Igor Pak, Greta Panova
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.
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We provide necessary and sufficient conditions on the unimodality of a convolution of two sequences of binomial coefficients preceded by a finite number of ones. These convolution sequences arise as as rank sequences of posets of vertex-induced subtrees for a particular class of trees. The number of such trees whose poset of vertex-induced subgraphs containing the root is not rank unimodal is determined for a fixed number of vertices $i$.
On unimodality problems in Pascal's triangle
许多二项式系数序列共享各种单峰性质。本文研究了杨辉三角形的一条射线或一条截线上的二项式系数序列的单峰性问题。我们的研究结果给出了一个肯定的答案，特别是Belbachir等人的猜想断言，这样的序列的二项式系数必须单峰。我们还提出了两个更一般的解释。
Unimodality and the reflection principle
我们展示了如何格路径和反射原理可以用来给单峰性结果的简单证明。特别地，我们给出了二项式系数单峰性的“单行”组合证明。其他例子包括二项式系数的乘积、与勒让德多项式相关的多项式，以及与Simion猜想相关的结果。
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 coefficients of representations of S_n. Other applications of this approach include strict unimodality of the diagonal q-binomial coefficients and unimodality of certain partition statistics.
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 their result by characterizing when a product of chains is strictly unimodal and then applying O'Hara's structure theorem for the partition lattice $L(m,n)$. In fact, we prove a stronger result: if $m, n \geq 8d$, and $2d \leq r \leq mn/2$, then the $r$-th rank of $L(m,n)$ has at least $d$ more elements that the next lower rank.
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 representation theory and Kronecker coefficients, the authors also asked for an argument that would employ Zeilberger's KOH theorem. In this note, we give such a proof. Then, as a further application of our method, we also provide a short proof of their conjecture that the difference between consecutive coefficients of $\binom{a+b}{b}_q$ can get arbitrarily large…
We classify rooted trees which have strictly unimodal q-polynomials (plucking polynomial). We also give criteria for a trapezoidal shape of a plucking polynomial. We generalize results of Pak and Panova on strict unimodality of q-binomial coefficients. We discuss which polynomials can be realized as plucking polynomials and whether or not different rooted trees can have the same plucking polynomial.
For a polynomial with palindromic coefficients, unimodality is equivalent to having a nonnegative $g$-vector. A sufficient condition for unimodality is having a nonnegative $γ$-vector, though one can have negative entries in the $γ$-vector and still have a nonnegative $g$-vector. In this paper we provide combinatorial models for three families of $γ$-vectors that alternate in sign. In each case, the $γ$-vectors come from unimodal polynomials with straightforward combinatorial descriptions, but for which there is no straightforward combinatorial proof of unimodality. By using the transformation…
We present yet another algebraic proof of the unimodality of the binomial coefficients.
We find a combinatorial interpretation of Shareshian and Wachs' $q$-binomial-Eulerian polynomials, which leads to an alternative proof of their $q$-$γ$-positivity using group actions. Motivated by the sign-balance identity of Désarménien--Foata--Loday for the $(\mathrm{des}, \mathrm{inv})$-Eulerian polynomials, we further investigate the sign-balance of the $q$-binomial-Eulerian polynomials. We show the unimodality of the resulting signed binomial-Eulerian polynomials by exploiting their continued fraction expansion and making use of a new quadratic recursion for the $q$-binomial-Eulerian poly…
Unimodality of Kazhdan--Lusztig polynomials of sparse paving matroids
本文证明无环稀疏铺砌拟阵的Kazhdan--Lusztig多项式单峰，并刻画了严格单峰性失效的例外情形，同时确定众数并分类退化情形。
The symmetric difference of the $q$-binomial coefficients $F_{n,k}(q)={n+k\brack k}-q^{n}{n+k-2\brack k-2}$ was introduced by Reiner and Stanton. They proved that $F_{n,k}(q)$ is symmetric and unimodal for $k \geq 2$ and $n$ even by using the representation theory for Lie algebras. Based on Sylvester's proof of the unimodality of the Gaussian coefficients, as conjectured by Cayley, we find an interpretation of the unimodality of $F_{n,k}(q)$ in terms of semi-invariants. In the spirit of the strict unimodality of the Gaussian coefficients due to Pak and Panova, we prove the strict unimodality o…
Ooms spectra of Frobenius maximal parabolics: strict unimodality and Euclidean log-concavity
研究 Frobenius 极大抛物子代数的 Ooms 谱，证明严格单峰性并借助欧几里得算法和有限截断验证对数凹性，覆盖较小块至多 128 的情形。
A Unified Approach to Unimodality of Gaussian Polynomials
提出基于计算机代数的新方法，通过闭式系数和柱形代数分解精确判定$q$-二项式系数的严格单峰性，并证明Stanley和Zanello猜想的两个额外情形。
As is well-known, a generalization of the classical concept of the factorial $n!$ for a real number $x\in {\mathbb R}$ is the value of Euler's gamma function $Γ(1+x)$. In this connection, the notion of a binomial coefficient naturally arose for admissible values of the real arguments. By elementary means, it is proved a number of properties of binomial coefficients $\binom{r}α$ of real arguments $r,\,α\in {\mathbb R}$ such as analogs of unimodality, symmetry, Pascal's triangle, etc. for classical binomial coefficients. The asymptotic behavior of such generalized binomial coefficients of a spec…
We present a lower bound on the Kronecker coefficients for tensor squares of the symmetric group via the characters of~$S_n$, which we apply to obtain various explicit estimates. Notably, we extend Sylvester's unimodality of $q$-binomial coefficients $\binom{n}{k}_q$ as polynomials in~$q$ to derive sharp bounds on the differences of their consecutive coefficients. We then derive effective asymptotic lower bounds for a wider class of Kronecker coefficients.
