Saharon Shelah
We prove that double exponentiation is an upper bound to Ramsey theorem for colouring of pairs when we want to predetermine the order of the differences of successive members of the homogeneous set.
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We give a product formula for the number of shifted plane partitions of shifted double staircase shape with bounded entries. This is the first new example of a family of shapes with a plane partition product formula in many years. The proof is based on the theory of lozenge tilings; specifically, we apply the "free boundary" Kuo condensation due to Ciucu.
It is well known that the Bell numbers represent the total number of partitions of an n-set. Similarly, the Stirling numbers of the second kind, represent the number of k-partitions of an n-set. In this paper we introduce a certain partitioning process that gives rise to a sequence of sets of "nested" partitions. We prove that at stage m, the cardinality of the resulting set will equal the m-th order Bell number. This set-theoretic interpretation enables us to make a natural definition of higher order Stirling numbers and to study the combinatorics of these entities. The cardinality of the ele…
MacMahon's Double Vision: Partition Diamonds Revisited
本文基于Stanley的P-划分理论，提出d重划分钻石的闭式公式，利用Euler-Mahonian多项式编码排列的下降统计信息。
Positive Weighted Partitions Generated by Double Series
本文研究生成函数为二重级数的加权整数分拆，通过建立 $q$-二重级数的闭式公式证明其系数非负，进而导出整数分拆间的不等式。
Sequences of odd length in strict partitions I: the combinatorics of double sum Rogers-Ramanujan type identities
本文将严格分拆的三元生成函数表为双重和 $q$-级数，据此为近期多个双重和 Rogers-Ramanujan 型恒等式提供分拆理论解释及 Franklin 型对合证明。
Symmetric noncrossing partitions of an annulus with double points
对于仿射型 $\tilde D$ 和 $\tilde B$ 的仿射 Coxeter 群，通过带一个或两个双点的环面上的对称非交叉划分，对绝对序中的区间 $[1,c]_T$ 进行建模，并推广到 McCammond 和 Sulway 定义的更大格。
On Cayley algorithm for double partition
针对双分拆问题中 Cayley 原始算法因条件不满足而失效的情况，本文提出其修改版本，以完成向标量分拆之和的归约。
The notion of exponential Dowling structures is introduced, generalizing Stanley's original theory of exponential structures. Enumerative theory is developed to determine the Möbius function of exponential Dowling structures, including a restriction of these structures to elements whose types satisfy a semigroup condition. Stanley's study of permutations associated with exponential structures leads to a similar vein of study for exponential Dowling structures. In particular, for the extended r-divisible partition lattice we show the Möbius function is, up to a sign, the number of permutations …
Classical Gončarov polynomials arose in numerical analysis as a basis for the solutions of the Gončarov interpolation problem. These polynomials provide a natural algebraic tool in the enumerative theory of parking functions. By replacing the differentiation operator with a delta operator and using the theory of finite operator calculus, Lorentz, Tringali and Yan introduced the sequence of generalized Gončarov polynomials associated to a pair $(Δ, Z)$ of a delta operator $Δ$ and an interpolation grid $Z$. Generalized Gončarov polynomials share many nice algebraic properties and have a connecti…
We focus on writing closed forms of generating functions for the number of partitions with gap conditions as double sums starting from a combinatorial construction. Some examples of the sets of partitions with gap conditions to be discussed here are the set of Rogers--Ramanujan, Göllnitz--Gordon, and little Göllnitz partitions. This work also includes finding the finite analogs of the related generating functions and the discussion of some related series and polynomial identities. Additionally, we present a different construction and a double sum representation for the products similar to the …
Double boxes and double dimers
本文通过建立双盒构型与六边形格上双二聚体构型的对应关系，并运用图形凝聚与双二聚体凝聚方法，给出了秩2 Donaldson-Thomas理论中双盒构型生成函数等于MacMahon平面分拆生成函数乘积的组合证明。
We use intermediate symplectic characters to give a proof and variations of Hopkins' conjecture, now proved by Hopkins and Lai, on the number of shifted plane partitions of shifted double staircase shape with bounded entries. In fact, we prove some character identities involving intermediate symplectic characters, and find generating functions for such shifted plane partitions. The key ingredients of the proof are a bialternant formula for intermediate symplectic characters, which interpolates between those for Schur functions and symplectic characters, and the Ishikawa-Wakayama minor-summatio…
