The Delta Conjecture at $q=1$

On Delta Graphs and Delta Conjecture

In this paper we define two infinite families of graphs called C-$δ$ graphs and $δ$- graph and prove that $δ$-graphs satisfy $δ$ conjecture. Also we introduce a family of C-$δ$ graphs from which we can identify $δ$ graphs as their complements. Finally we give a list of C-$δ$ graphs and the relationship with their minimum semidefinite rank.

方形路径猜想与一元 Delta 猜想的矩形类似物

Rectangular analogues of the square paths conjecture and the univariate Delta conjecture

本文提出方形路径猜想的矩形类似物以扩展 shuffle 猜想的矩形侧，并给出组合对象与统计量作为 Delta 及 Delta 方形猜想矩形扩展的初步基础，同时证明了互素情形下的新矩形路径猜想。

"Pushing" our way from the valley Delta to the generalised valley Delta

In [Haglund, Remmel, Wilson 2018] the authors state two versions of the so called Delta conjecture, the rise version and the valley version. Of the former, they also give a more general statement in which zero labels are also allowed. In [Qiu, Wilson 2020], the corresponding generalisation of the valley version is also formulated. In [D'Adderio, Iraci, Vanden Wyngaerd 2020], the authors use a pushing algorithm to prove the generalised version of the shuffle theorem. An extension of that argument is used in [Iraci, Vanden Wyngaerd 2020] to formulate a valley version of the (generalised) Delta s…

A Proof of Delta Conjecture

By finding orthogonal representation for a family of simple connected called $δ$-graphs it is possible to show that $δ$-graphs satisfy delta conjecture. An extension of the argument to graphs of the form $\overline{P_{Δ(G)+2}\sqcup G}$ where $P_{Δ(G)+2}$ is a path and $G$ is a simple connected graph it is possible to find an orthogonal representation of $\overline{P_{Δ(G)+2}\sqcup G}$ in $\mathbb{R}^{Δ(G)+1}$. As a consequence we prove delta conjecture.

The Delta square conjecture

We conjecture a formula for the symmetric function $\frac{[n-k]_t}{[n]_t}Δ_{h_m}Δ_{e_{n-k}}ω(p_n)$ in terms of decorated partially labelled square paths. This can be seen as a generalization of the square conjecture of Loehr and Warrington (Loehr, Warrington 2007), recently proved by Sergel (Sergel 2017) after the breakthrough of Carlsson and Mellit (Carlsson, Mellit 2018). Moreover, it extends to the square case the combinatorics of the generalized Delta conjecture of Haglund, Remmel and Wilson (Haglund, Remmel, Wilson 2015), answering one of their questions. We support our conjecture by prov…

A proof of the Delta Conjecture when $q=0$

In [The Delta Conjecture, Trans. Amer. Math. Soc., to appear] Haglund, Remmel, Wilson introduce a conjecture which gives a combinatorial prediction for the result of applying a certain operator to an elementary symmetric function. This operator, defined in terms of its action on the modified Macdonald basis, has played a role in work of Garsia and Haiman on diagonal harmonics, the Hilbert scheme, and Macdonald polynomials [A. M. Garsia and M. Haiman. A remarkable $q,t$-Catalan sequence and $q$-Lagrange inversion, J. Algebraic Combin. 5 (1996), 191--244], [M. Haiman, Vanishing theorems and char…

A proof of the $4$-variable Catalan polynomial of the Delta conjecture

In The Delta Conjecture (arxiv:1509.07058), Haglund, Remmel and Wilson introduced a four variable $q,t,z,w$ Catalan polynomial, so named because the specialization of this polynomial at the values $(q,t,z,w) = (1,1,0,0)$ is equal to the Catalan number $\frac{1}{n+1}\binom{2n}{n}$. We prove the compositional version of this conjecture (which implies the non-compositional version) that states that the coefficient of $s_{r,1^{n-r}}$ in the expression $Δ_{h_\ell} \nabla C_α$ is equal to a weighted sum over decorated Dyck paths.

