Rekha Natarajan
We present a bijection between non-crossing partitions of the set $[2n+1]$ into $n+1$ blocks such that no block contains two consecutive integers, and the set of sequences $\{s_{i}\}_{1}^{n}$ such that $1 \leq s_{i} \leq i$, and if $s_{i}=j$, then $s_{i-r} \leq j-r$ for $1 \leq r \leq j-1$.
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For a subclass of matchings, set partitions, and permutations, we describe a direct bijection involving only arc annotated diagrams that not only interchanges maximum nesting and crossing numbers, but also all refinements of crossing and nesting numbers. Furthermore, we show that the bijection cannot be applied to a similar class of coloured arc annotated diagrams.
A k-triangulation of a convex polygon is a maximal set of diagonals so that no k+1 of them mutually cross in their interiors. We present a bijection between 2-triangulations of a convex n-gon and pairs of non-crossing Dyck paths of length 2(n-4). This solves the problem of finding a bijective proof of a result of Jonsson for the case k=2. We obtain the bijection by constructing isomorphic generating trees for the sets of 2-triangulations and pairs of non-crossing Dyck paths.
We solve two problems regarding the enumeration of lattice paths in $\mathbb{Z}^2$ with steps $(1,1)$ and $(1,-1)$ with respect to the major index, defined as the sum of the positions of the valleys, and to the number of certain crossings. The first problem considers crossings of a single path with a fixed horizontal line. The second one counts pairs of paths with respect to the number of times they cross each other. Our proofs introduce lattice path bijections with convenient visual descriptions, and the answers are given by remarkably simple formulas involving $q$-binomial coefficients.
For each positive integer $n$, we construct a bijection between the odd partitions and the distinct partitions of $n$ which extends Bressoud's bijection between the odd-and-distinct partitions of $n$ and the splitting partitions of $n$. We compare our bijection with the classical bijections of Glaisher and Sylvester, and also with a recent bijection due to Chen, Gao, Ji and Li.
Using the theory of Properly Embedded Graphs developed in an earlier work we define an involutory duality on the set labeled non-crossing trees that lifts the obvious duality in the set of unlabeled non-crossing trees. The set of non-crossing trees is a free ternary magma with one generator and this duality is an instance of a duality that is defined in any such magma. Any two free ternary magmas with one generator are isomorphic via a unique isomorphism that we call the structural bijection. Besides the set of non-crossing trees we also consider as free ternary magmas with one generator the s…
We find bijections on 2-distant noncrossing partitions, 12312-avoiding partitions, 3-Motzkin paths, UH-free Schr{ö}der paths and Schr{ö}der paths without peaks at even height. We also give a direct bijection between 2-distant noncrossing partitions and 12312-avoiding partitions.
In this article, we investigate bijections on various classes of set partitions of classical types that preserve openers and closers. On the one hand we present bijections that interchange crossings and nestings. For types B and C, they generalize a construction by Kasraoui and Zeng for type A, whereas for type D, we were only able to construct a bijection between non-crossing and non-nesting set partitions. On the other hand we generalize a bijection to type B and C that interchanges the cardinality of the maximal crossing with the cardinality of the maximal nesting, as given by Chen, Deng, D…
We present an elementary type preserving bijection between noncrossing and nonnesting partitions for all classical reflection groups, answering a question of Athanasiadis.
In this note a bijection is constructed between the set of partitions of n simultaneously s-regular and t-distinct, and those simultaneously t-regular and s-distinct. Some implications of the map are discussed. As a generalized version of Glaisher's bijection, the map may be widely useful in other partition combinatorics. A previous conjecture concerning iterations of Glaisher's bijection is given a counterexample.
