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In this paper we consider doubly symmetric Dyck words, i.e. Dyck words which are fixed by two symmetry operations α and β introduced in [1] . We study combinatorial properties of doubly symmetric Dyck words, leading to the definition of two recursive algorithms to build these words. As a consequence we have a representation of doubly symmetric Dyck words as vectors of integers, called track vectors. Finally, we show some bijections between a subfamily of doubly symmetric Dyck words and a subfamily of integer partitions. The computation of the sequence f n of doubly symmetric Dyck words of semi-length n shows surprising properties giving rise to some conjectures.

We study fields of values of the restriction to Sylow subgroups of irreducible characters of the normalizers of Sylow subgroups in symmetric and alternating groups. As an application, we show that these classes of groups admit a McKay bijection that respects fields of values and restriction to Sylow subgroups.

The degree of symmetry of a combinatorial object, such as a lattice path, is a measure of how symmetric the object is. It typically ranges from zero, if the object is completely asymmetric, to its size, if it is completely symmetric. We study the behavior of this statistic on Dyck paths and grand Dyck paths, with symmetry described by reflection along a vertical line through their midpoint; partitions, with symmetry given by conjugation; and certain compositions interpreted as bargraphs. We find expressions for the generating functions for these objects with respect to their degree of symmetry, and their semilength or semiperimeter, deducing in most cases that, asymptotically, the degree of symmetry has a Rayleigh or half-normal limiting distribution. The resulting generating functions are often algebraic, with the notable exception of Dyck paths, for which we conjecture that it is D-finite (but not algebraic), based on a functional equation that we obtain using bijections to walks in the plane., Comment: 31 pages, 10 figures, 4 tables. The main additions to this version are the analysis of the limiting distributions of the degree of symmetry in the various cases, a new subsection 5.2 relating certain bargraphs to peakless Motzkin paths, minor corrections and improvements, and additional references

This article is the second part of a series of three articles, in which we develop a higher covering theory of racks and quandles. This project is rooted in M. Eisermann’s work on quandle coverings, and the categorical perspective brought to the subject by V. Even, who characterizes coverings as those surjections which are categorically central, relatively to trivial quandles. We extend this work by applying the techniques from higher categorical Galois theory, in the sense of G. Janelidze, and in particular we identify meaningful higher-dimensional centrality conditions defining our higher coverings of racks and quandles. In this second article (Part II), we show that categorical Galois theory applies to the inclusion of the category of coverings into the category of surjective morphisms of racks or of quandles. We characterise the induced Galois-theoretic concepts of trivial covering, normal covering and covering in this two-dimensional context. The latter is described via our definition and study of double coverings, also called algebraically central double extensions of racks (and quandles). We define a suitable and well-behaved commutator which captures the zero-, one- and two-dimensional concepts of centralization in the category of quandles. We keep track of the links with the corresponding concepts in the category of groups.

We define and study the totally nonnegative part of the Chow quotient of the Grassmannian, or more simply the nonnegative configuration space. This space has a natural stratification by positive Chow cells, and we show that nonnegative configuration space is homeomorphic to a polytope as a stratified space. We establish bijections between positive Chow cells and the following sets: (a) regular subdivisions of the hypersimplex into positroid polytopes, (b) the set of cones in the positive tropical Grassmannian, and (c) the set of cones in the positive Dressian. Our work is motivated by connections to super Yang-Mills scattering amplitudes, which will be discussed in a sequel., 46 pages; v2: citations added; v3: correction (Remark 7.1)

Generalising previous results on classical braid groups by Artin and Lin, we determine the values of m, n ∈ $\mathbb N$ for which there exists a surjection between the n- and m-string braid groups of an orientable surface without boundary. This result is essentially based on specific properties of their lower central series, and the proof is completely combinatorial. We provide similar but partial results in the case of orientable surfaces with boundary components and of non-orientable surfaces without boundary. We give also several results about the classification of different representations of surface braid groups in symmetric groups.