Graph Labeling with Distance Conditions and the Delta Squared Conjecture

We give bounds on the L(2,1)-labeling number of a simple graph in terms of its order and its maximum degree. We also describe an infinite class of graphs of which the elements have the highest L(2,1)-labeling numbers in terms of their maximum degrees of any known infinite class of graphs.

Smirnov字与Delta猜想

Smirnov words and the Delta Conjectures

本文用分段Smirnov字给出了对称函数的组合解释，该函数与对角余不变环及Delta猜想相关，并在$t=0$处证明了统一的Delta定理。

Theta operators, refined Delta conjectures, and coinvariants

We introduce the family of Theta operators $Θ_f$ indexed by symmetric functions $f$ that allow us to conjecture a compositional refinement of the Delta conjecture of Haglund, Remmel and Wilson for $Δ_{e_{n-k-1}}'e_n$. We show that the $4$-variable Catalan theorem of Zabrocki is precisely the Schröder case of our compositional Delta conjecture, and we show how to relate this conjecture to the Dyck path algebra introduced by Carlsson and Mellit, extending one of their results. Again using the Theta operators, we conjecture a touching refinement of the generalized Delta conjecture for $Δ_{h_m}Δ_{…

The Delta Conjecture

We conjecture two combinatorial interpretations for the symmetric function $Δ_{e_k} e_n$, where $Δ_f$ is an eigenoperator for the modified Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler. Both interpretations can be seen as generalizations of the Shuffle Conjecture of Haglund, Haiman, Remmel, Loehr, and Ulyanov, which was proved recently by Carlsson and Mellit. We show how previous work of the third author on Tesler matrices and ordered set partitions can be used to verify several cases of our conjectures. Furthermore, we use a reciprocity identity and LLT polynomials to …

The Schröder case of the generalized Delta conjecture

We prove the Schröder case, i.e. the case $\langle \cdot,e_{n-d}h_d \rangle$, of the conjecture of Haglund, Remmel and Wilson (Haglund et al. 2018) for $Δ_{h_m}Δ_{e_{n-k-1}}'e_n$ in terms of decorated partially labelled Dyck paths, which we call \emph{generalized Delta conjecture}. This result extends the Schröder case of the Delta conjecture proved in (D'Adderio, Vanden Wyngaerd 2017), which in turn generalized the $q,t$-Schröder of Haglund (Haglund 2004). The proof gives a recursion for these polynomials that extends the ones known for the aforementioned special cases. Also, we give another …

Decorated Dyck paths, the Delta conjecture, and a new q,t-square

We discuss the combinatorics of the decorated Dyck paths appearing in the Delta conjecture framework of Haglund, Remmel and Wilson (2015) and Zabrocki (2016), by introducing two new statistics, bounce and bounce'. We then provide plethystic formulae for their q,t-enumerators, by proving in this way a decorated version of Haglund's q,t-Schroeder theorem, answering a question in Haglund, Remmel and Wilson (2015). In particular we provide both an algebraic and a combinatorial explanation of a symmetry conjectured in Haglund, Remmel and Wilson (2015) and Zabrocki (2016). Finally, we show a link be…

图补猜想弱版本及Delta猜想的部分结果

The Weak Version of the Graph Complement Conjecture and Partial Results for the Delta Conjecture

本文利用Mader经典结果建立了图补猜想的弱版本，提出了$δ$-猜想的极值解，结合图退化改进了界限，并列出了Colin de Verdière数半正定变体的猜想界限。

山谷Delta猜想的一些推论

Some consequences of the valley Delta conjectures

本文针对山谷版Delta猜想研究较少的问题，证明了其Schröder情形、平方版本的Schröder情形及扩展版本的Catalan情形，并在对称性假设下推导出平方版本的Catalan情形。