A bijection between strongly stable and totally symmetric partitions
本文定义d维强稳定分拆并建立其与d元强稳定理想的对应，进而构造强稳定分拆与全对称分拆间保持最小包围盒边长的双射。
In the study of the algebra $\mathrm{NCSym}$ of symmetric functions in noncommutative variables, Bergeron and Zabrocki found a free generating set consisting of power sum symmetric functions indexed by atomic partitions. On the other hand, Bergeron, Reutenauer, Rosas, and Zabrocki studied another free generating set of $\mathrm{NCSym}$ consisting of monomial symmetric functions indexed by unsplitable partitions. Can and Sagan raised the question of finding a bijection between atomic partitions and unsplitable partitions. In this paper, we provide such a bijection.
Recently, Schneider and Schneider defined a new class of partitions called sequentially congruent partitions, in which each part is congruent to the next part modulo its index, and they proved two partition bijections involving these partitions. We introduce a new partition notation specific to sequentially congruent partitions which allows us to more easily study these bijections and their compositions, and we reinterpret them in terms of Young diagram transformations. We also define a generalization of sequentially congruent partitions, and we provide several new partition bijections for the…
In this thesis, we introduced and carried out a combinatorial study of permutations that avoid one or two patterns of length 3 according to the statistic number of crossings. For this purpose, we manipulated a bijection of Elizalde and Pak and constructed other bijections that preserve the number of crossings. As results, we found, throughout these bijections, various relationships on the distributions of the number of crossings on restricted permutations as well as combinatorial interpretations in terms of the number of crossings on permutations with forbidden patterns of some well known tria…
We present a direct bijection between descending plane partitions with no special parts and permutation matrices. This bijection has the desirable property that the number of parts of the descending plane partition corresponds to the inversion number of the permutation. Additionally, the number of maximum parts in the descending plane partition corresponds to the position of the one in the last column of the permutation matrix. We also discuss the possible extension of this approach to finding a bijection between descending plane partitions and alternating sign matrices.
We present a bijection between permutation matrices and descending plane partitions without special parts, which respects the quadruple of statistics considered by Behrend, Di Francesco and Zinn--Justin. This bijection involves the inversion words of permutations and the "usual" representation of descending plane partitions as families of non--intersec\-ting lattice paths.
This paper has been withdrawn by the author due to an error in Lemma 3, making the (bijective) proof of Theorem 4 and Corollary 5 invalid (symmetry of k-nonnesting and k-noncrossing set partitions).
We give a construction for the d-dimensional simplices with all distances in {1,2} from the set of partitions of d+1.
A bijection between symmetric plane partitions and quasi transpose complementary plane partitions
本文提出一种双射，将对称平面划分与准转置互补平面划分联系起来，并解决了一个35年的开放问题。
We construct a direct natural bijection between descending plane partitions without any special part and permutations. The directness is in the sense that the bijection avoids any reference to nonintersecting lattice paths. The advantage of the bijection is that it provides an interpretation for the seemingly long list of conditions needed to define descending plane partitions. Unfortunately, the bijection does not relate the number of parts of the descending plane partition with the number of inversions of the permutation as one might have expected from the conjecture of Mills, Robbins and Ru…
Part Size Count in Cylindric Partitions and Bijections for Small Profiles
本文研究圆柱分拆的部分大小统计量，建立普通分拆与小轮廓圆柱分拆间的双射，并扩展相关生成函数公式与函数方程。
We present a simple bijection between permutation matrices and descending plane partitions without special parts. This bijection is already mentioned in work of P. Lalonde (without giving the details); it involves the inversion words of permutations and the (well-known) representation of descending plane partitions as families of non--intersecting lattice paths. (Taking a short detour, we will also exhibit how the (well--known) enumeration of descending plane partitions follows easily from the evaluation of Andrew's determinant.)
We construct weight-preserving bijections between column strict shifted plane partitions with one row and alternating sign trapezoids with exactly one column in the left half that sums to $1$. Amongst other things, they relate the number of $-1$s in the alternating sign trapezoids to certain elements in the column strict shifted plane partitions that generalise the notion of special parts in descending plane partitions. The advantage of these bijections is that they include configurations with $-1$s, which is a feature that many of the bijections in the realm of alternating sign arrays lack.
