For a given prime , by studying -dissection identities for Ramanujanʼs theta functions and , we derive infinite families of congruences modulo 2 for some -regular partition functions, where .

We consider a special case of a WP-Bailey chain of George Andrews, and use it to derive a number of curious transformations of basic hypergeometric series. We also derive two new WP-Bailey pairs, and use them to derive some additional new transformations for basic hypergeometric series. Finally, we briefly consider the implications of WP-Bailey pairs , in which is independent of , for generalizations of identities of the Rogers–Ramanujan type.

We study Wronskians of Appell polynomials indexed by integer partitions. These families of polynomials appear in rational solutions of certain Painlevé equations and in the study of exceptional orthogonal polynomials. We determine their derivatives, their average and variance with respect to Plancherel measure, and introduce several recurrence relations. In addition, we prove an integrality conjecture for Wronskian Hermite polynomials previously made by the first and last authors. Our proofs all exploit strong connections with the theory of symmetric functions.

We consider two extensions of free probability that have been studied in the research literature, and are based on the notions of and respectively of for noncommutative random variables. In a 2012 paper, Belinschi and Shlyakhtenko pointed out a connection between these two frameworks, at the level of their operations of 1-dimensional free additive convolution. Motivated by that, we propose a construction which produces a multi-variate version of the Belinschi-Shlyakhtenko result, together with a result concerning free products of multi-variate noncommutative distributions. Our arguments are based on the combinatorics of the specific types of cumulants used in c-free and in infinitesimal free probability. They work in a rather general setting, where the initial data consists of a vector space given together with a linear map . In this setting, all the needed brands of cumulants live in the guise of families of multilinear functionals on , and our main result concerns a certain transformation on such families of multilinear functionals.

A general family of matrix valued Hermite type orthogonal polynomials is introduced as the matrix orthogonal polynomials with respect to a weight. The matrix polynomials are eigenfunctions of a matrix differential equation. For the weight we derive Pearson equations, which allow us to derive many explicit properties of these matrix polynomials. In particular, the matrix polynomials are eigenfunctions to another matrix differential equation. We also obtain for these polynomials shift operators, a Rodrigues formula, explicit expressions for the squared norm, explicit three term recurrence relations, etc. The matrix entries of the matrix polynomials can be expressed in terms of scalar Hermite and dual Hahn polynomials. We also derive a connection formula for the matrix Hermite polynomials. Next we show that operational Burchnall formulas extend to matrix polynomials. We make this explicit for the matrix Hermite polynomials and for previously introduced matrix Gegenbauer type orthogonal polynomials. The Burchnall approach gives two descriptions of the matrix valued orthogonal polynomials for the Toda modification of the matrix Hermite weight. In particular, we obtain an explicit non-trivial solution to the non-abelian Toda lattice equations.

A systematic study of of mesh patterns of length 2 was conducted by Hilmarsson et al., where 25 out of 65 non-equivalent cases were solved. In this paper, we give 27 results for these patterns including 14 distributions for which avoidance was not known. Moreover, for the unsolved cases, we prove an equidistribution result (out of 6 equidistribution results we prove in total), and conjecture 6 more equidistributions. Finally, we find seemingly unknown distribution of the well known permutation statistic “strict fixed point”, which plays a key role in many of our enumerative results. This paper is the first systematic study of distributions of mesh patterns. Our techniques to obtain the results include, but are not limited to, obtaining functional relations for generating functions, and finding recurrence relations and bijections.

Given a ruled surface defined in the standard parametric form , we present an algorithm that determines the singularities (and their multiplicities) of from the parametrization . More precisely, from we construct an auxiliary parametric curve and we show how the problem can be simplified to determine the singularities of this auxiliary curve. Only one univariate resultant has to be computed and no elimination theory techniques are necessary. These results improve some previous algorithms for detecting singularities for the special case of parametric ruled surfaces.

We study the relation between the cluster automorphisms and the quasi-automorphisms of a cluster algebra . We prove that under some mild condition, satisfied for example by every skew-symmetric cluster algebra, the quasi-automorphism group of is isomorphic to a subgroup of the cluster automorphism group of , and the two groups are isomorphic if has principal or universal coefficients; here is the cluster algebra with trivial coefficients obtained from by setting all frozen variables equal to the integer 1. We also compute the quasi-automorphism group of all finite type and all skew-symmetric affine type cluster algebras, and show in which types it is isomorphic to the cluster automorphism group of .

In this paper we investigate the complexity of model selection and model testing for dynamical systems with toric steady states. Such systems frequently arise in the study of chemical reaction networks. We do this by formulating these tasks as a constrained optimization problem in Euclidean space. This optimization problem is known as a Euclidean distance problem; the complexity of solving this problem is measured by an invariant called the Euclidean distance (ED) degree. We determine closed-form expressions for the ED degree of the steady states of several families of chemical reaction networks with toric steady states and arbitrarily many reactions. To illustrate the utility of this work we show how the ED degree can be used as a tool for estimating the computational cost of solving the model testing and model selection problems.