Twist polynomials of delta-matroids

Recently, Gross, Mansour and Tucker introduced the partial duality polynomial of a ribbon graph and posed a conjecture that there is no orientable ribbon graph whose partial duality polynomial has only one non-constant term. We found an infinite family of counterexamples for the conjecture and showed that essentially these are the only counterexamples. This is also obtained independently by Chumutov and Vignes-Tourneret and they posed a problem: it would be interesting to know whether the partial duality polynomial and the related conjectures would make sence for general delta-matroids. In thi…

Exploring a Delta Schur Conjecture

In \cite{HRW15}, Haglund, Remmel, Wilson state a conjecture which predicts a purely combinatorial way of obtaining the symmetric function $Δ_{e_k}e_n$. It is called the Delta Conjecture. It was recently proved in \cite{GHRY} that the Delta Conjecture is true when either $q=0$ or $t=0$. In this paper we complete a work initiated by Remmel whose initial aim was to explore the symmetric function $Δ_{s_ν} e_n$ by the same methods developed in \cite{GHRY}. Our first need here is a method for constructing a symmetric function that may be viewed as a "combinatorial side" for the symmetric function $Δ…

Decorated Dyck paths, polyominoes, and the Delta conjecture

We discuss the combinatorics of decorated Dyck paths and decorated parallelogram polyominoes, extending to the decorated case the main results of both [Haglund 2004] and [Aval et al. 2014]. This settles in particular the cases $\langle\cdot,e_{n-d}h_d\rangle$ and $\langle\cdot,h_{n-d}h_d\rangle$ of the Delta conjecture of Haglund, Remmel and Wilson (2018). Along the way, we introduce some new statistics, formulate some new conjectures, prove some new identities of symmetric functions, and answer a few open problems in the literature (e.g. from [Haglund et al. 2018], [Zabrocki 2016], [Aval et a…

扩展Delta猜想的谷型版本

The valley version of the Extended Delta Conjecture

本文提出扩展Delta猜想的谷型版本，并通过研究扩展有序多重集划分的组合学，证明当t或q等于0时该猜想与上升版本等价。

Parallelogram polyominoes, partially labelled Dyck paths, and the Delta conjecture

We introduce area, bounce and dinv statistics on decorated parallelogram polyominoes, and prove that some of their q,t-enumerators match $\langle Δ_{h_m} e_{n+1}, s_{k+1,1^{n-k}} \rangle$, extending in this way the work in (Aval et al. 2014). Also, we provide a bijective connection between decorated parallelogram polyominoes and decorated labelled Dyck paths, which allows us to prove the combinatorial interpretation of the coefficient $\langleΔ_{e_{m+n-k-1}}'e_{m+n}, h_m h_n \rangle$ predicted by the Delta conjecture in (Haglund et al. 2015). Finally, we define a statistic pmaj on partially la…

Parallelogram polyominoes, partially labelled Dyck paths, and the Delta conjecture (FULL VERSION)

We introduce area, bounce and dinv statistics on decorated parallelogram polyominoes, and prove that some of their q,t-enumerators match $\langle Δ_{h_m} e_{n+1},s_{k+1,1^{n-k}}\rangle$, extending in this way the work in (Aval et al. 2014). Also, we provide a bijective connection between decorated parallelogram polyominoes and decorated labelled Dyck paths, which allows us to prove the combinatorial interpretation of the coefficient $\langle Δ_{e_{m+n-k-1}}'e_{m+n},h_m h_n\rangle$ predicted by the Delta conjecture in (Haglund et al. 2015). Finally, we define a statistic pmaj on partially label…

Schedules and the Delta Conjecture

In a recent preprint, Carlsson and Oblomkov (2018) obtain a long sought after monomial basis for the ring $\operatorname{DR}_n$ of diagonal coinvariants. Their basis is closely related to the "schedules" formula for the Hilbert series of $\operatorname{DR}_n$ which was conjectured by the first author and Loehr (2005) and first proved by Carlsson and Mellit (2018), as a consequence of their proof of the famous Shuffle Conjecture. In this article we obtain a schedules formula for the combinatorial side of the Delta Conjecture, a conjecture introduced by the first author, Remmel and Wilson (2018)…