This note introduces some bijections relating core partitions and tuples of integers. We apply these bijections to count the number of cores with various types of restriction, including fixed number of parts, limited size of parts, parts divisible by some integer, and distinct parts. For example, we prove that the number of $2t$-core partitions into $l$ even parts equals the number of $t$-core partitions into $l$ parts. We also generalize one expression for simultaneous cores, which was given by Baek, Nam and Yu, recently. Subsequently, we use this expression to obtain recurrence satisfied by …
A theorem of Andrews equates partitions in which no part is repeated more than 2k-1 times to partitions in which, if j appears at least k times, all parts less than j also do so. This paper proves the theorem bijectively, with some of the generalizations that usually arise from such proofs.
An optimal refinement-compatible bijection between singleton-free partitions and partitions without cyclic adjacencies
本文针对Bernhart的论断，构造了无单元素划分与无循环相邻对划分之间兼容精化序的最优直接单轮双射，该双射还可推广至非交叉划分对应类。
It is well-known that Catalan numbers $C_n = \frac{1}{n+1} \binom{2n}{n}$ count the number of dominant regions in the Shi arrangement of type $A$, and that they also count partitions which are both $n$-cores as well as $(n+1)$-cores. These concepts have natural extensions, which we call here the $m$-Catalan numbers and $m$-Shi arrangement. In this paper, we construct a bijection between dominant regions of the $m$-Shi arrangement and partitions which are both $n$-cores as well as $(mn+1)$-cores. The bijection is natural in the sense that it commutes with the action of the affine symmetric grou…
We say two posets are "doppelgängers" if they have the same number of $P$-partitions of each height $k$. We give a uniform framework for bijective proofs that posets are doppelgängers by synthesizing $K$-theoretic Schubert calculus techniques of H. Thomas and A. Yong with M. Haiman's rectification bijection and an observation of R. Proctor. Geometrically, these bijections reflect the rational equivalence of certain subvarieties of minuscule flag manifolds. As a special case, we provide the first bijective proof of a 1983 theorem of R. Proctor---that plane partitions of height $k$ in a rectangl…
The total number of noncrossing partitions of type $Ψ$ is the $n$th Catalan number $\frac{1}{n+1}\binom{2n}{n}$ when $Ψ=A_{n-1}$, and the binomial $\binom{2n}{n}$ when $Ψ=B_n$, and these numbers coincide with the correspondent number of nonnesting partitions. For type A, there are several bijective proofs of this equality, being the intuitive map that locally converts each crossing to a nesting one of them. In this paper we present a bijection between nonnesting and noncrossing partitions of types A and B that generalizes the type A bijection that locally converts each crossing to a nesting.
We provide a bijective map from the partitions enumerated by the series side of the Rogers-Selberg mod 7 identities onto partitions associated with a special case of Basil Gordon's combinatorial generalization of the Rogers-Ramanujan identities. The implications of applying the same map to a special case of David Bressoud's even modulus analog of Gordon's theorem are also explored.
Bijections between colored compositions, Dyck paths, and polygon partitions
该研究通过识别并装饰合成、Dyck 路径与凸多边形剖分中的构建块，构造了保持部分且支持彩色情形的双射，从而统一刻画三类组合对象。
Johnson recently proved Armstrong's conjecture which states that the average size of an $(a,b)$-core partition is $(a+b+1)(a-1)(b-1)/24$. He used various coordinate changes and one-to-one correspondences that are useful for counting problems about simultaneous core partitions. We give an expression for the number of $(b_1,b_2,\cdots, b_n)$-core partitions where $\{b_1,b_2,\cdots,b_n\}$ contains at least one pair of relatively prime numbers. We also evaluate the largest size of a self-conjugate $(s,s+1,s+2)$-core partition.