We prove the cancellation-free formula for the antipode of the noncrossing partition lattice in the reduced incidence Hopf algebra of posets due to Einziger. The proof is based on a map from chains in the noncrossing partition lattice to noncrossing hypertrees and expressing the alternating sum over these fibers as an Euler characteristic.

This paper introduces , a generalization of graphic hyperplane arrangements arising from electrical networks and order polytopes of finite posets. We generalize combinatorial results on graphic arrangements to Dirichlet arrangements, addressing characteristic polynomials and supersolvability in particular. We apply these results to visibility sets of order polytopes and fixed-energy harmonic functions on electrical networks.

A Ramanujan-type formula involving the squares of odd zeta values is obtained. The crucial part in obtaining such a result is to conceive the correct analogue of the Eisenstein series involved in Ramanujan's formula for . The formula for is then generalized in two different directions, one, by considering the generalized divisor function , and the other, by studying a more general analogue of the aforementioned Eisenstein series, consisting of one more parameter . A number of important special cases are derived from the first generalization. For example, we obtain a series representation for , where is a non-trivial zero of . We also evaluate a series involving the modified Bessel function of the second kind in the form of a rational linear combination of and for .

The Turán inequality and its higher order analog arise in the study of Maclaurin coefficients of an entire function in the Laguerre-Pólya class. It is well known that if a real entire function is in the class, the Maclaurin coefficients satisfy both the Turán inequality and the higher order Turán inequality. Chen, Jia and Wang proved that for , the higher order Turán inequality holds for the partition function and the 3-rd associated Jensen polynomials have only real zeros. Recently, Griffin, Ono, Rolen and Zagier showed that Jensen polynomials for a large family of functions, including those associated to and the partition function, are hyperbolic for sufficiently large . This result gave evidence for Riemann hypothesis. In this paper, we give a unified approach to investigate the higher order Turán inequality for the sequences , where satisfy a three-term recurrence relation. In particular, we prove higher order Turán inequality for the sequences , where are the Motzkin numbers, the Fine number, the Franel numbers of order 3 and the Domb numbers. As a consequence, for these combinatorial sequences, the 3-rd associated Jensen polynomials have only real zeros. Furthermore, for these combinatorial sequences we conjecture that for any given integer , there exists an integer such that for , the -th associated Jensen polynomials have only real zeros.

The group of basis-conjugating automorphisms of the free group of rank , also known as the McCool group or the welded braid group , contains a much-studied subgroup, called the upper McCool group . Starting from the cohomology ring of , we find, by means of a Gröbner basis computation, a simple presentation for the infinitesimal Alexander invariant of this group, from which we determine the resonance varieties and the Chen ranks of the upper McCool groups. These computations reveal that, unlike for the pure braid group and the full McCool group , the Chen ranks conjecture does not hold for , for any . Consequently, is not isomorphic to in that range, thus answering a question of Cohen, Pakianathan, Vershinin, and Wu. We also determine the scheme structure of the resonance varieties , and show that these schemes are not reduced for .

Let [DELTA] be a connected, pure 2-dimensional simplicial complex embedded in R.sup.2 and let C.sup.r([[circumflex accent] over [DELTA]]) be the homogenized spline module of [DELTA] with smoothness r as in . To study C.sup.r([[circumflex accent] over [DELTA]]), Schenck and Stillman developed in the quotient complex S.sub.*/J.sub.*. In , Schenck and Stiller conjectured that the regularity of H.sub.1(S.sub.*/J.sub.*) is less than 2r+1. In this article, we pose a counterexample to this "2r+1" conjecture. We also propose some modifications to the conjecture.

Let Δ be a connected, pure 2-dimensional simplicial complex embedded in and let be the homogenized spline module of Δ with smoothness as in . To study , Schenck and Stillman developed in the quotient complex . In , Schenck and Stiller conjectured that the regularity of is less than . In this article, we pose a counterexample to this “ ” conjecture. We also propose some modifications to the conjecture.

Let the polynomial be defined as the -th coefficient in the power series expansion (in variable ) of the function The polynomial has the following combinatorial interpretation: the -th coefficient counts the number of representations of as sums of exactly powers of 2. The can be seen as a polynomial analogue of the number , which counts the number of binary partitions of . In the present paper we obtain several results concerning arithmetic properties of the polynomials as well as its coefficients. Moreover, we show an interesting connection between coefficients of and the number counting so called -partitions, i.e., the representations of as sums of numbers of the form .