Partition functors and universal exponential relations
本文引入划分函子并构造单子PD，建立了乘法与加法幂运算间的普适指数关系，统一了表示环中对称幂与Adams运算及Morava E-理论中对称幂与Hecke算子的经典关系。
Double-Partition Coloring problem: a unified approach for conflict-free coloring of hypergraphs and other coloring problems in graphs
本文研究双划分着色问题（DPCP），提出结构性质、启发式方法及增强的分支定价精确算法，并在CFCP实例上首次进行计算评估，与紧凑整数规划公式互补求解。
Equivariant quasisymmetry and noncrossing partitions
本文引入等变拟对称性定义，构造双Schur与双Schubert多项式的拟对称推广（双基本与双森林多项式），以非交叉划分替代$S_n$，并预示拟对称旗簇将主导其几何。\n
In this paper, we give a simple criterion to verify that functions of the form $e^g$ are in the Hayman class when $g$ is a power series with nonnegative coefficients. Thus, using the Hayman and Báez-Duarte formulas, we obtain asymptotics for the coefficients of generating functions that arise in many examples of set construction in analytic combinatorics. This new criterion greatly simplifies that obtained previously by the authors.
In this paper we use the Klazar-Marcus-Tardos method to prove that if a hereditary property of partitions P has super-exponential speed, then for every k-permutation pi, P contains the partition of [2k] with parts {i, pi(i) + k}, where 1 <= i <= k. We also prove a similar jump, from exponential to factorial, in the possible speeds of monotone properties of ordered graphs, and of hereditary properties of ordered graphs not containing large complete, or complete bipartite ordered graphs. Our results generalize the Stanley-Wilf Conjecture on the number of n-permutations avoiding a fixed permutati…
The mean values of non-homogeneously parameterized generating exponential are obtained and investigated for the periodic Heisenberg XX model. The norm-trace generating function of boxed plane partitions with fixed volume of their diagonal parts is obtained as N-particles average of the generating exponential. The generating function of self-avoiding walks of random turns vicious walkers is obtained in terms of the circulant matrices that leads to generalizations of the Ramus's identity. Under various specifications of the generating exponential, the N-particles averages arise for a set of inco…
The squish map and the $\text{SL}_2$ double dimer model
本文将平面分拆的粗化操作联系于 squish 映射，构造蜂窝图二聚体模型到粗图 $\ ext{SL}_2$ 双二聚体模型的保测映射，并据此得到平面分拆对的 $(-1)$-枚举生成函数。
Schur's partition theorem states that the number of partitions of n into distinct parts congruent 1, 2 (mod 3) equals the number of partitions of n into parts which differ by >= 3, where the inequality is strict if a part is a multiple of 3. We establish a double bounded refined version of this theorem by imposing one bound on the parts congruent 0,1 (mod 3) and another on the parts congruent 2 (mod 3), and by keeping track of the number of parts in each of the residue classes (mod 3). Despite the long history of Schur's theorem, our result is new, and extends earlier work of Andrews, Alladi-G…
We prove that the partition function for tripartite double-dimer configurations of a planar bipartite graph satisfies a recurrence related to the Desnanot-Jacobi identity from linear algebra. A similar identity for the dimer partition function was established nearly 20 years ago by Kuo and has applications to random tiling theory and the theory of cluster algebras. This work was motivated in part by the potential for applications in these areas. Additionally, we discuss an application to Donaldson-Thomas and Pandharipande-Thomas theory which will be the subject of a forthcoming paper. The proo…
Given integers i,j,k,L,M, we establish a new double bounded q-series identity from which the three parameter (i,j,k) key identity of Alladi-Andrews-Gordon for Goellnitz's (big) theorem follows if L, M tend to infinity. When L = M, the identity yields a strong refinement of Goellnitz's theorem with a bound on the parts given by L. This is the first time a bounded version of Goellnitz's (big) theorem has been proved. This leads to new bounded versions of Jacobi's triple product identity for theta functions and other fundamental identities.