Let $\ell,k$ be fixed positive integers. In an earlier work, the first and third authors established a bijection between $\ell$-cores with first part equal to $k$ and $(\ell-1)$-cores with first part less than or equal to $k$. This paper gives several new interpretations of that bijection. The $\ell$-cores index minimal length coset representatives for $\widetilde{S_{\ell}} / S_{\ell}$ where $\widetilde{S_{\ell}}$ denotes the affine symmetric group and $S_{\ell}$ denotes the finite symmetric group. In this setting, the bijection has a beautiful geometric interpretation in terms of the root lat…
In 2007, D.I. Panyushev defined a remarkable map on the set of nonnesting partitions (antichains in the root poset of a finite Weyl group). In this paper we identify Panyushev's map with the Kreweras complement on the set of noncrossing partitions, and hence construct the first uniform bijection between nonnesting and noncrossing partitions. Unfortunately, the proof that our construction is well-defined is case-by-case, using a computer in the exceptional types. Fortunately, the proof involves new and interesting combinatorics in the classical types. As consequences, we prove several conjectur…
Set partitions avoiding $k$-crossing and $k$-nesting have been extensively studied from the aspects of both combinatorics and mathematical biology. By using the generating tree technique, the obstinate kernel method and Zeilberger's algorithm, Lin confirmed a conjecture due independently to the author and Martinez-Savage that asserts inversion sequences with no weakly decreasing subsequence of length 3 and enhanced 3-nonnesting partitions have the same cardinality. In this paper, we provide a bijective proof of this conjecture. Our bijection also enables us to provide a new bijective proof of …
How many matchings on the vertex set V={1,2,...,2n} avoid a given configuration of three edges? Chen, Deng and Du have shown that the number of matchings that avoid three nesting edges is equal to the number of matchings avoiding three pairwise crossing edges. In this paper, we consider other forbidden configurations of size three. We present a bijection between matchings avoiding three crossing edges and matchings avoiding an edge nested below two crossing edges. This bijection uses non-crossing pairs of Dyck paths of length 2n as an intermediate step. Apart from that, we give a bijection tha…
A simple bijection for enhanced, classical, and 2-distant k-noncrossing partitions
本文提出一个简单的扩展映射，证明了增强、经典和2-距离k-非交叉分区的数量通过二项式变换依次关联，解决了Zhicong Lin的猜想并推广了早期归约恒等式。
We give a bijection between the set of self-conjugate partitions and that of ordinary partitions. Also, we show the relation between hook lengths of self conjugate partition and corresponding partition via the bijection. As a corollary, we give new combinatorial interpretations for the Catalan number and the Motzkin number in terms of self-conjugate simultaneous core partitions.
In this paper, we study classes of subexcedant functions enumerated by the Bell numbers and present bijections on set partitions. We present a set of permutations whose transposition arrays are the restricted growth functions, thus defining Bell permutations of the second kind. We describe a bijection between Bell permutations of the first kind (introduced by Ponti and Vajnovzski) and the second kind. We present two other Bell number enumerated classes of subexcedant functions. Further, we present bijections on set partitions, in particular, an involution that interchanges the set of merging b…
It is a classical result in combinatorics that among lattice paths with 2m steps U=(1,1) and D=(1,-1) starting at the origin, the number of those that do not go below the x-axis equals the number of those that end on the x-axis. A much more unfamiliar fact is that the analogous equality obtained by replacing single paths with k-tuples of non-crossing paths holds for every k. This result has appeared in the literature in different contexts involving plane partitions (where it was proved by Proctor), partially ordered sets, Young tableaux, and lattice walks, but no bijective proof for k>1 seems …
A pipe dream perspective on totally symmetric self-complementary plane partitions
本文通过管道梦视角刻画完全对称自补平面划分，建立其与交替符号矩阵的双射，并保持偏序结构。
A bijection related to Bressoud's conjecture
本文引入了一个与Bressoud划分函数生成函数猜想相关的双射，并以此为基础给出了Göllnitz-Gordon恒等式的新伴式。
We study a curious class of partitions, the parts of which obey an exceedingly strict congruence condition we refer to as "sequential congruence": the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part congruent to zero modulo the length of the partition. It turns out these obscure-seeming objects are embedded in a natural way in partition theory. We show that sequentially congruent partitions with largest part $n$ are in bijection with the partitions of $n$. Moreover, we show sequentially congruent partitions induce a bijection between partitions of $n$ and parti…
We give short elementary expositions of combinatorial proofs of some variants of Euler's partitition problem that were first addressed analytically by George Andrews, and later combinatorially by others. Our methods, based on ideas from a previous paper by the author , enable us to state and prove new generalizations of two of these results.