We define and study an effective version of the Wadge hierarchy in computable quasi-Polish spaces which include most spaces of interest for computable analysis. Along with hierarchies of sets we study hierarchies of k-partitions which are interesting on their own. We show that levels of such hierarchies are preserved by the computable effectively open surjections, that if the effective Hausdorff-Kuratowski theorem holds in the Baire space then it holds in every computable quasi-Polish space, and we extend the effective Hausdorff theorem to k-partitions.

Let M = ( T , C , P ) be a security model, where T is a rooted tree, C is a multiset of costs and P is a multiset of prizes and let ( T , c , p ) be a security system, where c and p are bijections of costs and prizes. The problems of computing an optimal attack on a security system and of determining an edge e ∈ E ( T ) such that the maximum sum of prizes obtained from an optimal attack in ( T , c , p ) is minimized when c ( e ) = ∞ are considered. An O ( G 2 n ) -time algorithm to compute an optimal attack as well as an O ( G 2 n 2 ) -time algorithm to determine such an edge are given, in addition to a (1-ϵ) FPTAS with the time bound O ( 1 ϵ 2 n 3 log ⁡ G ) and a (1+ϵ) FPTAS with the time bound O ( 1 ϵ 2 n 4 log ⁡ G ) for the first and second problems, respectively.

Flajolet and Francon [European. J. Combin. 10 (1989) 235-241] gave a combinatorial interpretation for the Taylor coefficients of the Jacobian elliptic functions in terms of doubled permutations. We show that a multivariable counting of the doubled permutations has also an explicit continued fraction expansion generalizing the continued fraction expansions of Rogers and Stieltjes. The second goal of this paper is to study the expansion of the Taylor coefficients of the generalized Jacobian elliptic functions, which implies the symmetric and unimodal property of the Taylor coefficients of the generalized Jacobian elliptic functions. The main tools are the combinatorial theory of continued fractions due to Flajolet and bijections due to Francon-Viennot, Foata-Zeilberger and Clarke-Steingrimsson-Zeng.

We examine whether some well-known properties of partial recursive functions are valid for ITBM-computable functions, i.e., functions which can be computed with infinite time Blum–Shub–Smale machines. It is shown that properties of graphs of ITBM-computable functions differ from the usual properties of graphs of partial recursive functions. All possible ranges of ITBM-computable functions are described, and we consider the existence problem for ITBM-computable bijections between ITBM-computable sets of the same cardinality.

The Euler number $E_n$ (resp. Entringer number $E_{n,k}$) enumerates the alternating (down-up) permutations of $\{1,\dots,n\}$ (resp. starting with $k$). The Springer number $S_n$ (resp. Arnold number $S_{n,k}$) enumerates the type $B$ alternating permutations (resp. starting with $k$). In this paper, using bijections we first derive the counterparts in {\em Andr\'e permutations} and {\em Simsun permutations} for the Entringer numbers $(E_{n,k})$, and then the counterparts in {\em signed Andr\'e permutations} and {\em type $B$ increasing 1-2 trees} for the Arnold numbers $(S_{n,k})$., Comment: 21 pages, 3 figures, 6 tables

This paper is the 2nd in a series of planned papers that provide 1st bijective proofs of alternating sign matrix (ASM) results. Based on the main result from the 1st paper, we construct a bijective proof of the enumeration formula for ASMs and of the fact that ASMs are equinumerous with descending plane partitions. We are also able to refine these bijections by including the position of the unique $1$ in the top row of the matrix. Our constructions rely on signed sets and related notions. The starting point for these constructions were known “computational” proofs, but the combinatorial point of view led to several drastic modifications. We also provide computer code where all of our constructions have been implemented.

We obtain a compact Sobolev embedding for H-invariant functions in compact metric-measure spaces, where H is a subgroup of the measure preserving bijections. In Riemannian manifolds, H is a subgroup of the volume preserving diffeomorphisms: a compact embedding for the critical exponents follows. The results can be viewed as an extension of Sobolev embeddings of functions invariant under isometries in compact manifolds.