Using ultrafilter techniques we show that in any partition of $\mathbb{N}$ into 2 cells there is one cell containing infinitely many exponential triples, i.e. triples of the kind $a,b,a^b$ (with $a,b>1$). Also, we will show that any multiplicative $IP^*$ set is an "exponential $IP$ set", the analogue of an $IP$ set with respect to exponentiation.
Asymptotic results on Klazar set partition avoidance
本文建立了 Klazar 意义下避免给定划分的划分数的渐近界，对块情形给出指数级精确答案，并推广到一般下界，同时提出图论重述及若干猜想。
Reduced Groebner Bases With Double Exponential Cardinality
本文提出了一种多项式族和准则，使得在特定单项式顺序下，缩减格罗伯纳基的基数呈现双指数增长特性。
Understanding the ratio of the partition sum to its Bethe approximation via double covers
本文论证了图形模型中配分函数与Bethe近似之比接近其与2阶Bethe近似之比的平方，并分析了两类对数超模模型的这些比值。
We resolve an open conjecture from algebraic geometry, which states that two generating functions for plane partition-like objects (the "box-counting" formulae for the Calabi-Yau topological vertices in Donaldson-Thomas theory and Pandharipande-Thomas theory) are equal up to a factor of MacMahon's generating function for plane partitions. The main tools in our proof are a Desnanot-Jacobi-type condensation identity, and a novel application of the tripartite double-dimer model of Kenyon-Wilson.
Twists of Gr(3,n) Cluster Variables as Double and Triple Dimer Partition Functions
本文为 Gr(3,n) 簇代数中可表示为 Plücker 坐标多项式的簇变量提供了基于双重和三重二聚体配置的组合解释，并给出其在扭曲映射下像的生成函数，推进了对该簇代数的理解。
We consider the problem of generating uniformly random partitions of the vertex set of a graph such that every piece induces a connected subgraph. For the case where we want to have partitions with linearly many pieces of bounded size, we obtain approximate sampling algorithms based on Glauber dynamics which are fixed-parameter tractable with respect to the bandwidth of $G$, with simple-exponential dependence on the bandwidth. For example, for rectangles of constant or logarithmic width this gives polynomial-time sampling algorithms. More generally, this gives sub-exponential algorithms for bo…
We prove some variants of the exponential formula and apply them to the multivariate Tutte polynomials (also known as Potts-model partition functions) of graphs. We also prove some further identities for the multivariate Tutte polynomial, which generalize an identity for counting connected graphs found by Riordan, Nijenhuis, Wilf and Kreweras and in more general form by Leroux and Gessel, and an identity for the inversion enumerator of trees found by Mallows, Riordan and Kreweras. Finally, we prove a generalization of Mobius inversion on the partition lattice.
Congruences for spin characters of the double covers of the symmetric and alternating groups
本文从算术角度研究对称群和交错群双重覆盖的旋特征生成函数，利用经典解析工具和初等生成函数操作，推导出许多 Ramanujan-like 同余。
Combining tree decomposition and transfer matrix techniques provides a very general algorithm for computing exact partition functions of statistical models defined on arbitrary graphs. The algorithm is particularly efficient in the case of planar graphs. We illustrate it by computing the Potts model partition functions and chromatic polynomials (the number of proper vertex colourings using Q colours) for large samples of random planar graphs with up to N=100 vertices. In the latter case, our algorithm yields a sub-exponential average running time of ~ exp(1.516 sqrt(N)), a substantial improvem…
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