无限乘积实幂的符号模式：Schlosser 与 Zhou 四个猜想的解决

Sign patterns of real powers of infinite products: resolution of four conjectures of Schlosser and Zhou

本文完全解决了Schlosser与Zhou关于无限乘积实幂系数符号模式的四个猜想，精确刻画了三个猜想失败的确切区间，并揭示了圆法主导项消失的失败机制。

Ordered set partitions, generalized coinvariant algebras, and the Delta Conjecture

The symmetric group $\mathfrak{S}_n$ acts on the polynomial ring $\mathbb{Q}[\mathbf{x}_n] = \mathbb{Q}[x_1, \dots, x_n]$ by variable permutation. The invariant ideal $I_n$ is the ideal generated by all $\mathfrak{S}_n$-invariant polynomials with vanishing constant term. The quotient $R_n = \frac{\mathbb{Q}[\mathbf{x}_n]}{I_n}$ is called the coinvariant algebra. The coinvariant algebra $R_n$ has received a great deal of study in algebraic and geometric combinatorics. We introduce a generalization $I_{n,k} \subseteq \mathbb{Q}[\mathbf{x}_n]$ of the ideal $I_n$ indexed by two positive integers $…

A flag variety for the Delta Conjecture

The Delta Conjecture of Haglund, Remmel, and Wilson predicts the monomial expansion of the symmetric function $Δ'_{e_{k-1}} e_n$, where $k \leq n$ are positive integers and $Δ'_{e_{k-1}}$ is a Macdonald eigenoperator. When $k = n$, the specialization $Δ'_{e_{n-1}} e_n|_{t = 0}$ is the Frobenius image of the graded $S_n$-module afforded by the cohomology ring of the {\em flag variety} consisting of complete flags in $\mathbb{C}^n$. We define and study a variety $X_{n,k}$ which carries an action of $S_n$ whose cohomology ring $H^{\bullet}(X_{n,k})$ has Frobenius image given by $Δ'_{e_{k-1}} e_n|…

Computing Node Polynomials for Plane Curves

According to the Göttsche conjecture (now a theorem), the degree N^{d, delta} of the Severi variety of plane curves of degree d with delta nodes is given by a polynomial in d, provided d is large enough. These "node polynomials" N_delta(d) were determined by Vainsencher and Kleiman-Piene for delta <= 6 and delta <= 8, respectively. Building on ideas of Fomin and Mikhalkin, we develop an explicit algorithm for computing all node polynomials, and use it to compute N_delta(d) for delta <= 14. Furthermore, we improve the threshold of polynomiality and verify Göttsche's conjecture on the optimal th…

A proof of the Extended Delta Conjecture

We prove the Extended Delta Conjecture of Haglund, Remmel, and Wilson, a combinatorial formula for $Δ_{h_l}Δ' _{e_k} e_{n}$, where $Δ' _{e_k}$ and $Δ_{h_l}$ are Macdonald eigenoperators and $e_n$ is an elementary symmetric function. We actually prove a stronger identity of infinite series of $GL_m$ characters expressed in terms of LLT series. This is achieved through new results in the theory of the Schiffmann algebra and its action on the algebra of symmetric functions.

Rise Delta 定理的一个组合斜化公式

A combinatorial skewing formula for the Rise Delta Theorem

本文证明了 Delta Conjecture 中的对称函数可通过 Schur 斜化算子从 Rational Shuffle Theorem 获得，并给出该斜化恒等式的纯组合证明，从而为 Rise Delta Theorem 提供了新证明。

山谷Delta猜想的交换恒等式与对称切片

Exchange identities and symmetric slices of the valley Delta conjecture

本文证明了山谷Delta猜想中生成函数在t^0和t^1项的对称性，通过相邻交换恒等式并引入q-变形双变量代数中的转移算子演算，将交换简化为两个标量对称级数恒等式。