Significance In the early 1980s, it was discovered that alternating sign matrices (ASMs), which are also commonly encountered in statistical mechanics, are counted by the same numbers as two classes of plane partitions. Since then it has been a major open problem in this area to construct explicit bijections between the three classes of objects. In this paper we describe a bijective proof that relates ASMs to one of the two classes of plane partitions as well as a bijective proof of the fact that the numbers can be expressed by a simple product formula. The constructions are complicated, but they are completely explicit and also provided in the form of computer code.

Explicit bijections are given between the weights associated to a block of a symmetric group and the irreducible Brauer characters in the block. Similar bijections are given between the weights associated to a block of a general linear group and a basic set of the block.

The rowmotion action on order ideals or on antichains of a finite partially ordered set has been studied (under a variety of names) by many authors. Depending on the poset, one finds unexpectedly interesting orbit structures, instances of (small order) periodicity, cyclic sieving, and homomesy. Many of these nice features still hold when the action is extended to $[0,1]$-labelings of the poset or (via detropicalization) to labelings by rational functions (the birational setting). In this work, we parallel the birational lifting already done for order-ideal rowmotion to antichain rowmotion. We give explicit equivariant bijections between the birational toggle groups and between their respective liftings. We further extend all of these notions to labellings by noncommutative rational functions, setting an unpublished periodicity conjecture of Grinberg in a broader context., 33 pages

In this paper we study the combinatorial consequences of the relationship between rational Cherednik algebras of type G ( l , 1 , n ) , cyclic quiver varieties and Hilbert schemes. We classify and explicitly construct C ⁎ -fixed points in cyclic quiver varieties and calculate the corresponding characters of tautological bundles. Furthermore, we give a combinatorial description of the bijections between C ⁎ -fixed points induced by the Etingof-Ginzburg isomorphism and Nakajima reflection functors. We apply our results to obtain a new proof as well as a generalization of the q-hook formula.

We present a general bijective approach to planar hypermaps with two main results. First we obtain unified bijections for all classes of maps or hypermaps defined by face-degree constraints and girth constraints. To any such class we associate bijectively a class of plane trees characterized by local constraints. This unifies and greatly generalizes several bijections for maps and hypermaps. Second, we present yet another level of generalization of the bijective approach by considering classes of maps with non-uniform girth constraints. More precisely, we consider \emph{well-charged maps}, which are maps with an assignment of ''charges'' (real numbers) on vertices and faces, with the constraints that the length of any cycle of the map is at least equal to the sum of the charges of the vertices and faces enclosed by the cycle. We obtain a bijection between charged hypermaps and a class of plane trees characterized by local constraints.

The results of Zarzycki for the cardinality of the sets of all bijections, surjections, and injections are generalized to the case when the domains and codomains are infinite and different. The elementary proofs the cardinality of the sets of bijections and surjections are given within the framework of the Zermelo-Fraenkel set theory with the axiom of choice. The case of the set of all injections is considered in detail and more explicit an expression is obtained when the Generalized Continuum Hypothesis is assumed.

We introduce bijections between generalized type An noncrossing partitions (that is, associated to arbitrary standard Coxeter elements) and fully commutative elements of the same type. The latter index the diagram basis of the classical Temperley–Lieb algebra, while for each choice of standard Coxeter element the corresponding noncrossing partitions also index a basis, given by the images in the Temperley–Lieb algebra of the simple elements of the dual Garside structure (associated to this choice of standard Coxeter element) of the Artin braid group on n + 1 strands. We then show that our bijections come from triangular base changes between the diagram basis and the various bases indexed by noncrossing partitions, by explicitly describing the orders giving triangularity. These orders were introduced in a joint paper with Williams and provide exotic lattice structures on noncrossing partitions. Several combinatorial objects are introduced along the way, including an involution on the set of noncrossing partitions.