Linked partitions are introduced by Dykema in the study of transforms in free probability theory, whereas permutation tableaux are introduced by Steingrímsson and Williams in the study of totally positive Grassmannian cells. Let $[n]=\{1,2,\ldots,n\}$. Let $L(n,k)$ denote the set of linked partitions of $[n]$ with $k$ blocks, let $P(n,k)$ denote the set of permutations of $[n]$ with $k$ descents, and let $T(n,k)$ denote the set of permutation tableaux of length $n$ with $k$ rows. Steingrímsson and Williams found a bijection between the set of permutation tableaux of length $n$ with $k$ rows an…
For any integer $k\geq2$, we prove combinatorially the following Euler (binomial) transformation identity $$ \NC_{n+1}^{(k)}(t)=t\sum_{i=0}^n{n\choose i}\NW_{i}^{(k)}(t), $$ where $\NC_{m}^{(k)}(t)$ (resp.~$\NW_{m}^{(k)}(t)$) is the sum of weights, $t^\text{number of blocks}$, of partitions of $\{1,\ldots,m\}$ without $k$-crossings (resp.~enhanced $k$-crossings). The special $k=2$ and $t=1$ case, asserting the Euler transformation of Motzkin numbers are Catalan numbers, was discovered by Donaghey 1977. The result for $k=3$ and $t=1$, arising naturally in a recent study of pattern avoidance in …
Plumbing bijections
该研究通过展示水管布局的双射性，证明不同设置方式下水系统结果一致，揭示了管道设计中的对称性特性。
In this paper, we give a purely bijective proof that two different partition classes that are both combinatorial interpretations of the partition function $p_ν(n)$, a partition function related to the third order mock theta function $ν(q)$, are equinumerous. In doing so, we give a partial solution to a combinatorial problem proposed in a paper by Andrews.
Although both noncrossing partitions and nonnesting partitions are uniformly enumerated for Weyl groups, the exact relationship between these two sets of combinatorial objects remains frustratingly mysterious. In this paper, we give a precise combinatorial answer in the case of the symmetric group: for any standard Coxeter element, we construct an equivariant bijection between noncrossing partitions under the Kreweras complement and nonnesting partitions under a Coxeter-theoretically natural cyclic action we call the Kroweras complement. Our equivariant bijection is the unique bijection that i…
We provide new Schmidt-type results through an investigation of two bijections, which are results involving partitions with parts counted only at given indices. Mork's bijection, the first of these, was originally given as a proof of Schmidt's theorem. We show that a version of Sylvester's bijection is equivalent to Mork's bijection applied to 2-modular diagrams, which implies refinements of existing results and new generating function identities. We then develop a bijection based on an idea appearing in a recent paper of Andrews and Keith, that places partitions counted at the indices $r$, $t…
Phylogenetic trees are binary nonplanar trees with labelled leaves, and plane oriented recursive trees are planar trees with an increasing labelling. Both families are enumerated by double factorials. A bijection is constructed, using the respective representations a 2-partitions and trapezoidal words.
