R. M. Green
Motivated by work of Stembridge, we study rank functions for Viennot's heaps of pieces. We produce a simple and sufficient criterion for a heap to be a ranked poset and apply the results to the heaps arising from fully commutative words in Coxeter groups.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Heaps of pieces for lattice paths
本文引入格路径的I型与II型片段堆，建立对偶、双射及高维对应，并给出生成函数的行列式与显式公式。
We analyze the Sprague-Grundy functions for a class of almost disjoint selective compound games played on Nim heaps. Surprisingly, we find that these functions behave chaotically for smaller Sprague-Grundy values of each component game yet predictably when any one heap is sufficiently large.
Characteristic Polynomials and Hypergraph Generating Functions via Heaps of Pieces
本文利用Viennot的块堆框架，将Jacobi关于图中闭游走生成函数的经典结论推广至超图，通过分析与邻接超矩阵相关的多项式系统的多元结式，证明了超图对应的有理函数也是特定组合对象的生成函数。
We investigate the integrability of the discrete non-linear equation governing the dependence on geodesic distance of planar graphs with inner vertices of even valences. This equation follows from a bijection between graphs and blossom trees and is expressed in terms of generating functions for random walks. We construct explicitly an infinite set of conserved quantities for this equation, also involving suitable combinations of random walk generating functions. The proof of their conservation, i.e. their eventual independence on the geodesic distance, relies on the connection between random w…
Power sum expansions for Kromatic symmetric functions using Lyndon heaps
该研究给出Kromatic对称函数及其对合像的幂和基系数的组合描述，提出更简洁的K-类似幂和基，基于Lyndon堆实现展开，并证明其与诱导子图独立多项式等价，还给出特定图下Kromatic拟对称函数的幂和展开。
We consider the cluster algebra associated to the $Q$-system for $A_r$ as a tool for relating $Q$-system solutions to all possible sets of initial data. We show that the conserved quantities of the $Q$-system are partition functions for hard particles on particular target graphs with weights, which are determined by the choice of initial data. This allows us to interpret the simplest solutions of the Q-system as generating functions for Viennot's heaps on these target graphs, and equivalently as generating functions of weighted paths on suitable dual target graphs. The generating functions tak…
Chromatic quasisymmetric functions and noncommutative $P$-symmetric functions
本文针对自然单位区间序，利用堆和局部翻转操作研究色拟对称函数，建立其与非交换对称函数的对偶，并得到正展开及e-正性猜想的部分结果。
We characterize which graph invariants are partition functions of a spin model over the complex numbers, in terms of the rank growth of associated `connection matrices'.
Rank functions and invariants of delta-matroids
本文通过秩函数公理化δ-拟阵，研究其秩生成函数，并利用洛伦兹多项式理论证明相关取值的对数凹性。
The generating function and an explicit expression is derived for the (colored) Motzkin numbers of higher rank introduced recently. Considering the special case of rank one yields the corresponding results for the conventional colored Motzkin numbers for which in addition a recursion relation is given.
We define the rank-metric zeta function of a code as a generating function of its normalized $q$-binomial moments. We show that, as in the Hamming case, the zeta function gives a generating function for the weight enumerators of rank-metric codes. We further prove a functional equation and derive an upper bound for the minimum distance in terms of the reciprocal roots of the zeta function. Finally, we show invariance under suitable puncturing and shortening operators and study the distribution of zeroes of the zeta function for a family of codes.
Upper homogeneous finite type (upho) posets are a large class of partially ordered sets with the property that the principal order filter at every vertex is isomorphic to the whole poset. Well-known examples include k-array trees, the grid graphs, and the Stern poset. Very little is known about upho posets in general. In this paper, we construct upho posets with Schur-positive Ehrenborg quasisymmetric functions, whose rank-generating functions have rational poles and zeros. We also categorize the rank-generating functions of all planar upho posets. Finally, we prove the existence of an upho po…
In this paper we study generating functions resembling the rank of strongly unimodal sequences. We give combinatorial interpretations, identities in terms of mock modular forms, asymptotics, and a parity result. Our functions imitate a relation between the rank of strongly unimodal sequences and the rank of integer partitions.
We derive generating functions for the ranks of pre-modular categories associated to quantum groups at roots of unity.
It is well known that Ramanujan conjectured congruences modulo powers of 5, 7 and and 11 for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo powers of 5 for the crank parity function. The generating function for rank parity function is f(q), which is the first example of a mock theta function that Ramanujan mentioned in his last letter to Hardy. We prove congruences modulo powers of 5 for the rank parity function.
We characterize which graph invariants are partition functions of an edge-coloring model over the complex numbers, in terms of the rank growth of associated `connection matrices'.
Supermodular Rank: Set Function Decomposition and Optimization
本文定义了格上函数的超模秩与子模秩，并提出一种利用子模分解将优化问题拆分为子模子问题的算法，从而在取决于子模秩的计算开销下，有效提升了单调集合函数最大化等问题的近似比保证。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。