Yi Ming Zou
The purpose of this short communication is to make some observations on the connections between various existing formulas of counting the number of sublattices of a fixed index in an $n$-dimensional lattice and their connection with the Gaussian binomials.
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Loeb showed that a natural extension of the usual binomial coefficient to negative (integer) entries continues to satisfy many of the fundamental properties. In particular, he gave a uniform binomial theorem as well as a combinatorial interpretation in terms of choosing subsets of sets with a negative number of elements. We show that all of this can be extended to the case of Gaussian binomial coefficients. Moreover, we demonstrate that several of the well-known arithmetic properties of binomial coefficients also hold in the case of negative entries. In particular, we show that Lucas' Theorem …
The q-binomial coefficients are the polynomial cousins of the traditional binomial coefficients, and a number of identities for binomial coefficients can be translated into this polynomial setting. For instance, the familiar vanishing of the alternating sum across row n of Pascal's triangle is captured by the so-called Gaussian Formula. In this paper, we find a q-binomial congruence which synthesizes this result and Fleck's congruence for 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-Ramaujan type partition theorem.
There is no field with only one element, yet there is a well-defined notion of what projective geometry over such a field means. This notion is familiar to experts and plays an interesting role behind the scenes in combinatorics and algebra, but it is rarely discussed as such. The purpose of this article is to bring it to the attention of a broader audience, as the solution to a puzzle about Gaussian binomial coefficients.
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 parts, obtaining a two-parameter generalization $\overline{m+n \brack n}_{q,t}$ of Gaussian polynomials, which is also a $(q,t)$-analogue of Delannoy numbers. First we obtain finite versions of classical $q$-series identities such as the $q$-binomial theorem and the Lebesgue identity, as well as two-variable generalizations of classical identities involving Gauss…
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 compact than the one discovered by Fu, Reiner, Stanton and Thiem. Underlying our $q$-$(1+q)$-analogue is a Boolean algebra decomposition of an associated poset. These ideas are extended to the Birkhoff transform of any finite poset. We end with a discussion of higher analogues of the $q$-binomial.
On a conjecture of Han and Xiong for fractional Gaussian binomial coefficients
本文针对Han和Xiong提出的分数高斯二项式系数猜想，通过证明支持度优势定理解决了大部分情形，将猜想简化为有限单位分数情形，结合计算机计算和Lean中的自动验证，最终验证了所有正有理数$r$及$k\neq200$时猜想成立。
Bergeron's conjecture & a tale of two binomial coefficients
研究伯杰龙猜想，即整数\(1\leq a<b<c<d\)且\(ad=bc\)时高斯多项式系数关系\({\binom{b+c}b}_q \ge {\binom{a+d}a}_q\)。给出两个证明并推广，还给出特殊情况的组合证明。
We look at the asymptotic behavior of the coefficients of the $q$-binomial coefficients (or Gaussian polynomials) $\binom{a+k}{k}_q$, when $k$ is fixed. We give a number of results in this direction, some of which involve Eulerian polynomials and their generalizations.
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 permutations and in representation theory. Stanley conjectured that the function $f_{k,R}(n) = \#\left\{i : [q^{i}] \left[{n \atop k}\right]_q \equiv R \pmod{N}\right\}$ is quasipolynomial for $N=2$. We generalize, showing that this is in fact true for any integer $N\in \mathbb{N}$ and determine a quasi-period $π'_N(k)$ derived from the minimal period $π_N(k…
Gaussian inequality
本文证明了涉及两个高斯多项式（或$q$-二项式）的Bergeron不等式的若干特殊情况。
Binary binomial equivalence via hyperplane arrangements
通过构造三维空间中的n个平面排列，其区域自然对应于二元2-二项式等价类，给出了该等价类计数与蛋糕数相等的几何解释，并推广到高维。
In this expository article, we discuss the relation between the Gaussian binomial and multinomial coefficients and ordinary binomial and multinomial coefficients from a combinatorial viewpoint, based on expositions by Butler, Knuth and Stanley.
Introducing q-deformed binomial coefficients of words
本文引入词的二项式系数的q-类似物，推广经典恒等式并揭示词结构信息，同时定义非交换形式幂级数的q-shuffle与q-infiltration乘积，最终推广Eilenberg关于p-群语言的刻画定理。
It is known that Bernoulli scheme of independent trials with two outcomes is connected with the binomial coefficients. The aim of this paper is to indicate stochastic processes which are connected with the $q$-polynomial coefficients (in particular, with the $q$-binomial coefficients, or the Gaussian polynomials), Stirling numbers of the first and the second kind, and Euler numbers in a natural way. A probabilistic approach allows us to give very simple proofs of some identities for these coefficients.
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