Recognizing when a ring is a complete matrix ring is of significant importance in algebra. It is well-known folklore that a ring $R$ is a complete $n\times n$ matrix ring, so $R\cong M_{n}(S)$ for some ring $S$, if and only if it contains a set of $n\times n$ matrix units $\{e_{ij}\}_{i,j=1}^n$. A more recent and less known result states that a ring $R$ is a complete $(m+n)\times(m+n)$ matrix ring if and only if, $R$ contains three elements, $a$, $b$, and $f$, satisfying the two relations $af^m+f^nb=1$ and $f^{m+n}=0$. In many instances the two elements $a$ and $b$ can be replaced by appropriate powers $a^i$ and $a^j$ of a single element $a$ respectively. In general very little is known about the structure of the ring $S$. In this article we study in depth the case $m=n=1$ when $R\cong M_2(S)$. More specifically we study the universal algebra over a commutative ring $A$ with elements $x$ and $y$ that satisfy the relations $x^iy+yx^j=1$ and $y^2=0$. We describe completely the structure of these $A$-algebras and their underlying rings when $\gcd(i,j)=1$. Finally we obtain results that fully determine when there are surjections onto $M_2({\mathbb F})$ when ${\mathbb F}$ is a base field ${\mathbb Q}$ or ${\mathbb Z}_p$ for a prime number $p$., 32 pages

In the paper, we consider a two-symbol system of encoding for real numbers with two bases having different signs \({g_0

We define an admissible decomposition of a graph $E$ into subgraphs $F_1$ and $F_2$, and consider the intersection graph $F_1\cap F_2$ as a subgraph of both $F_1$ and $F_2$. We prove that, if the graph $E$ is row finite and its decomposition into the subgraphs $F_1$ and $F_2$ is admissible, then the graph C*-algebra $C^*(E)$ of $E$ is the pullback C*-algebra of the canonical surjections from $C^*(F_1)$ and $C^*(F_2)$ onto $C^*(F_1\cap F_2)$., Comment: 10 pages; acknowledgments changed

In this paper some new refinements of the discrete Jensen’s inequality are obtained in real vector spaces. The idea comes from some former refinements determined by cyclic permutations. We essentially generalize and extend these results by using permutations of finite sets and bijections of the set of positive numbers. We get refinements of the discrete Jensen’s inequality for infinite convex combinations in Banach spaces. Similar results are rare. Finally, some applications are given on different topics.

A Dyck path is non-decreasing if the y -coordinates of its valleys form a non-decreasing sequence. In this paper we give enumerative results and some statistics of several aspects of non-decreasing Dyck paths. We give the number of pyramids at a fixed level that the paths of a given length have, count the number of primitive paths, count how many of the non-primitive paths can be expressed as a product of primitive paths, and count the number of paths of a given height and a given length. We present and prove our results using combinatorial arguments, generating functions (using the symbolic method) and parameterize the results studied here using the Riordan arrays. We use known bijections to connect direct column-convex polyominoes, Elena trees, and non-decreasing Dyck paths.

In 1986, Shi derived the famous formula $(n+1)^{n-1}$ for the number of regions of the Shi arrangement, a hyperplane arrangement in $\mathbb{R}^n$. There are at least two different bijective explanations of this formula, one by Pak and Stanley, another by Athanasiadis and Linusson. In 1996, Athanasiadis used the finite field method to derive a formula for the number of $k$-dimensional faces of the Shi arrangement for any $k$. Until now, the formula of Athanasiadis did not have a bijective explanation. In this paper, we extend a bijection for regions defined by Bernardi to obtain a bijection between the $k$-dimensional faces of the Shi arrangement for any $k$ and a set of decorated binary trees. Furthermore, we show how these trees can be converted to a simple set of functions of the form $f: [n-1] \to [n+1]$ together with a marked subset of $\text{Im}(f)$. This correspondence gives the first bijective proof of the formula of Athanasiadis. In the process, we also obtain a bijection and counting formula for the faces of the Catalan arrangement. All of our results generalize to both extended arrangements., Comment: 45 pages, 13 figures

