An overpartition analogue of the <Emphasis Type="Italic">q</Emphasis>-binomial coefficients

We define an overpartition analogue of Gaussian polynomials (also known as q-binomial coefficients) as a generating function for the number of overpartitions fitting inside the \(M \times N\) rectangle. We call these new polynomials over Gaussian polynomials or over q-binomial coefficients. We investigate basic properties and applications of over q-binomial coefficients. In particular, via the recurrences and combinatorial interpretations of over q-binomial coefficients, we prove a Rogers–Ramanujan type partition theorem.     

The homogeneous q-difference operator

We introduce a q-differential operator D_xy on functions in two variables which turns out to be suitable for dealing with the homogeneous form of the q-binomial theorem as studied by Andrews, Goldman, and Rota, Roman, Ihrig, and Ismail, et al. The homogeneous versions of the q-binomial theorem and the Cauchy identity are often useful for their specializations of the two parameters. Using this operator, we derive an equivalent form of the Goldman–Rota binomial identity and show that it is a homogeneous generalization of the q-Vandermonde identity. Moreover, the inverse identity of Goldman and Rota also follows from our unified identity. We also obtain the q-Leibniz formula for this operator. In the last section, we introduce the homogeneous Rogers–Szegö polynomials and derive their generating function by using the homogeneous q-shift operator.     

Undetected error probability of q-ary constant weight codes

In this paper, we introduce a new combinatorial invariant called q-binomial moment for q-ary constant weight codes. We derive a lower bound on the q-binomial moments and introduce a new combinatorial structure called generalized (s, t)-designs which could achieve the lower bounds. Moreover, we employ the q-binomial moments to study the undetected error probability of q-ary constant weight codes. A lower bound on the undetected error probability for q-ary constant weight codes is obtained. This lower bound extends and unifies the related results of Abdel-Ghaffar for q-ary codes and Xia-Fu-Ling for binary constant weight codes. Finally, some q-ary constant weight codes which achieve the lower bounds are found.      

Bender's generalizedq-binomial Vandermonde convolution

A simple arithmetical proof and a generalization of Bender's generalizedq-binomial Vandermonde convolution are given.     

q-Discrete Painlevé equations for recurrence coefficients of modified q-Freud orthogonal polynomials

We present an asymmetric q-Painlevé equation. We will derive this using q-orthogonal polynomials with respect to generalized Freud weights: their recurrence coefficients will obey this q-Painlevé equation (up to a simple transformation). We will show a stable method of computing a special solution, which gives the recurrence coefficients. We establish a connection with α-q-P_V.     

( q,t)-analogues and $GL_{n}({\ma thbb{F}}_{ q})$

We start with a (q,t)-generalization of a binomial coefficient. It can be viewed as a polynomial in t that depends upon an integer q, with combinatorial interpretations when q is a positive integer, and algebraic interpretations when q is the order of a finite field. These (q,t)-binomial coefficients and their interpretations generalize further in two directions, one relating to column-strict tableaux and Macdonald’s “7 ^th variation” of Schur functions, the other relating to permutation statistics and Hilbert series from the invariant theory of GL_n(\mathbbF_q)GL_{n}({\mathbb{F}}_{q}) .     

Parity in partition identities

This paper considers a variety of parity questions connected with classical partition identities of Euler, Rogers, Ramanujan and Gordon. We begin by restricting the partitions in the Rogers-Ramanujan-Gordon identities to those wherein even parts appear an even number of times. We then take up questions involving sequences of alternating parity in the parts of partitions. This latter study leads to: (1) a bi-basic q-binomial theorem and q-binomial series, (2) a new interpretation of the Rogers-Ramanujan identities, and (3) a new natural interpretation of the fifth-order mock theta functions f ₀(q) along with a new proof of the Hecke-type series representation.     

A short note on the overpartition function

In this brief note, we give a combinatorial proof of a variation of Gauss’s q-binomial theorem, and we determine arithmetic properties of the overpartition function modulo 8.     

New series representations for Jacobi?s triple product identity and more via the q-Markov method

We derive new series representations for Jacobi?s triple product identity, the q-binomial theorem, q-analogs of the exponential function, and more with several special cases using the q-Markov-WZ method.     

Schanuel's Conjecture and Algebraic Roots of Exponential Polynomials

In this article, I will prove that assuming Schanuel's conjecture, an exponential polynomial with algebraic coefficients can have only finitely many algebraic roots. Furthermore, this proof demonstrates that there are no unexpected algebraic roots of any such exponential polynomial. This implies a special case of Shapiro's conjecture: if p(x) and q(x) are two exponential polynomials with algebraic coefficients, each involving only one iteration of the exponential map, and they have common factors only of the form exp (g) for some exponential polynomial g, then p and q have only finitely many common zeros.     

An overpartition analogue of partitions with bounded differences between largest and smallest parts

We study the generating function for overpartitions with bounded differences between largest and smallest parts, which is analogous to a result of Breuer and Kronholm on integer partitions. We also connect this problem with over $q$-binomial coefficients introduced by Dousse and Kim.     

Generalizations of two identities of Guo and Yang

In this paper, we provide generalizations of two identities of Guo and Yang [2] for the q-binomial coe?cients. This approach allows us to derive new convolution identities for the complete and elementary symmetric functions. New identities involving q-binomial coe?cients are obtained as very special cases of these results. A new relationship between restricted partitions and restricted partitions into parts of two kinds is derived in this context.     

