T. Kyle Petersen
We consider a bivariate polynomial that generalizes both the length and reflection length generating functions in a finite Coxeter group. In seeking a combinatorial description of the coefficients, we are led to the study of a new Mahonian statistic, which we call the sorting index. The sorting index of a permutation and its type B and type D analogues have natural combinatorial descriptions which we describe in detail.
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We prove that the Mahonian-Stirling pairs of permutation statistics $(\sor, \cyc)$ and $(\inv, \mathrm{rlmin})$ are equidistributed on the set of permutations that correspond to arrangements of $n$ non-atacking rooks on a Ferrers board with $n$ rows and $n$ columns. The proofs are combinatorial and use bijections between matchings and Dyck paths and a new statistic, sorting index for matchings, that we define. We also prove a refinement of this equidistribution result which describes the minimal elements in the permutation cycles and the right-to-left minimum letters. Moreover, we define a sor…
Björner and Wachs defined a major index for labeled plane forests and showed that it has the same distribution as the number of inversions. We define and study the distributions of a few other natural statistics on labeled forests. Specifically, we introduce the notions of bottom-to-top maxima, cyclic bottom-to-top maxima, sorting index, and cycle minima. Then we show that the pairs (inv, Bt-max), (sor, Cyc), and (maj, Cbt-max) are equidistributed. Our results extend the result of Björner and Wachs and generalize results for permutations. We also introduce analogous statistics for signed label…
Patience Sorting is a combinatorial algorithm that can be viewed as an iterated, non-recursive form of the Schensted Insertion Algorithm. In recent work the authors extended Patience Sorting to a full bijection between the symmetric group and certain pairs of combinatorial objects (called pile configurations) that are most naturally defined in terms of generalized permutation pattern and barred pattern avoidance. This Extended Patience Sorting Algorithm is very similar to the Robinson-Schensted-Knuth (or RSK) Correspondence, which is itself built from repeated application of the Schensted Inse…
We exhibit a bijection between recently-introduced combinatorial objects known as valid hook configurations and certain weighted set partitions. When restricting our attention to set partitions that are matchings, we obtain three new combinatorial interpretations of Lassalle's sequence. One of these interpretations involves permutations that have exactly one preimage under the (West) stack-sorting map. We prove that the sequences obtained by counting these permutations according to their first entries are symmetric, and we conjecture that they are log-concave. We also obtain new recurrence rel…
Recently Petersen defined a new Mahonian index sor over the symmetric group $\mathfrak{S}_n$ and proved that $(\text{inv}, \text{rlmin})$ and $(\text{sor}, \text{cyc})$ have the same joint distribution. Foata and Han proved that the pairs of set-valued statistics $(\text{Cyc}, \text{Rmil}), (\text{Cyc}, \text{Lmap}), (\text{Rmil}, \text{Lmap})$ have the same joint distribution over $\mathfrak{S}_n$. In this paper we introduce the set-valued statistics $\text{Inv}, \text{Lmil}, \text{Sor}$ and $\text{Lmicycl}_1$ and generalize simultaneously results of Petersen and Foata-Han and find many equid…
Flip-sort and combinatorial aspects of pop-stack sorting
本文研究自然排序过程 flip-sort 的组合性质，刻画单次迭代后的置换形状及最佳/最差情形相关族，并建立与 pop-stack 排序、自动机、格路径的联系，提出若干双射证明技巧。
In the combinatorial study of the coefficients of a bivariate polynomial that generalizes both the length and the reflection length generating functions for finite Coxeter groups, Petersen introduced a new Mahonian statistic $sor$, called the sorting index. Petersen proved that the pairs of statistics $(sor,cyc)$ and $(inv,rl\textrm{-}min)$ have the same joint distribution over the symmetric group, and asked for a combinatorial proof of this fact. In answer to the question of Petersen, we observe a connection between the sorting index and the B-code of a permutation defined by Foata and Han, a…
The sorting indices $\text{sor}_B$ and $\text{sor}_D$ on the Coxeter groups of type $B$ and $D$ respectively are defined by Petersen and it is proved that $(\text{inv}_B, \text{rlmin})$ and $(\text{sor}_B, \ell'_B)$ have the same joint distribution for type $B$ while $\text{inv}_D$ and $\text{sor}_D$ have the same distribution for type $D$. These results, including a set-valued extension of type $B$ involving two equildistributed pairs of three statistics, are proved combinatorially by Chen et al. via two mappings $φ:=\text{(B-code)}^{-1}\circ \text{(A-code)}$ and $ψ:=\text{(D-code)}^{-1}\circ…
We define a new statistic $\mathsf{sor}$ on the set of colored permutations $\mathsf{G}_{r,n}$ and prove that it has the same distribution as the length function. For the set of restricted colored permutations corresponding to the arrangements of $n$ non-attacking rooks on a fixed Ferrers shape we show that the following two sequences of set-valued statistics are joint equidistributed: $(\ell,\mathsf{Rmil}^0,\mathsf{Rmil}^1,...,\mathsf{Rmil}^{r-1}$, $\mathsf{Lmil}^0,\mathsf{Lmil}^1,...,\mathsf{Lmil}^{r-1}$, $\mathsf{Lmal}^0,\mathsf{Lmal}^1,...,\mathsf{Lmal}^{r-1}$, $\mathsf{Lmap}^0,\mathsf{Lma…
