Zhiyun Cheng, Sujoy Mukherjee, Jozef H. Przytycki, Xiao Wang, Seung Yeop Yang
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.
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This is a survey on recent developments on unimodality, log-concavity and real-rootedness in combinatorics. Stanley and Brenti have written extensive surveys of various techniques that can be used to prove real-rootedness, log-concavity or unimodality. After a brief introduction, we will complement these surveys with a survey over some new techniques that have been developed, as well as problems and conjectures that have been solved. This is a draft of a chapter to appear in Handbook of Enumerative Combinatorics, published by CRC Press.
Exploring unimodality of the plucking polynomial with delay function
本文研究带根树不变量拔除多项式及其带延迟函数形式，给出刺猬型带根树的公式并探索其单峰性，通过反单峰延迟函数和特定像集延迟函数提供例子与推测。
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 algebraic, topological, and geometric combinatorics inequalities among the coefficients of combinatorial polynomials are frequently studied. Recently a notion called the alternatingly increasing property, which is stronger than unimodality, was introduced. In this paper, we relate the alternatingly increasing property to real-rootedness of the symmetric decomposition of a polynomial to develop a systematic approach for proving the alternatingly increasing property for several classes of polynomials. We apply our results to strengthen and generalize real-rootedness, unimodality, and alternat…
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.
For a poset whose Hasse diagram is a rooted plane forest $F$, we consider the corresponding tree descent polynomial $A_F(q)$, which is a generating function of the number of descents of the labelings of $F$. When the forest is a path, $A_F(q)$ specializes to the classical Eulerian polynomial. We prove that the coefficient sequence of $A_F(q)$ is unimodal and that if $\{T_{n}\}$ is a sequence of trees with $|T_{n}| = n$ and maximal down degree $D_{n} = O(n^{0.5-ε})$ then the number of descents in a labeling of $T_{n}$ is asymptotically normal.
In the study of Kostka numbers and Catalan numbers, Kirillov posed a unimodality conjecture for the rectangular Narayana polynomials. We prove that the rectangular Narayana polynomials have only real zeros, and thereby confirm Kirillov's unimodality conjecture with the help of Newton's inequality. By using an equidistribution property between descent numbers and ascent numbers on ballot paths due to Sulanke and a bijection between lattice words and standard Young tableaux, we show that the rectangular Narayana polynomial is equal to the descent generating function on standard Young tableaux of…
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…
It is well known that the coefficients of the matching polynomial are unimodal. Unimodality of the coefficients (or their absolute values) of other graph polynomials have been studied as well. One way to prove unimodality is to prove real-rootedness.` Recently I. Beaton and J. Brown (2020) proved the for almost all graphs the coefficients of the domination polynomial form a unimodal sequence, and C. Barton, J. Brown and D. Pike (2020) proved that the forest polynomial (aka acyclic polynomial) is real-rooted iff $G$ is a forest. Let $\mathcal{A}$ be a graph property, and let $a_i(G)$ be the num…
On the Asymptotic Normality and Unimodality of Genus Distributions of Wheels
本文结合联合树模型与特征理论推导轮图亏格多项式的显式公式并证明其实根性，证实了轮图亏格分布的单峰性猜想，还肯定回答了Zhang等人提出的渐近正态性问题。
Given two graphs $G$ and $H$, assume that $\mathscr{C}=\{C_1,C_2,\ldots, C_q\}$ is a clique cover of $G$ and $U$ is a subset of $V(H)$. We introduce a new graph operation called the clique cover product, denoted by $G^{\mathscr{C}}\star H^U$, as follows: for each clique $C_i\in \mathscr{C}$, add a copy of the graph $H$ and join every vertex of $C_i$ to every vertex of $U$. We prove that the independence polynomial of $G^{\mathscr{C}}\star H^U$ $$I(G^{\mathscr{C}}\star H^U;x)=I^q(H;x)I(G;\frac{xI(H-U;x)}{I(H;x)}),$$ which generalizes some known results on independence polynomials of corona and …
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。
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。
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。
Real-rootedness and ultra log-concavity of rank-two matroid Ehrhart $h^*$-polynomials
本文证明秩二拟阵在特定条件下Ehrhart $h^*$-多项式的实根性，给出反例否证猜想，并证明超对数凹性及单峰性。
Determining whether two graphs are isomorphic is an important and difficult problem in graph theory. One way to make progress towards this problem is by finding and studying graph invariants that distinguish large classes of graphs. Stanley conjectured that his chromatic symmetric function distinguishes all trees, which has remained unresolved. Recently, Hasebe and Tsujie introduced an analogue of Stanley's function for posets, called the strict order quasisymmetric function, and proved that it distinguishes all rooted trees. In this paper, we devise a procedure to explicitly reconstruct a roo…
On the real-rootedness of the Eulerian transformation
本文证明 Brändén–Jochemko 关于非负系数多项式 Eulerian 变换实根性的猜想，并借助单纯复形一致三角剖分推广变换，建立强单峰性与 gamma 正性。
Real-rootedness of the $Z$-polynomials of sparse paving matroids and strict interlacing for uniform matroids
本文证明稀疏铺砌拟阵的$Z$-多项式仅有负实零点，证实实根性猜想，并证明固定余秩下均匀拟阵的$Z$-多项式严格交错，方法是通过$\gamma$-多项式转移性质。
We introduce and study s-lecture hall P-partitions which is a generalization of s-lecture hall partitions to labeled (weighted) posets. We provide generating function identities for s-lecture hall P-partitions that generalize identities obtained by Savage and Schuster for s-lecture hall partitions, and by Stanley for P-partitions. We also prove that the corresponding (P,s)-Eulerian polynomials are real-rooted for certain pairs (P,s), and speculate on unimodality properties of these polynomials.