Gene activation is usually a non-Markovian process that has been modeled as various frameworks that consist of multiple rate-limiting steps. Understanding the exact activation framework for a gene of interest is a central problem for single-cell studies. In this paper, we focus on the dynamical data of the average transcription level M (t), which is typically neglected when deciphering gene activation. Firstly, the smooth trend lines of M (t) data present rich, visually dynamic features. Secondly, tractable analysis of M (t) allows the establishment of bijections between M (t) dynamics and system parameter regions. Because of these two clear advantages, we can rule out frameworks that fail to capture M (t) features and we can further test potential competent frameworks by fitting M (t) data. We implemented this procedure to determine an exact activation framework for a large number of mouse fibroblast genes under tumor necrosis factor induction; the cross-talk between the signaling and basal pathways is crucial to trigger the first peak of M (t), while the following damped gentle M (t) oscillation is regulated by the multi-step basal pathway. Moreover, the fitted parameters for the mouse genes tested revealed two distinct regulation scenarios for transcription dynamics. Taken together, we were able to develop an efficient procedure for using traditional M (t) data to estimate the gene activation frameworks and system parameters. This procedure, together with sophisticated single-cell transcription data, may facilitate a more accurate understanding of stochastic gene activation.Author SummaryIt has been suggested that genes randomly transit between inactive and active states, with mRNA produced only when a gene is active. The gene activation process has been modeled as a framework of multiple rate-limiting steps listed sequentially, parallel, or in combination. The system step numbers and parameters can be predicted by computationally fitting sophisticated single-cell transcription data. However, current algorithms require a prior hypothetical framework of gene activation. We found that the prior estimation of the framework can be achieved using the traditional dynamical data of mRNA average level M (t) which present easily discriminated dynamical features. The theory regarding M (t) profiles allows us to confidently rule out other frameworks and to determine optimal frameworks by fitting M (t) data. We successfully applied this procedure to a large number of mouse fibroblast genes and confirmed that M (t) is capable of providing a reliable estimation of gene activation frameworks and system parameters.

Let R be a commutative Noetherian ring. Denote by modR the category of finitely generated R-modules. In this paper, we study dominant resolving subcategories of modR. We introduce a new N-valued function on SpecR which we call a moderate function. Under an acceptable assumption, we construct explicit bijections between the set of dominant resolving subcategories of modR and the set of moderate functions on SpecR.

This article concerns Ehresmann structures in the partition monoid P X . Since P X contains the symmetric and dual symmetric inverse monoids on the same base set X, it naturally contains the semilattices of idempotents of both submonoids. We show that one of these semilattices leads to an Ehresmann structure on P X while the other does not. We explore some consequences of this (structural/combinatorial and representation theoretic), and in particular characterise the largest left-, right- and two-sided restriction submonoids. The new results are contrasted with known results concerning relation monoids, and a number of interesting dualities arise, stemming from the traditional philosophies of inverse semigroups as models of partial symmetries (Vagner and Preston) or block symmetries (FitzGerald and Leech): “surjections between subsets” for relations become “injections between quotients” for partitions. We also consider some related diagram monoids, including rook partition monoids, and state several open problems.

In this paper we introduce abstract string modules and give an explicit bijection between the submodule lattice of an abstract string module and the perfect matching lattice of the corresponding abstract snake graph. In particular, we make explicit the direct correspondence between a submodule of a string module and the perfect matching of the corresponding snake graph. For every string module, we define a Coxeter element in a symmetric group, and we establish a bijection between these lattices and the interval in the weak Bruhat order determined by the Coxeter element. Using the correspondence between string modules and snake graphs, we give a new concise formulation of snake graph calculus., 17 pages

