Michele D'Adderio, Anton Mellit
We prove a compositional refinement of the Delta conjecture (rise version) of Haglund, Remmel and Wilson (2018) for $Δ_{e_{n-k-1}}'e_n$ which was stated by D'Adderio, Iraci and Vanden Wyngaerd (2020) in terms of Theta operators.
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The nonsymmetric compositional Delta theorem
本文基于Blasiak等人对标志性LLT多项式理论，推导出非对称身份，并引入非对称∇和τ*算子，提出新的算子形式，通过Weyl对称化恢复原始组合Delta定理，并提出关于稳定原子正性猜想。
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.
In 2008, Haglund, Morse and Zabrocki formulated a Compositional form of the Shuffle Conjecture of Haglund et al. In very recent work, Gorsky and Negut by combining their discoveries with the work of Schiffmann-Vasserot on the symmetric function side and the work of Hikita and Gorsky-Mazin on the combinatorial side, were led to formulate an infinite family of conjectures that extend the original Shuffle Conjecture of Haglund et al. In fact, they formulated one conjecture for each pair (m,n) of coprime integers. This work of Gorsky-Negut leads naturally to the question as to where the Compositio…
We define a module that is an extension of the diagonal harmonics and whose graded Frobenius characteristic is conjectured to be the symmetric function expression which appears in `the Delta conjecture' of Haglund, Remmel and Wilson [arXiv:1509.07058].
Inspired by [Qiu, Wilson 2019] and [D'Adderio, Iraci, Vanden Wyngaerd 2019 - Delta Square], we formulate a generalised Delta square conjecture (valley version). Furthermore, we use similar techniques as in [Haglund, Sergel 2019] to obtain a schedule formula for the combinatorics of our conjecture. We then use this formula to prove that the (generalised) valley version of the Delta conjecture implies our (generalised) valley version of the Delta square conjecture. This implication broadens the argument in [Sergel 2016], relying on the formulation of the touching version in terms of the $Θ_f$ op…
We prove the cases q=0 and t=0 of the generalized Delta conjecture of Haglund, Remmel and Wilson involving the symmetric function $Δ_{h_m}Δ_{e_{n-k-1}}'e_n$. Our theorem generalizes recent results by Garsia, Haglund, Remmel and Yoo. This proves also the case q=0 of our recent generalized Delta square conjecture.
The Delta Conjecture of Haglund, Remmel, and Wilson is a recent generalization of the Shuffle Conjecture in the field of diagonal harmonics. In this paper we give evidence for the Delta Conjecture by proving a pair of conjectures of Wilson and Haglund-Remmel-Wilson which give equidistribution results for statistics related to inversion count and major index on objects related to ordered set partitions. Our results generalize the famous result of MacMahon that major index and inversion number share the same distribution on permutations.
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.
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。
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…
We use a weight-preserving, sign-reversing involution to find a combinatorial expansion of $Δ_{e_k} e_n$ at $q=1$ in terms of the elementary symmetric function basis. We then use a weight-preserving bijection to prove the Delta Conjecture at $q=1$. The method of proof provides a variety of structures which can compute the inner product of $Δ_{e_k} e_n|_{q=1}$ with any symmetric function.
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.
Our main result here is that the specialization at $t=1/q$ of the $Q_{km,kn}$ operators studied in [4] may be given a very simple plethystic form. This discovery yields elementary and direct derivations of several identities relating these operators at $t=1/q$ to the Rational Compositional Shuffle conjecture of [3]. In particular we show that if $m,n $ and $k$ are positive integers and $(m,n)$ is a coprime pair then $$ q^{(km-1)(kn-1)+k-1\over 2} Q_{km,kn}(-1)^{kn}\Big|_{t=1/q} \,=\, \textstyle{[k]_q\over [km]_q} e_{km}\big[ X[km]_q\big] $$ where as customarily, for any integer $s \geq 0$ and …
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…
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…
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 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。
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_{…
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 …
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 …
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…
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。
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。
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…
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…
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…
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…
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…
We introduce a $q,t$-enumeration of Dyck paths which are forced to touch the main diagonal at specific points and forbidden to touch elsewhere and conjecture that it describes the action of the Macdonald theory $\nabla$ operator applied to a Hall-Littlewood polynomial. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the "shuffle conjecture" (Duke J. Math. $\mathbf {126}$ (2005), pp. 195-232) for $\nabla e_n[X]$. We bring to light that certain generalized Hall-Littlewood polynomials indexed by compositions are the building blocks for the a…
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。
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。
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)…
We prove here that the polynomial <nabla(C_p(1)), e_a h_b h_c> q, t-enumerates, by the statistics dinv and area, the parking functions whose supporting Dyck path touches the main diagonal according to the composition p of size a + b + c and have a reading word which is a shuffle of one decreasing word and two increasing words of respective sizes a, b, c. Here Cp(1) is a rescaled Hall-Littlewood polynomial and "nabla" is the Macdonald eigenoperator introduced in [1]. This is our latest progress in a continued effort to settle the decade old shuffle conjecture of [14]. It includes as special cas…
In the early 2000's the first and second named authors worked for a period of six years in an attempt of proving the Compositional Shuffle Conjecture [1]. Their approach was based on the discovery that all the Combinatorial properties predicted by the Compositional Shuffle Conjecture remain valid for each family of Parking Functions with prescribed diagonal cars. The validity of this property was reduced to the proof of a functional equation satisfied by a Catalan family of univariate polynomials. The main result in this paper is a proof of this functional equation. The Compositional Shuffle C…
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 $…
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|…
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。
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。
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.
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…
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.
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。
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…
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。
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 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。
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…
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。
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.
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 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。
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.
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。
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…
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…
We present a proof of the compositional shuffle conjecture, which generalizes the famous shuffle conjecture for the character of the diagonal coinvariant algebra. We first formulate the combinatorial side of the conjecture in terms of certain operators on a graded vector space $V_*$ whose degree zero part is the ring of symmetric functions $Sym[X]$ over $\mathbb{Q}(q,t)$. We then extend these operators to an action of an algebra $\tildeÅ$ acting on this space, and interpret the right generalization of the $\nabla$ using an involution of the algebra which is antilinear with respect to the conju…
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。
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…
We prove that, for any prime $p$ and positive integer $r$ with $p^r>2$, the number of multinomial coefficients such that $$ {k\choose k_1,k_2,\ldots,k_n}=p^r,\quad \text{and}\quad k_1+2k_2+\cdots+nk_n=n, $$ is given by $$ δ_{p^r,\,k}\left(\left\lfloor\frac{n-1}{p^r-1}\right\rfloor -δ_{0,\,n\bmod p^r} \right), $$ where $δ_{i,j}$ is the Kronecker delta and $\lfloor x\rfloor$ stands for the largest integer not exceeding $x$. This confirms a recent conjecture of Mircea Merca.
The Theta Conjecture
本文提出双变量Theta猜想，通过定义带标记Dyck路径上的对角反转统计量，细化并支持已证明的单变量Theta猜想，并证明其Schröder情形。
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