There is currently a growing interest in understanding which lattice simplices have unimodal local $h^\ast$-polynomials (sometimes called box polynomials); specifically in light of their potential applications to unimodality questions for Ehrhart $h^\ast$-polynomials. In this note, we compute a general form for the local $h^\ast$-polynomial of a well-studied family of lattice simplices whose associated toric varieties are weighted projective spaces. We then apply this formula to prove that certain such lattice simplices, whose combinatorics are naturally encoded using common systems of numerat…
Algebraic and Combinatorial Stability of Independence Polynomials in Iterated Strong Products of Cycles
本文通过代数和组合方法研究了迭代强积循环图中独立多项式的稳定性，揭示了其对数凹性和单峰性特性，并探讨了其在图论和统计力学中的应用。
We refine a technique used in a paper by Schur on real-rooted polynomials. This amounts to an extension of a theorem of Wagner on Hadamard products of Toeplitz matrices. We also apply our results to polynomials for which the Neggers-Stanley Conjecture is known to hold. More precisely, we settle interlacing properties for $E$-polynomials of series-parallel posets and column-strict labelled Ferrers posets.
Real-rooted Eulerian polynomials from permutations, words, and paths
本文研究六个Eulerian型多项式族，证明错排下降多项式实根性等猜想，并加强或解决多个相关实根性与交错性问题。
In this paper, we present monotone sequences of lower and upper bounds on the Perron value of a nonngeative matrix, and we study their strict monotonicity. Using those sequences, we provide two combinatorial applications. One is to improve bounds on Perron values of rooted trees in combinatorial settings, in order to find characteristic sets of trees. The other is to generate log-concave and log-convex sequences through the monotone sequences.
Extremal ranks of unlabeled multifurcating rooted trees in a bijective encoding by the positive integers
针对无标签多叉有根树，研究固定叶子数下获得最大和最小排名的树结构，发现最大平衡树得最小排名，最小平衡树得最大排名，并推导了递归关系及增长渐近。
A transversal in a rooted tree is any set of nodes that meets every path from the root to a leaf. We let c(T,k) denote the number of transversals of size k in a rooted tree T. We define a partial order on the set of all rooted trees with n nodes by saying that a tree T succeeds a tree T' if c(T,k) is at least c(T',k) for all k and strictly greater than c(T',k) for at least one k. We prove that, for every choice of positive integers d and n, the set of all rooted trees on n nodes where each node has at most d children has a unique minimal element with respect to this partial order and we descri…
Let $A$ be a commutative $k$-algebra over a field of $k$ and $Ξ$ a linear operator defined on $A$. We define a family of $A$-valued invariants $Ψ$ for finite rooted forests by a recurrent algorithm using the operator $Ξ$ and show that the invariant $Ψ$ distinguishes rooted forests if (and only if) it distinguishes rooted trees $T$, and if (and only if) it is {\it finer} than the quantity $α(T)=|\text{Aut}(T)|$ of rooted trees $T$. We also consider the generating function $U(q)=\sum_{n=1}^\infty U_n q^n$ with $U_n =\sum_{T\in \bT_n} \frac 1{α(T)} Ψ(T)$, where $\bT_n$ is the set of rooted trees …
Closing Trees into Unicyclic Counterexamples
本文通过一种基于家族的方法，构造了单循环图，其独立多项式是单峰但非对数凹的。通过分离闭包多项式为主导卷积项和实根修正项，证明了非对数凹性和单峰性，从而得到一个显式的无限家族。
Let $K$ be any unital commutative $\bQ$-algebra and $W$ any non-empty subset of $\bN^+$. Let $z=(z_1, ..., z_n)$ be commutative or noncommutative free variables and $t$ a formal central parameter. % Denote uniformly by $\kzz$ and $\kttzz$ the formal power series algebras % of $z$ over $K$ and $K[[t]]$, respectively. Let $\cDazz$ $(α\geq 1)$ be the unital algebra generated by the differential operators of $\kzz$ which increase the degree in $z$ by at least $α-1$ and $ \ataz $ the group of automorphisms $F_t(z)=z-H_t(z)$ of $\kttzz$ with $o(H_t(z))\geq α$ and $H_{t=0}(z)=0$. First, we study a co…
In this paper we present a CAT generation algorithm for Dyck paths with a fixed length n. It is the formalization of a method for the exhaustive generation of this kind of paths which can be described by means of two equivalent strategies. The former is described by a rooted tree, the latter lists the paths by means of three operations which, as we are going to see, are equivalent to visit the tree. These constructions are strictly connected with ECO method and can be encoded by a rule, very similar to the succession rule in ECO, with a finite number of labels for each n. Moreover with a sligh…
Noetherian forms of free non-symmetric operads
本文研究带标签有限有根有序树范畴，通过正合性性质刻画其同构类，并证明附加严格初始对象后具有诺特形式。
Refined Horton-Strahler numbers I: a discrete bijection
本文提出一种新的双射证明，通过引入插值完全二叉树的树列来定义精细化Horton-Strahler数，证明了具有特定内部顶点数和该数的满二叉树与特定长度和高度的Dyck路径数量相同。
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