections of polygons. Our main result is that weak friezes are characterised by satisfying an expansion formula which we call the T -path formula. We also show that weak friezes can be glued together, and that the resulting weak frieze is a frieze if and only if so was each of the weak friezes being glued. [ABSTRACT FROM AUTHOR]

Abstract: It would appear that minor-closed classes of matroids that are representable over any given finite field are very well behaved. This paper explores what happens when we go a little further to minor-closed classes of matroids that exclude a uniform minor. Numerous open problems of varying difficulty are posed. [Copyright &y& Elsevier]

Abstract: Determining the location of the maximum of Stirling numbers is a well-developed area. In this paper we give the same results for the so-called r-Stirling numbers which are natural generalizations of Stirling numbers. [Copyright &y& Elsevier]

Abstract: A standard matrix representation A of a matroid M represents M relative to a fixed basis B. Deleting rows and columns of A correspond to contracting elements of B and deleting elements of . If M is 3-connected, it is often desirable to perform such an element removal from M while maintaining 3-connectivity. This paper proves that this is always possible provided M has no 4-element fans. We also show that, subject to a mild essential restriction, this element removal can be done so as to retain a copy of a specified 3-connected minor of M. [Copyright &y& Elsevier]

Abstract: In this paper we prove that two quantities relating to the length of permutations defined on trees are independent of the structures of trees. We also find that these results are closely related to the results obtained by Graham and Pollak [R.L. Graham, H.O. Pollak, On the addressing problem for loop switching, Bell System Tech. J. 50 (1971) 2495–2519] and by Bapat, Kirkland, and Neumann [R. Bapat, S.J. Kirkland, M. Neumann, On distance matrices and Laplacians, Linear Algebra Appl. 401 (2005) 193–209]. [Copyright &y& Elsevier]

Abstract: The gap function of an infinite word over the binary alphabet gives the distances between consecutive 1''s in this word. In this paper we study infinite binary words whose gap function is injective or “almost injective.” A method for computing the subword complexity of such words is given. A necessary and sufficient condition for a function to be the subword complexity function of a binary word whose gap function is increasing is obtained. [Copyright &y& Elsevier]

Abstract: Generalizing a sequence of Lambert, Cayley and Ramanujan, Chapoton has recently introduced a polynomial sequence defined by In this paper we prove Chapoton''s conjecture on the duality formula: , and answer his question about the combinatorial interpretation of . Actually we give combinatorial interpretations of these polynomials in terms of plane trees, forests of half-mobile trees, and forests of plane trees. Our approach also leads to a general formula that unifies several known results for enumerating trees and plane trees. [Copyright &y& Elsevier]

Abstract: Two kinds of decompositions of a -module are introduced in this article, a Loewy-decomposition and a primary decomposition. Algorithms are described which allow to construct decompositions for finite-dimensional -modules. Both decompositions rely on the concept of the -module of relative syzygies introduced in the paper. Furthermore an algorithm is described which tests whether two given -modules are isomorphic under the relevant type of transformations. [Copyright &y& Elsevier]

Abstract: We study a fundamental family of root-finding iteration functions in the context of symmetric functions. This family, which we refer to as the Basic Family, goes back to Schröder''s 1870 paper, and admits numerous different representations. In one representation, it is known as König''s family. A purely algebraic derivation by Kalantari et al. leads to the discovery of many minimality and uniqueness properties of this family. Our new perspective reveals a symmetric algebraic structure of the Basic Family, which gives rise to simple combinatorial proofs of many important properties of this family and two of its variants. The first variant maintains high order of convergence for multiple roots. The second variant, called the Truncated Basic Family, is an infinite family of mth order methods for every , using only the first derivatives. Our result extends Kalantari''s analysis of Halley''s family, the special case of the Truncated Basic Family where . Finally, we give a recipe for constructing new high order root-finding algorithms, and use it to derive an interesting family of iteration functions that have higher orders of convergence for multiple roots than for simple roots. [Copyright &y& Elsevier]

