Related papers: $q$-rious unimodality

We prove a continued fraction expansion for the reciprocal of a certain $q$-series. All the specialists in the world are asked whether it is new or not.

We provide various counter-examples to the long-standing so-called "Omnibus Conjecture" in Rational Homotopy Theory. That is, we show that a space with finite dimensional even-degree rational cohomology and finite dimensional spherical…

The q-binomial coefficients were assumed to be unimodal as early as the 1850's, but it remained unproven until Sylvester's 1878 proof using invariant theory. In 1982, Proctor gave an "elementary" proof using linear algebra. Finally, in…

We present a multivariable generalization of the digital binomial theorem from which a q-analog is derived as a special case.

An interesting, and still wide open, conjecture of Reiner and Stanton predicts that certain "strange" symmetric differences of $q$-binomial coefficients are always nonnegative and unimodal. We extend their conjecture to a broader, and…

We propose a conjectural $q$-analogue of the classical duality for iterated integrals on $\mathbb{P}^{1}$ minus four points, arising from the involutive M\"{o}bius transformation which exchanges the four marked points in pairs. To this end,…

We provide a new proof of the Kac positivity conjecture for an arbitrary quiver $Q$. The ingredients are the cohomological integrality theorem in Donaldson-Thomas theory, dimensional reduction, and an easy purity result. These facts imply…

In 1878, Sylvester proved Cayley's Conjecture that the coefficients of the Gaussian $q$-binomial coefficients are unimodal. In 1990, O'Hara famously discovered a constructive combinatorial proof, and in 2013, Pak and Panova proved the…

The $q$-binomial coefficients $\qbinom{n}{m}=\prod_{i=1}^m(1-q^{n-m+i})/(1-q^i)$, for integers $0\le m\le n$, are known to be polynomials with non-negative integer coefficients. This readily follows from the $q$-binomial theorem, or the…

We establish a natural translation from word rewriting systems to strictly positive polymodal logics. Thereby, the latter can be considered as a generalization of the former. As a corollary we obtain examples of undecidable strictly…

We obtain similar types of conclusions as that of Br\"{u}ck [1] for two differential polynomials which in turn radically improve and generalize several existing results. Moreover, a number of examples have been exhibited to justify the…

Let U be an open subset of a unirational variety (or more generally of a separably rationally connected variety). We prove that there is rational curve C in U such that the fundamental group of C surjects onto the fundamental group of U.…

If the odd and even parts of a continued fraction converge to different values, the continued fraction may or may not converge in the general sense. We prove a theorem which settles the question of general convergence for a wide class of…

We give an $n$-space generalized $q$-binomial theorem, and some new $q$ series identities that resemble the traditional $q$ series partition generating functions. These identities enumerate stepping stone weighted vector partitions.

We prove inversion of adjunction for higher rational singularities.

We prove a theorem which implies a quantum (multiplicative) analogue of the Horn conjecture, and also of the saturation conjecture. We obtain transversality statements for quantum schubert calculus in any characteristic and also determine…

We prove the validity of the strong version of the union of uniform closed balls conjecture, formulated in 2011 as [4, Conjecture 2.5], in the plane.

The q-monopole bundle introduced previously is extended to a general construction for quantum group bundles with non-universal differential calculi. We show that the theory applies to several other classes of bundles as well, including…

Transformation formulas for four-parameter refinements of the q-trinomial coefficients are proven. The iterative nature of these transformations allows for the easy derivation of several infinite series of q-trinomial identities, and can be…

We prove a rigidity theorem for morphisms from products of open subschemes of the projective line into solvable groups not containing a copy of $\Ga$ (for example, wound unipotent groups). As a consequence, we deduce several structural…