Hall-Littlewood expansions of Schur delta operators at $t = 0$

For any Schur function $s_ν$, the associated {\em delta operator} $Δ'_{s_ν}$ is a linear operator on the ring of symmetric functions which has the modified Macdonald polynomials as an eigenbasis. When $ν= (1^{n-1})$ is a column of length $n-1$, the symmetric function $Δ'_{e_{n-1}} e_n$ appears in the Shuffle Theorem of Carlsson-Mellit. More generally, when $ν= (1^{k-1})$ is any column the polynomial $Δ'_{e_{k-1}} e_n$ is the symmetric function side of the Delta Conjecture of Haglund-Remmel-Wilson. We give an expansion of $ωΔ'_{s_ν} e_n$ at $t = 0$ in the dual Hall-Littlewood basis for any part…

From $q$-Stirling numbers to the Delta Conjecture: a viewpoint from vincular patterns

The distribution of certain Mahonian statistic (called $\mathrm{BAST}$) introduced by Babson and Steingrímsson over the set of permutations that avoid vincular pattern $1\underline{32}$, is shown bijectively to match the distribution of major index over the same set. This new layer of equidistribution is then applied to give alternative interpretations of two related $q$-Stirling numbers of the second kind, studied by Carlitz and Gould. Moreover, extensions to an Euler-Mahonian statistic over ordered set partitions, and to statistics over ordered multiset partitions present themselves naturall…

关于Las Vergnas的一个猜想

On a conjecture of Las Vergnas

本文证明Las Vergnas关于平面图对角Tutte多项式在z=-1处导数可被2的幂整除的猜想成立，通过引入更强的LV性质并分析Δ-w约化图类。

关于 Gross、Mansour 和 Tucker 对 $\Delta$-拟阵的一个猜想的注记

On a conjecture of Gross, Mansour and Tucker for $Δ$-matroids

本文研究 Gross、Mansour 和 Tucker 关于 delta-拟阵部分对偶多项式的猜想，证明偶非二元 delta-拟阵均无改变宽度的扭转，扩展了 Yan 和 Jin 对偶正规二元情形的结论。

Delta 猜想的一个几何解释

A geometric interpretation of the Delta Conjecture

本文通过引入仿射 Δ-Springer 纤维簇 Y_{n,k}，为 Delta 猜想及整数斜率情形的有理 Shuffle 定理提供了统一的几何解释，并借助 Schur 斜移算子公式完成证明。

Derangements in finite classical groups for actions related to extension field and imprimitive subgroups and the solution of the Boston-Shalev conjecture

This is the fourth paper in a series. We prove a conjecture made independently by Boston et al and Shalev. The conjecture asserts that there is an absolute positive constant delta such that if G is a finite simple group acting transitively on a set of size n > 1, then the proportion of derangements in G is greater than delta. We show that with possibly finitely many exceptions, one can take delta = .016. Indeed, we prove much stronger results showing that for many actions, the proportion of derangements goes to 1 as n increases and prove similar results for families of permutation representati…

Delta 猜想在 q=-1 处的组合学

Combinatorics of the Delta conjecture at q=-1

本文证明了 Delta 猜想在 q=-1 处的观察，即 q,t-analog 变为 Euler 数的 t-analog，通过 Delta 算子与排列峰谷组合统计量的一般性陈述，并利用 parking function 的 schedule numbers 作为关键工具。

A proof of the Theta Operator Conjecture

In the context of the (generalized) Delta Conjecture and its compositional form, D'Adderio, Iraci, and Wyngaerd recently stated a conjecture relating two symmetric function operators, $D_k$ and $Θ_k$. We prove this Theta Operator Conjecture, finding it as a consequence of the five-term relation of Mellit and Garsia. As a result, we find surprising ways of writing the $D_k$ operators.

delta定理：忠实正交图表示的维度界

The Delta Theorem: a dimension bound for faithful orthogonal graph representations

本文证明了Maehara的猜想及相关猜想，通过概率构造和运算符推理，展示了图的忠实正交表示的维度界及SAP属性。

On a conjecture of Gowers and Long

We show that rounding to a delta-net in SO(3) is not close to a group operation, thus confirming a conjecture of Gowers and Long.