We establish a poset structure on combinatorial bases of multivariate polynomials defined by positive expansions, and study properties common to bases in this poset. Included are the well-studied bases of Schubert polynomials, Demazure characters and Demazure atoms; the quasi-key, fundamental and monomial slide bases introduced in 2017 by Assaf and the author; and a new basis we introduce completing this poset structure. We show the product of a Schur polynomial and an element of a basis in this poset expands positively in that basis; in particular, we give the first Littlewood-Richardson rule for the product of a Schur polynomial and a quasi-key polynomial. This rule simultaneously extends Haglund, Luoto, Mason and van Willigenburg's (2011) Littlewood-Richardson rule for quasi-Schur polynomials and refines their Littlewood-Richardson rule for Demazure characters. We also establish bijections connecting combinatorial models for these polynomials including semi-skyline fillings and quasi-key tableaux., Comment: 26 pages, 9 figures. Section 5 updated and expanded with new results, minor updates throughout

In this paper, we expand on the work of Guo and Zeng (Adv Appl Math 39(1):96–115, 2007) on a generalisation of the Ramanujan polynomials and planar trees. We manage to find combinatorial interpretations of this family of polynomials in terms of Greg trees, Cayley trees and planar trees by constructing bijections that preserve relevant tree statistics.

Let a(n) be the number of partitions of n, such that the set of even parts has exactly one element, b(n) be the difference between the number of parts in all odd partitions of n and the number of parts in all distinct partitions of n, and c(n) be the number of partitions of n in which exactly one part is repeated. Beck conjectured that a(n) = b(n) and Andrews, using generating functions, proved that a(n) = b(n) = c(n). We give a combinatorial proof of Andrews’ result. Our proof relies on bijections between a set and a multiset, where the partitions in the multiset are decorated with bit strings. We prove combinatorially Beck’s second conjecture, which was also proved by Andrews using generating functions. Let c1(n) be the number of partitions of n, such that there is exactly one part occurring three times, while all other parts occur only once and let b1(n) be the difference between the total number of parts in the partitions of n into distinct parts and the total number of different parts in the partitions of n into odd parts. Then, c1(n) = b1(n).

In this work, we first present new bijections for the composition analogues of the partition theorems of Euler, Glaisher, and Franklin respectively. Then we generalize one composition analogue of Beck’s conjectures found by Huang recently, and give a purely combinatorial proof applying our bijection. Finally, a new and general composition analogue of Beck’s conjectures is established.

We prove the Schröder case, i.e. the case 〈⋅,e n−d h d 〉, of the conjecture of 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 generalized Delta conjecture. This result extends the Schröder case of the Delta conjecture proved in (D'Adderio, 2017), which in turn generalized the q,t-Schröder in Haglund (2004). The proof gives a recursion for these polynomials that extends the ones known for the aforementioned special cases. Also, we give another combinatorial interpretation of the same polynomial in terms of a new bounce statistic. Moreover, we give two more interpretations of the same polynomial in terms of doubly decorated parallelogram polyominoes, extending some of the results in D'Adderio (2017), which in turn extended results in Aval et al. (2014). Also, we provide combinatorial bijections explaining some of the equivalences among these interpretations., SCOPUS: ar.j, info:eu-repo/semantics/published

A loop L is strongly admissible if there exists a bijection ϕ : L → L for which the mappings g ↦ g ϕ ( g ) and g ↦ g ( g ϕ ( g ) ) are both bijections. We derive some results on the strong admissibility of finite nonassociative Moufang loops, constructed from groups using the Chein constructions.

A systematic study of avoidance of mesh patterns of length 2 was conducted by Hilmarsson et al., where 25 out of 65 non-equivalent cases were solved. In this paper, we give 27 distribution results for these patterns including 14 distributions for which avoidance was not known. Moreover, for the unsolved cases, we prove an equidistribution result (out of 6 equidistribution results we prove in total), and conjecture 6 more equidistributions. Finally, we find seemingly unknown distribution of the well known permutation statistic ``strict fixed point'', which plays a key role in many of our enumerative results. This paper is the first systematic study of distributions of mesh patterns. Our techniques to obtain the results include, but are not limited to, obtaining functional relations for generating functions, and finding recurrence relations and bijections., Comment: Advances in Applied Mathematics 110 (2019) 1-32; there is a mistake in the proof of Theorem 5.1 in version 1 of the paper; its statement is now a conjecture