Factors of the Gaussian coefficients

We present some simple observations on factors of the q-binomial coefficients, the q-Catalan numbers, and the q-multinomial coefficients. Writing the Gaussian coefficient with numerator n and denominator k in a form such that 2k?n by the symmetry in k, we show that this coefficient has at least k factors. Some divisibility results of Andrews, Brunetti and Del Lungo are also discussed.     

Some aspects of groups acting on finite posets

Let P be a finite poset and G a group of automorphisms of P. The action of G on P can be used to define various linear representations of G, and we investigate how these representations are related to one another and to the structure of P. Several examples are analyzed in detail, viz., the symmetric group $\text{G}$_n acting on a boolean algebra, GL_n(q) acting on subspaces of an n-dimensional vector space over GF(q), the hyperoctahedral group B_n acting on the lattice of faces of a cross-polytope, and $\text{G}$_n acting on the lattice Π_n of partitions of an n-set. Several results of a general nature are also proved. These include a duality theorem related to Alexander duality, a special property of geometric lattices, the behavior of barycentric subdivision, and a method for showing that certain sequences are unimodal. In particular, we give what seems to be the simplest proof to date that the q-binomial coefficient $\text{k+l}\text{k}$ has unimodal coefficients.     

A q-rious positivity

The q-binomial coefficients ${\genfrac{[}{]}{0pt}{}{n}{m}= \prod_{i=1}^m (1-q^{n-m+i})/(1-q^i)}$ , for integers 0??? m??? n, are known to be polynomials with non-negative integer coefficients. This readily follows from the q-binomial theorem, or the many combinatorial interpretations of ${\genfrac{[}{]}{0pt}{}{n}{m}}$ . In this note we conjecture an arithmetically motivated generalisation of the non-negativity property for products of ratios of q-factorials that happen to be polynomials.     

A convolution for complete and elementary symmetric functions

In this paper we give a convolution identity for complete and elementary symmetric functions. This result can be used to prove and discover some combinatorial identities involving r-Stirling numbers, r-Whitney numbers and q-binomial coefficients. As a corollary we derive a generalization of the quantum Vandermonde’s convolution identity.     

Order Ideals in Weak Subposets of Young’s Lattice and Associated Unimodality Conjectures

The k-Young lattice Y^k is a weak subposet of the Young lattice containing partitions whose first part is bounded by an integer k > 0. The Y^k poset was introduced in connection with generalized Schur functions and later shown to be isomorphic to the weak order on the quotient of the affine symmetric group S_{k + 1} by a maximal parabolic subgroup. We prove a number of properties for Y^k including that the covering relation is preserved when elements are translated by rectangular partitions with hook-length k. We highlight the order ideal generated by an m x n rectangular shape. This order ideal, L^k(m, n), reduces to L(m, n) for large k, and we prove it is isomorphic to the induced subposet of L(m, n) whose vertex set is restricted to elements with no more than k - m + 1 parts smaller than m. We provide explicit formulas for the number of elements and the rank-generating function of L^k(m, n). We conclude with unimodality conjectures involving q-binomial coefficients and discuss how implications connect to recent work on sieved q-binomial coefficients.AMS Subject Classification: 06A06, 05A17, 05A10, 05E05.     

The unsatisfiability threshold revisited

The problem of determining the unsatisfiability threshold for random 3-SAT formulas consists in determining the clause to variable ratio that marks the experimentally observed abrupt change from almost surely satisfiable formulas to almost surely unsatisfiable. Up to now, there have been rigorously established increasingly better lower and upper bounds to the actual threshold value. In this paper, we consider the problem of bounding the threshold value from above using methods that, we believe, are of interest on their own right. More specifically, we show how the method of local maximum satisfying truth assignments can be combined with results for the occupancy problem in schemes of random allocation of balls into bins in order to achieve an upper bound for the unsatisfiability threshold less than 4.571. In order to obtain this value, we establish a bound on the q-binomial coefficients (a generalization of the binomial coefficients). No such bound was previously known, despite the extensive literature on q-binomial coefficients. Finally, to prove our result we had to establish certain relations among the conditional probabilities of an event in various probabilistic models for random formulas. It turned out that these relations were considerably harder to prove than the corresponding ones for unconditional probabilities, which were previously known.     

Identities involving Farey fractions

The rational numbers a/q in [0, 1] can be counted by increasing height H(a/q) = max(a, q), or ordered as real numbers. Franel’s identity shows that the Riemann hypothesis is equivalent to a strong bound for a measure of the independence of these two orderings. We give a proof using Dedekind sums that allows weights w(q). Taking w(q) = χ(q) we find an extension to Dirichlet L-functions.     

Beyond Zudilin's conjectured q-analogue of Schmidt's problem

Using the methodology of (rigorous) experimental mathematics, we give a simple and motivated solution to Zudilin's question concerning a q-analogue of a problem posed by Asmus Schmidt about a certain binomial coefficients sum. Our method is based on two simple identities that can be automatically proved using the Zeilberger and q-Zeilberger algorithms. We further illustrate our method by proving two further binomial coefficient sums.