This paper continues the analysis of the pattern-avoiding sorting machines recently introduced by Cerbai, Claesson and Ferrari [CCF]. These devices consist of two stacks, through which a permutation is passed in order to sort it, where the content of each stack must at all times avoid a certain pattern. Here we characterize and enumerate the set of permutations that can be sorted when the first stack is $132$-avoiding, solving one of the open problems proposed in [CCF]. To that end we present several connections with other well known combinatorial objects, such as lattice paths and restricted …
We present new combinatorial and probabilistic identities relating three random processes: the oriented swap process on $n$ particles, the corner growth process, and the last passage percolation model. We prove one of the probabilistic identities, relating a random vector of last passage percolation times to its dual, using the duality between the Robinson-Schensted-Knuth and Burge correspondences. A second probabilistic identity, relating those two vectors to a vector of "last swap times" in the oriented swap process, is conjectural. We give a computer-assisted proof of this identity for $n\l…
We introduce operators $\mathsf{hare}$ and $\mathsf{tortoise}$, which act on words as natural generalizations of West's stack-sorting map. We show that the heuristically slower algorithm $\mathsf{tortoise}$ can sort words arbitrarily faster than its counterpart $\mathsf{hare}$. We then generalize the combinatorial objects known as valid hook configurations in order to find a method for computing the number of preimages of any word under these two operators. We relate the question of determining which words are sortable by $\mathsf{hare}$ and $\mathsf{tortoise}$ to more classical problems in pa…
This paper makes a combinatorial study of the two monoids and the two types of tableaux that arise from the two possible generalizations of the Patience Sorting algorithm from permutations (or standard words) to words. For both types of tableaux, we present Robinson--Schensted--Knuth-type correspondences (that is, bijective correspondences between word arrays and certain pairs of semistandard tableaux of the same shape), generalizing two known correspondences: a bijective correspondence between standard words and certain pairs of standard tableaux, and an injective correspondence between words…
Type A affine shuffles are compared with riffle shuffles followed by a cut. Although these probability measures on the symmetric group S_n are different, they both satisfy a convolution property. Strong evidence is given that when the underlying parameter $q$ satisfies $gcd(n,q-1)=1$, the induced measures on conjugacy classes of the symmetric group coincide. This gives rise to interesting combinatorics concerning the modular equidistribution by major index of permutations in a given conjugacy class and with a given number of cyclic descents. It is proved that the use of cuts does not speed up …
The mapping index through the lens of the cross-index
本文研究自由G-偏序的交叉指数作为同调拓扑指数的组合类比，揭示了Z₂群作用下的交叉指数与拓扑指数的差异，并展示了交叉指数与单纯指数之间的关系。
We present a decomposition of the generalized binomial coefficients associated with Jack polynomials into two factors: a stem, which is described explicitly in terms of hooks of the indexing partitions, and a leaf, which inherits various recurrence properties from the binomial coefficients and depends exclusively on the skew diagram. We then derive a direct combinatorial formula for the leaf in the special case where the two indexing partitions differ by at most two rows. This formula also exhibits an unexpected symmetry with respect to the lengths of the two rows.
Analogous to the types A, B, and C cases, we address the computation of the index of seaweed subalgebras in the type-D case. Formulas for the algebra's index can be computed by counting the connected components of its associated meander. We focus on a set of distinguished vertices of the meander, called the tail of the meander, and using the tail, we provide comprehensive combinatorial formulas for the index of a seaweed in all the classical types. Using these formulas, we provide all general closed-form index formulas where the index is given by a polynomial greatest common divisor formula in…
Combinatorial explanation of the weighted Kirchhoff index of graphs
本文对任意连通图的加权基尔霍夫指数给出组合解释，通过加权细分图及其子图的匹配权重和表示该指数，解决了相关文献提出的问题，特例可恢复树等图的对应指数公式。
The notion of containment and avoidance provides a natural partial ordering on set partitions. Work of Sagan and of Goyt has led to enumerative results in avoidance classes of set partitions, which were refined by Dahlberg et al. through the use of combinatorial statistics. We continue this work by computing the distribution of the dimension index (a statistic arising from the supercharacter theory of finite groups) across certain avoidance classes of partitions. In doing so we obtain a novel connection between noncrossing partitions and 321-avoiding permutations, as well as connections to man…
The h-index was introduced by the physicist J.E. Hirsch in 2005 as measure of a researcher's productivity. We consider the "combinatorial Fermi problem" of estimating h given the citation count. Using the Euler-Gauss identity for integer partitions, we compute confidence intervals. An asymptotic theorem about Durfee squares, due to E.R. Canfield-S. Corteel-C.D. Savage from 1998, is reinterpreted as the rule of thumb h=0.54 x (citations)^{1/2}. We compare these intervals and the rule of thumb to empirical data (primarily using mathematicians).