Let denote the number of non-isomorphic matroids on an n-element set. In 1969, Welsh conjectured that, for all non-negative integers m and n, . In this paper, we prove this conjecture. [Copyright &y& Elsevier]

Let Wn,r denote the n-fold iterated wreath product of Z/rZ with itself. In this paper, we are interested in the tower of groups W1,r⊂W2,r⊂⋯. We show that the irreducible representations of Wn,r are indexed by a set of labeled rooted trees. By adding a partial order on this set of rooted trees, we obtain the Bratteli diagram for this tower of groups. In particular, we give the branching rules. This approach yields combinatorial rules for the decomposition of restricted and induced representations. [Copyright &y& Elsevier]

In this paper we split the defining relations of the two well known one-parameter quantizations of GL(2), Manin and Dipper–Donkin, into two bialgebras. We recombine these bialgebras to obtain two-parameter quantum groups. We find “one-sided preantipodes” for these bialgebras. By fixing the condition for these maps to be an antiautomorphism of the whole algebra and setting particular commuting relations for the corresponding determinant, we recover the Manin and Dipper–Donkin quantization plus an additional quantum group that is a particular case of the two-parameter Takeuchi quantum group. [Copyright &y& Elsevier]

For all integers m,n, such that 3⩽m⩽n, let r(S(m),S(n)) represent the least integer such that for every 2-coloring of the set {1,2,…,r(S(m),S(n))} there exists a monochromatic solution to either S(m): ∑i=1m−1xi=xm or S(n): ∑i=1n−1xi=xn. The integer r(S(m),S(n)) is called the disjunctive Rado number for the above two equations. In this paper it is determined thatrS(m),S(n)=m2−m−1for m⩽n⩽m+1,m2−2m+1for m+2⩽n⩽m2−2m+2,n−1for m2−2m+3⩽n⩽m2−m−1,m2−m−1for n⩾m2−m. [Copyright &y& Elsevier]

Let S={1,2,…,n}, and let S=S1∪S2 be a partition of S in two disjoint subsets. A triple (a,b,c), a,b,c∈S, is called a Schur triple if a+b=c. If in addition a, b, and c all lie in the same subset Si of S, we call the triple (a,b,c) monochromatic. In this paper we give a simple proof that the minimal number of monochromatic Schur triples is asymptotic to n2/11. We also show that the number of monochromatic Schur triples modulo n equals n2−3&z.sfnc;S1&z.sfnc;&z.sfnc;S2&z.sfnc;. [Copyright &y& Elsevier]

In his 1998 Fubini Lectures, Rota discusses twelve problems in probability that "no one likes to bring up". The first problem calls for a revision of the notion of a sample space, guided by the belief that mention of sample points in a probabilistic argument is bad form and that a "pointless" foundation of probability should be provided by algebras of random variables. In 1958 Chang introduced MV-algebras to prove the completeness theorem of Łukasiewicz logic Ł ∞. The aim of this paper is to show that MV-algebras provide a solution of Rota's first problem. The adjunction between MV-algebras and unital commutative C*-algebras equips every MV-algebra A with a natural ring structure, as advocated by Nelson for algebras of random variables. The closed compact set S (A) ⊆ [ 0 , 1 ] A of finitely additive probability measures on A (the states of A) coincides with the set of [ 0 , 1 ] -valued functions on A whose finite restrictions are consistent in de Finetti's sense. MV-algebras and Ł ∞ thus provide the framework for a generalization (known as ŁIPSAT) of Boole's probabilistic inference problem, and its modern reformulation known as probabilistic satisfiability, PSAT. We construct an affine homeomorphism γ A of S (A) onto the weakly compact space of regular Borel probability measures on the maximal spectral space μ (A). The latter is the most general compact Hausdorff space. As a consequence, for every Kolmogorov probability space (Ω , F Ω , P) , with F Ω the sigma-algebra of Borel sets of a compact Hausdorff space Ω, and P a regular probability measure on F Ω , there is an MV-algebra A and a state σ of A such that (Ω , F Ω , P) ≅ (μ (A) , F μ (A) , γ A (σ)). [ABSTRACT FROM AUTHOR]