Triangular Ladders $P_{d,2}$ are $e$-positive

In 1995 Stanley conjectured that the chromatic symmetric functions of the graphs $P_{d,2}$, which we call triangular ladders, were $e$-positive. In this paper we confirm this conjecture, which is also an unsolved case of the celebrated $(3+1)$-free conjecture. Our method is to follow the generalization of the chromatic symmetric functions by Gebhard and Sagan to symmetric functions in non-commuting variables. These functions satisfy a deletion-contraction property unlike the chromatic symmetric function in commuting variables. We do this by proving a new signed combinatorial formula for \emph{…

$\Delta$-Springer 模与 Delta 猜想的余荷与倾斜公式

Cocharge and skewing formulas for $Δ$-Springer modules and the Delta Conjecture

本文证明 Delta Conjecture 在 $t=0$ 时的对称函数是作用于 Hall-Littlewood 多项式的倾斜算子，将其推广至 $\Delta$-Springer 模，并基于 battery-powered tableau 给出几何与组合证明。

Springer纤维与t=0时的Delta猜想

Springer fibers and the Delta Conjecture at $t=0$

本研究引入推广A型Springer纤维的Δ-Springer簇，给出其上同调环的显式表示与对称群作用，为t=0时的Delta猜想提供几何实现，并推广了对角幂零矩阵概形的相关结果。

A minimaj-preserving crystal on ordered multiset partitions

We provide a crystal structure on the set of ordered multiset partitions, which recently arose in the pursuit of the Delta Conjecture. This conjecture was stated by Haglund, Remmel and Wilson as a generalization of the Shuffle Conjecture. Various statistics on ordered multiset partitions arise in the combinatorial analysis of the Delta Conjecture, one of them being the minimaj statistic, which is a variant of the major index statistic on words. Our crystal has the property that the minimaj statistic is constant on connected components of the crystal. In particular, this yields another proof of…

Staircase skew Schur functions are Schur P-positive

We prove Stanley's conjecture that, if delta_n is the staircase shape, then the skew Schur functions s_{delta_n / mu} are non-negative sums of Schur P-functions. We prove that the coefficients in this sum count certain fillings of shifted shapes. In particular, for the skew Schur function s_{delta_n / delta_{n-2}}, we discuss connections with Eulerian numbers and alternating permutations.

A Near Proof of Weak Graph Positivity, A New Property of Random Regular Graphs

One deals with r-regular bipartite graphs with 2n vertices. In a previous paper Butera, Pernici, and the author have introduced a quantity d(i), a function of the number of i-matchings, and conjectured that as n goes to infinity the fraction of graphs that satisfy Delta^k d(i) for all k and i, approaches 1. Here Delta is the finite difference operator. This conjecture we called the 'graph positivity conjecture'. In this paper it is formally shown that for each i and k the probability that Delta^k d(i) goes to 1 with n going to infinity. We call this weaker result the 'weak graph positivity con…

Delta and Theta Operator Expansions

We give an elementary symmetric function expansion for $MΔ_{m_γe_1}Πe_λ^{\ast}$ and $MΔ_{m_γe_1}Πs_λ^{\ast}$ when $t=1$ in terms of what we call $γ$-parking functions and lattice $γ$-parking functions. Here, $Δ_F$ and $Π$ are certain eigenoperators of the modified Macdonald basis and $M=(1-q)(1-t)$. Our main results in turn give an elementary basis expansion at $t=1$ for symmetric functions of the form $M Δ_{Fe_1} Θ_{G} J$ whenever $F$ is expanded in terms of monomials, $G$ is expanded in terms of the elementary basis, and $J$ is expanded in terms of the modified elementary basis $\{Πe_λ^\ast\…