Combinatorial Hodge Index Theorem for Polytopes
该研究将Maxim-Schuermann公式从顶点有理的多面体推广至任意凸多面体，依托BBFK的组合相交上同调框架，还探讨了多面体的霍奇指标定理。
In enumerative combinatorics, it is often a goal to enumerate both labeled and unlabeled structures of a given type. The theory of combinatorial species is a novel toolset which provides a rigorous foundation for dealing with the distinction between labeled and unlabeled structures. The cycle index series of a species encodes the labeled and unlabeled enumerative data of that species. Moreover, by using species operations, we are able to solve for the cycle index series of one species in terms of other, known cycle indices of other species. Section 3 is an exposition of species theory and Sect…
Combinatorial characterzations of $T$-designs in the nonbinary Johnson scheme
本文通过扩展Delsarte框架，提出非二元乔纳森方案中T-设计的更广泛组合刻画，并引入(r,s)-设计作为新组合对象，同时导出了其基数下界并构造了实例。
Many combinatorial proofs rely on induction. When these proofs are formulated in traditional language, they can be bulky and unmanageable. Coalgebras provide a language which can reduce reduce many inductive proofs in graded poset theory to comprehensible size. As a bonus, the visual form of the resulting recursive proofs suggests combinatorial interpretations for constants appearing in the longer arguments. We use the techniques of coalgebras to compute invariants of toric and affine arrangements as well as of poset products. In additional chapters we prove structure theorems for acyclic orie…
The purpose of this paper is to review some combinatorial ideas behind the mirror symmetry for Calabi-Yau hypersurfaces and complete intersections in Gorenstein toric Fano varieties. We suggest as a basic combinatorial object the notion of a Gorenstein polytope of index r. A natural combinatorial duality for d-dimensional Gorenstein polytopes of index r extends the well-known polar duality for reflexive polytopes (case r=1). We consider the Borisov duality between two nef-partitions as a duality between two Gorenstein polytopes P and P^* of index r together with selected special (r-1)-dimensio…
Toric log del Pezzo surfaces correspond to convex lattice polygons containing the origin in their interior and having only primitive vertices. An upper bound on the volume and on the number of boundary lattice points of these polygons is derived in terms of the index l. Techniques for classifying these polygons are also described: a direct classification for index two is given, and a classification for all l<17 is obtained.
Stabilization index of V-number of powers of edge ideals of graphs
本文针对任意图的边理想幂的v-函数，给出了其稳定化指数的组合刻画。
Conley-Morse-Forman theory for generalized combinatorial multivector fields on finite topological spaces
该研究推广拓展了组合多向量场的Conley-Morse-Forman理论，去掉多向量唯一极大元假设，将场景从Lefschetz复形拓展到有限拓扑空间，定义了相关概念并证明了Conley指标可加性与Morse不等式。
We derive a generating series for the number of free subgroups of finite index in $Δ^+ = \mathbb{Z}_p*\mathbb{Z}_q$ by using a connection between free subgroups of $Δ^+$ and certain hypermaps (also known as ribbon graphs or "fat" graphs), and show that this generating series is transcendental. We provide non-linear recurrence relations for the above numbers based on differential equations that are part of the Riccati hierarchy. We also study the generating series for conjugacy classes of free subgroups of finite index in $Δ^+$, which correspond to isomorphism classes of hypermaps. Asymptotic f…
A combinatorial construction of the moduli space of flowlines in discrete Morse theory
本研究在离散Morse理论领域，通过组合方式构造了离散Morse函数的指标2流线模空间，为离散Morse同调中Morse微分平方为零的结论提供了新的证明方法。
We extend the exponential formula by Bender and Canfield (1996), which relates log-concavity and the cycle index polynomials. The extension clarifies the log-convexity relation. The proof is by noticing the property of a compound Poisson distribution together with its moment generating function. We also give a combinatorial proof of extended "log-convex part" referring Bender and Canfield's approach, where the formula by Bruijn and Erdös (1953) is additionally exploited. The combinatorial approach yields richer structural results more than log-convexity. Furthermore, we consider normal and bin…
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