相关论文: Semi-invariants of Binary Forms Pertaining to a Un…
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…
Bergeron--Ceballos--K\"ustner introduced the $q$-Fibonomial coefficients \( \qfibonom{m+n}{n}\), gave a combinatorial interpretation of the $q$-Fibonomial coefficients via a weighted path-domino tiling model, and conjectured that these…
We prove strict unimodality of the q-binomial coefficients \binom{n}{k}_q as polynomials in q. The proof is based on the combinatorics of certain Young tableaux and the semigroup property of Kronecker coefficients of S_n representations.
In 2013, Pak and Panova proved the strict unimodality property of $q$-binomial coefficients $\binom{\ell+m}{m}_q$ (as polynomials in $q$) based on the combinatorics of Young tableaux and the semigroup property of Kronecker coefficients.…
$q$-Kaplansky numbers were considered by Chen and Rota. We find that $q$-Kaplansky numbers are connected to the symmetric differences of Gaussian polynomials introduced by Reiner and Stanton. Based on the work of Reiner and Stanton, we…
The $q$-analogue of the binomial coefficient, known as a $q$-binomial coefficient, is typically denoted $\left[{n \atop k}\right]_q$. These polynomials are important combinatorial objects, often appearing in generating functions related to…
We present new proofs and generalizations of unimodality of the q-binomial coefficients \binom{n}{k}_q as polynomials in q. We use an algebraic approach by interpreting the differences between numbers of certain partitions as Kronecker…
We give a new $q$-$(1+q)$-analogue of the Gaussian coefficient, also known as the $q$-binomial which, like the original $q$-binomial $\genfrac{[}{]}{0pt}{}{n}{k}_{q}$, is symmetric in $k$ and $n-k$. We show this $q$-$(1+q)$-binomial is more…
Pak and Panova recently proved that the $q$-binomial coefficient ${m+n \choose m}_q$ is a strictly unimodal polynomial in $q$ for $m,n \geq 8$, via the representation theory of the symmetric group. We give a direct combinatorial proof of…
We obtain a combinatorial formula related to the shear transformation for semi-invariants of binary forms, which implies the classical characterization of semi-invariants in terms of a differential operator. Then, we present a combinatorial…
A recent nice result due to I. Pak and G. Panova is the strict unimodality of the $q$-binomial coefficients $\binom{a+b}{b}_q$ (see \cite{PP} and also \cite{PP2} for a slightly revised version of their theorem). Since their proof used…
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 give a $q$-analogue of some binomial coefficient identities of Y. Sun [Electron. J. Combin. 17 (2010), #N20] as follows: {align*} \sum_{k=0}^{\lfloor n/2\rfloor}{m+k\brack k}_{q^2}{m+1\brack n-2k}_{q} q^{n-2k\choose 2} &={m+n\brack…
In this paper, by constructing some identities, we prove some $q$-analogues of some congruences. For example, for any odd integer $n>1$, we show that \begin{gather*} \sum_{k=0}^{n-1} \frac{(q^{-1};q^2)_k}{(q;q)_k} q^k \equiv (-1)^{(n+1)/2}…
Motivated by recent works of Sun and Tauraso, we prove some variations on the Green-Krammer identity involving central q-binomial coefficients, such as $$ \sum_{k=0}^{n-1}(-1)^kq^{-{k+1\choose 2}}{2k\brack k}_q \equiv (\frac{n}{5})…
We prove polynomial identities for the N=1 superconformal model SM(2,4\nu) which generalize and extend the known Fermi/Bose character identities. Our proof uses the q-trinomial coefficients of Andrews and Baxter on the bosonic side and a…
Let $N$ be a normal subgroup of a group $G$. A quasimorphism $f$ on $N$ is $G$-invariant if $f(gxg^{-1}) = f(x)$ for every $g \in G$ and every $x \in N$. The goal in this paper is to establish Bavard's duality theorem of $G$-invariant…
We present several upper and lower bounds on the Kronecker coefficients of the symmetric group. We prove $k$-stability of the Kronecker coefficients generalizing the (usual) stability, and giving a new upper bound. We prove a lower bound…
In this paper, we confirm the following conjecture of Guo and Schlosser: for any odd integer $n>1$ and $M=(n+1)/2$ or $n-1$, $$ \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…
In a previous paper, we studied an overpartition analogue of Gaussian polynomials as the generating function for overpartitions fitting inside an $m \times n$ rectangle. Here, we add one more parameter counting the number of overlined…