落星：一个秋季装饰的有理洗牌定理

Falling stars: a fall-decorated rational shuffle theorem

本文提出秋季Delta定理和Delta平方猜想的有理模拟，通过定义新的dinv统计量，证明秋季装饰矩形Dyck路径的q,t-生成函数的对称函数公式。

A PROOF of Weak Graph Positivity, for a Large Range of the Parameters

One deals with r-regular bipartite graphs with 2n vertices. In a previous paper Butera, Pernici, and the author have introduced a quantity d(i), a function of the number of i-matchings, and conjectured that as n goes to infinity the fraction of graphs that satisfy Delta^k( d(i)) is non-negative, for all k and i, approaches 1. Here Delta is the finite difference operator. This conjecture we called the "graph positivity conjecture". "Weak graph positivity" is the conjecture that for each i and k the probability that Delta^k (d(i) is non-negative goes to 1 as n goes to infinity. Here we prove thi…

The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions

We consider the groundstate wavefunction of the quantum symmetric antiferromagnetic XXZ chain with open and twisted boundary conditions at $Δ=-{1/2}$, along with the groundstate wavefunction of the corresponding O($n$) loop model at $n=1$. Based on exact results for finite-size systems, sums involving the wavefunction components, and in some cases the largest component itself, are conjectured to be directly related to the total number of alternating sign matrices and plane partitions in certain symmetry classes.

一类连接正畴（positroids）、Theta猜想、Delta猜想和欧拉拟对称函数的代数族

A family of algebras that connects positroids, the Theta conjecture, the Delta conjecture, and Eulerian quasisymmetric functions

该研究定义了一类连接正畴、Theta猜想、Delta猜想和欧拉拟对称函数的代数族，推导了其弗罗贝尼乌斯特征的递推关系，构造了对应开子簇并建立了与正畴分层的联系。

A new polynomial on compositions of integers: on distinguishing caterpillars from their symmetric chromatic function

In this paper, we propose an algebraic approach to determine whether two non-isomorphic caterpillar trees can have the same symmetric function generalization of the chromatic polynomial. On the set of all composition on integers, we introduce: An operation, which we call composition product; and a combinatorial polynomial, which we call the composition-lattice polynomial or L-polynomial, that mimics the weighted graph polynomial of Noble and Welsh. We prove a unique irreducible factorization theorem and establish a connection between the L-polynomial of a composition and its irreducible factor…

$Δ$-Springer varieties and Hall-Littlewood polynomials

The $Δ$-Springer varieties are a generalization of Springer fibers introduced by Levinson, Woo, and the author that have connections to the Delta Conjecture from algebraic combinatorics. We prove a positive Hall-Littlewood expansion formula for the graded Frobenius characteristic of the cohomology ring of a $Δ$-Springer variety. We do this by interpreting the Frobenius characteristic in terms of counting points over a finite field $\mathbb{F}_q$ and partitioning the $Δ$-Springer variety into copies of Springer fibers crossed with affine spaces. As a special case, our proof method gives a geome…

广义网的Schur-正性

Schur-positivity for generalized nets

本文引入特殊边钩G-tabloid组合对象，通过构造符号反转映射推导出广义网等无爪图族的Schur系数递推关系，证明了其Schur-正性，为推进无爪图猜想提供了新方法。

A weighted sum over generalized Tesler matrices

We generalize previous definitions of Tesler matrices to allow negative matrix entries and negative hook sums. Our main result is an algebraic interpretation of a certain weighted sum over these matrices, which we call the Tesler function. Our interpretation uses a new class of symmetric function specializations which are defined by their values on Macdonald polynomials. As a result of this interpretation, we obtain a Tesler function expression for the Hall inner product $\langle Δ_f e_n, p_{1^{n}}\rangle$, where $Δ_f$ is the delta operator introduced by Bergeron, Garsia, Haiman, and Tesler. W…