Bijections for rhombic alternative tableaux
本文将替代表格与排列之间的双射推广到rhombic替代表格与排列集合之间，发现双射关系并推导了RAT的细化计数公式，同时将RAT的统计量映射到排列集合。
The Stanley lattice, Tamari lattice and Kreweras lattice are three remarkable orders defined on the set of Catalan objects of a given size. These lattices are ordered by inclusion: the Stanley lattice is an extension of the Tamari lattice which is an extension of the Kreweras lattice. The Stanley order can be defined on the set of Dyck paths of size $n$ as the relation of \emph{being above}. Hence, intervals in the Stanley lattice are pairs of non-crossing Dyck paths. In a former article, the second author defined a bijection $Φ$ between pairs of non-crossing Dyck paths and the realizers of tr…
We interpret noncrossing partitions of type $B$ and type $D$ in terms of noncrossing partitions of type $A$. As an application, we get type-preserving bijections between noncrossing and nonnesting partitions of type $B$, type $C$ and type $D$ which are different from those in the recent work of Fink and Giraldo. We also define Catalan tableaux of type $B$ and type $D$, and find bijections between them and noncrossing partitions of type $B$ and type $D$ respectively.
We examine the enumeration of certain Motzkin objects according to the numbers of crossings and nestings. With respect to continued fractions, we compute and express the distributions of the statistics of the numbers of crossings and nestings over three sets, namely the set of $4321$-avoiding involutions, the set of $3412$-avoiding involutions, and the set of $(321,3\bar{1}42)$-avoiding permutations. To get our results, we exploit the bijection of Biane restricted to the sets of $4321$- and $3412$-avoiding involutions which was characterized by Barnabei et al.~ and the bijection between $(321,…
P.L. Erdos and L.A. Szekely [Adv. Appl. Math. 10(1989), 488-496] gave a bijection between rooted semilabeled trees and set partitions. L.H. Harper's results [Ann. Math. Stat. 38(1967), 410-414] on the asymptotic normality of the Stirling numbers of the second kind translates into asymptotic normality of rooted semilabeled trees with given number of vertices, when the number of internal vertices varies. The Erdos-Szekely bijection specializes to a bijection between phylogenetic trees and set partitions with classes of size \geq 2. We consider modified Stirling numbers of the second kind that en…
Let $st=\{st_1,\ldots,st_k\}$ be a set of $k$ statistics on permutations with $k\geq 1$. We say that two given subset of permutations $T$ and $T'$ are $st$-Wilf-equivalent if the joint distributions of all statistics in $st$ over the sets of $T$-avoiding permutations $S_n(T)$ and $T'$-avoiding permutations $S_n(T')$ are the same. The main purpose of this paper is the (cr,nes)-Wilf-equivalence classes for all single patterns in $S_3$, where cr and nes denote respectively the statistics number of crossings and nestings. One of the main tools that we use is the bijection $Θ:S_n(321)\rightarrow S_…
Recently, Chen et al. derived the generating function for partitions avoiding right nestings and posed the problem of finding the generating function for partitions avoiding right crossings. In this paper, we derive the generating function for partitions avoiding right crossings via an intermediate structure of partial matchings avoiding 2-right crossings and right nestings. We show that there is a bijection between partial matchings avoiding 2-right crossing and right nestings and partitions avoiding right crossings.
A Bijection Between All Shi Regions and Core Partitions
本文推广了Fishel-Vazirani关于$m$-Shi配置支配区域的双射，将Type $A_{n}$的$m$-Shi配置的所有极小房间与特定的$n$-core分拆等价类集合建立双射，并给出了$m$-极小房间$S_{n}$-轨道的分类。
Motivated by Andrews' recent work related to Euler's partition theorem, we consider the set of partitions of an integer $n$ where the set of even parts has exactly $j$ elements, versus the set of partitions of $n$ where the set of repeated parts has exactly $j$ elements. These two sets of partitions turn out to be equinumerous, and this naturally encloses Euler's theorem and Andrews' theorem as two special cases. We give two proofs, one using generating function, and the other is a direct bijection that builds on Glaisher's bijection
In this paper, the problem of pattern avoidance in generalized non-crossing trees is studied. The generating functions for generalized non-crossing trees avoiding patterns of length one and two are obtained. Lagrange inversion formula is used to obtain the explicit formulas for some special cases. Bijection is also established between generalized non-crossing trees with special pattern avoidance and the little Schröder paths.
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