Related papers: Strict unimodality of q-binomial coefficients

We explain how Teleman quantization can be applied to moduli spaces of quiver representations to compute the higher cohomology of the endomorphism bundle of the universal bundle. We use this to prove Schofield's partial tilting conjecture,…

In Pacific J. Math. 292 (2018), 223-238, Shareshian and Woodroofe asked if for every positive integer $n$ there exist primes $p$ and $q$ such that, for all integers $k$ with $1 \leq k \leq n-1$, the binomial coefficient $\binom{n}{k}$ is…

We establish a $q$-analogue of Sun--Zhao's congruence on harmonic sums. Based on this $q$-congruence and a $q$-series identity, we prove a congruence conjecture on sums of central $q$-binomial coefficients, which was recently proposed by…

Kronecker coefficients encode the tensor products of complex irreducible representations of symmetric groups. Their stability properties have been considered recently by several authors (Vallejo, Pak and Panova, Stembridge). In previous…

Recently, Ni and Pan proved a $q$-congruence on certain sums involving central $q$-binomial coefficients, which was conjectured by Guo. In this paper, we give a generalization of this $q$-congruence and confirm another $q$-congruence, also…

In this paper, we confirm the following conjecture of Guo and Schlosser: for any odd integer $n>1$ and $M=(n+1)/2$ or $n-1$, $$ \sum_{k=0}^{M}[4k-1]_{q^2}[4k-1]^2\frac{(q^{-2};q^4)_k^4}{(q^4;q^4)_k^4}q^{4k}\equiv…

Let F be a finite extension of Q_p. We show that every Schwartz function on F, with values in an algebraic closure of Q_p, is the uniform limit of a sequence of Schwartz functions, whose Fourier transforms tend uniformly to 0. The proof…

By examining asymptotic behavior of certain infinite basic ($q$-) hypergeometric sums at roots of unity (that is, at a "$q$-microscopic" level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial…

Suppose that $p$ is an odd prime and $m$ is an integer not divisible by $p$. Sun and Tauraso [Adv. in Appl. Math., 45(2010), 125--148] gave $\sum_{k=0}^{n-1}\binom{2k}{k+d}/m^k$ and $\sum_{k=0}^{n-1}\binom{2k}{k+d}/(km^k)$ modulo $p$ for…

Using a property of the q-shifted factorial, an identity for q-binomial coefficients is proved, which is used to derive the formulas for the q-binomial coefficient for negative arguments. The result is in agreement with an earlier paper…

We find a combinatorial interpretation of Shareshian and Wachs' $q$-binomial-Eulerian polynomials, which leads to an alternative proof of their $q$-$\gamma$-positivity using group actions. Motivated by the sign-balance identity of…

In this paper, we describe a congruence property of solvable polynomials with coefficients in the Gaussian field Q(i).

Motivated by the Saxl conjecture and the tensor square conjecture, which states that the tensor squares of certain irreducible representations of the symmetric group contain all irreducible representations, we study the tensor squares of…

Recently, Kim-Jang-Yi have introduced q-Bernstein polynomials. From these q-Berstein polynomials, we investigte some properties related to q-Stirling numbes and q-Bernoulli numbes.

We offer a new proof that a certain q-analogue of multinomial coeffi- cients furnishes a q-counting of the set of permutations of an associated multiset of positive integers, according to the number of inversions in such arrangements. Our…

Bloch-Okounkov studied certain functions on partitions $f$ called shifted symmetric polynomials. They showed that certain $q$-series arising from these functions (the so-called \emph{$q$-brackets} $\left<f\right>_q$) are quasimodular forms.…

We show that there are polynomials $p_N$ of arbitrarily large degree $N$, with coefficients equal to 0 or 1 (Newman polynomials), such that $$ \liminf_{N \to \infty} N \Linf{p_N^2} \bigl / p_N^2(1) < 1, $$ where $\Linf{q}$ denotes the…

We say a polynomial f having integer coefficients is strongly coefficient convex if the set of coefficients of f consists of consecutive integers only. We establish various results suggesting that the divisors of x^n-1 with integer…

We prove K-theoretic and shifted K-theoretic analogues of the bijection of Stanton and White between domino tableaux and pairs of semistandard tableaux. As a result, we obtain product formulas for pairs of stable Grothendieck polynomials…

We prove that for every positive integer $k$ there exist an inclusion-exclusion polynomial $Q_{\{q_1,q_2,...,q_k\}}$ with the height at least $c^{2^k}\prod_{j=1}^{k-2}q_j^{2^{k-j-1}-1}$, where $c$ is a positive constant and…