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 1989, Kathy O'Hara provided a combinatorial proof of the unimodality of the q-binomial coefficients. Very soon thereafter, Doron Zeilberger translated the argument into an elegant recurrence. We introduce several perturbations to the recurrence to create a larger family of unimodal polynomials. We analyze how these perturbations affect the final polynomial and a…
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 perhaps more natural, framework, by conjecturing that, for each $k\ge 5$, the polynomials $$f(k,m,b)(q)=\binom{m}{k}_q-q^{\frac{k(m-b)}{2}+b-2k+2}\cdot\binom{b}{k-2}_q$$ are nonnegative and unimodal for all $m\gg_k 0$ and $b\le \frac{km-4k+4}{k-2}$ such that $kb\equiv km$ (mod 2), with the only exception of $b=\frac{km-4k+2}{k-2}$ when this is an integer. Using t…
The main theme of this dissertation is retooling methods to work for different situations. I have taken the method derived by O'Hara and simplified by Zeilberger to prove unimodality of $q$-binomials and tweaked it. This allows us to create many more families of polynomials for which unimodality is not, a priori, given. I analyze how many of the tweaks affect the resulting polynomial. Ayyer and Zeilberger proved a result about bounded lattice walks. I employ their generating function relation technique to analyze lattice walks with a general step set in bounded, semi-bounded, and unbounded pla…
Graph binomial coefficients and a positivity conjecture of Guo and Krattenthaler
该研究针对Guo和Krattenthaler的正性猜想，通过分拆注入法、推广的Andrews引理结合鞍点分析，关联图二项式系数，证明了该猜想对大n成立并给出轨道计数解释。
We extend the polynomial approach to hook length formula proposed in a recent joint paper with Károlyi, Nagy and Volkov to several other problems of the same type, including number of paths formula in the Young graph of strict partitions.
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 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 via the characters of $S_n$. We apply these and other results to generalize Sylvester's unimodality of the $q$-binomial coefficients $\binom{n}{k}_q$ as polynomials in $q$: we derive explicit sharp bounds on the differences of their consecutive coefficients.
Hermite-Jensen limits and $d$ log-concavity of $q$-multinomials
研究$q$-多项式$ d $ log-concavity的性质，通过Hermite-Jensen极限和归一化Jensen多项式渐进行为，证明在特定条件下，$ q $-多项式满足更强的log-concavity不等式。
Unimodality of Forest Independence Polynomials
针对森林独立多项式序列的单峰性，给出第二个证明，结合分解、矩论证与计算机辅助验证，并附Lean 4形式化证明。
Lattice polytopes of large width have real-rooted Ehrhart $h^*$-polynomials
该研究证明固定维数下宽幅格多面体的Ehrhart h*-多项式为实根，还给出格单纯形局部h*-多项式的类似结论，解答了相关问题，证明借助ChatGPT 5.6 Sol完成。
We show that a well-known asymptotic series for the logarithm of the central binomial coefficient is strictly enveloping in the sense of Pólya and Szegö, so the error incurred in truncating the series is of the same sign as the next term, and is bounded in magnitude by that term. We consider closely related asymptotic series for Binet's function, for $\lnΓ(z+1/2)$, and for the Riemann-Siegel theta function, and make some historical remarks.
An algebra over the operad of posets and structural binomial identities
本文研究了序列平行 poset 的顺序序列生成函数，并基于其基的性质，推导了 Stanley 互易定理的新证明，同时发现新的二项式恒等式和负超几何分布的性质。
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 ``seventh variation'' of Schur functions, the other relating to permutation statistics and Hilbert series from the invariant theory of GL_n(F_q).
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 proof of an identity of Hilbert, which leads to a relation of Cayley on semi-invariants. This identity plays a crucial role in the original proof of Sylvester's theorem on semi-invariants in connection with the Gaussian coefficients. Moreover, we show that the additivity lemma of Pak and Panova which yields the strict unimodality of the Gaussian coefficient…
Inequalities for $f^*$-vectors of Lattice Polytopes
本文研究了格多面体 $f^*$-向量的系数不等式，揭示了其与单纯多面体 $f$-向量不等式的相似性，并证明了特定多面体族的单峰性及 $h^*$-向量相同条件下的单峰 $f^*$-向量存在性。
Interlacing on the Unit Circle via Coefficientwise Reciprocals
本文证明具有负实数零点的回文多项式之系数倒数在单位圆上严格交错，基于有限Blaschke乘积，并应用于二项式、欧拉、Narayana等多项式族。
Merged-log-concavity of rational functions, almost strictly unimodal sequences, and phase transitions of ideal boson-fermion gases
该研究提出有理函数的合并对数凹性概念，构造相关显式函数并研究单峰序列的临界点，同时将其应用于理想玻色-费米气体，揭示了金属比临界点对应的自由能相变。
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