Let G be a graph with undirected and directed edges. Its representation is given by assigning a vector space to each vertex, a bilinear form on the corresponding vector spaces to each directed edge, and a linear map to each directed edge. Two representations A and A' of G are called isomorphic if there is a system of linear bijections between the vector spaces corresponding to the same vertices that transforms A to A'. We prove that if two representations are isomorphic and close to each other, then their isomorphism can be chosen close to the identity., 12 pages

We explore inverses of disjointness preserving bijections in infinite dimensional normed pre-Riesz spaces by several methods. As in the case of Banach lattices, our aim is to show that such inverses are disjointness preserving. One method is extension of the operator to the Riesz completion, which works under suitable denseness and continuity conditions. Another method involves a condition on principle bands. Examples illustrate the differences to the Riesz space theory.

In the recent decades, core partitions and bar core partitions have attracted much attention from combinatorial researchers. In this paper, we build a bijection between $$\bar{t}$$ -cores and self-conjugate t-cores for odd integers $$t\ge 3$$ . To this end, we first establish a bijection between $$\bar{t}$$ -cores and self-conjugate $$(t-1)$$ -cores for odd integers $$t\ge 3$$ , then we prove that this bijection can be extended to a bijection between self-conjugate $$(t-1)$$ -cores and self-conjugate t-cores for odd integers $$t\ge 3$$ .

We study an alternating sign matrix analogue of the Chan–Robbins–Yuen polytope, which we call the ASM-CRY polytope. We show that this polytope has Catalan many vertices and its volume is equal to the number of standard Young tableaus of staircase shape; we also determine its Ehrhart polynomial. We achieve the previous by proving that the members of a family of faces of the alternating sign matrix polytope which includes ASM-CRY are both order and flow polytopes. Inspired by the above results, we relate three established triangulations of order and flow polytopes, namely Stanley’s triangulation of order polytopes, the Postnikov–Stanley triangulation of flow polytopes and the Danilov–Karzanov–Koshevoy triangulation of flow polytopes. We show that when a graph G is a planar graph, in which case the flow polytope $${{\mathcal {F}}}_G$$ is also an order polytope, Stanley’s triangulation of this order polytope is one of the Danilov–Karzanov–Koshevoy triangulations of $${{\mathcal {F}}}_G$$ . Moreover, for a general graph G we show that the set of Danilov–Karzanov–Koshevoy triangulations of $${{\mathcal {F}}}_G$$ equals the set of framed Postnikov–Stanley triangulations of $${{\mathcal {F}}}_G$$ . We also describe explicit bijections between the combinatorial objects labeling the simplices in the above triangulations.

When is a function from an abelian group to itself expressible as a difference of two bijections? Answering this question for finite cyclic groups solves a problem about juggling. A theorem of Mars...

We extend the result of B. Cascales et al. about expand-contract plasticity of the unit ball of strictly convex Banach space to those spaces whose unit sphere is the union of all its finite-dimensional polyhedral extreme subsets. We also extend the definition of expand-contract plasticity to uniform spaces and generalize the theorem on expand-contract plasticity of totally bounded metric spaces to this new setting.

Let $\mathcal{T}$ be a Krull-Schmidt, Hom-finite triangulated category with suspension functor $[1]$. Let $R$ be a basic rigid object, $\Gamma$ the endomorphism algebra of $R$, and $\operatorname{\mathsf{pr}}(R)\subseteq \mathcal{T}$ the subcategory of objects finitely presented by $R$. We investigate the relative rigid objects, \ie $R[1]$-rigid objects of $\mathcal{T}$. Our main results show that the $R[1]$-rigid objects in $\operatorname{\mathsf{pr}}(R)$ are in bijection with $\tau$-rigid $\Gamma$-modules, and the maximal $R[1]$-rigid objects with respect to $\operatorname{\mathsf{pr}}(R)$ are in bijection with support $\tau$-tilting $\Gamma$-modules. We also show that various previously known bijections involving support $\tau$-tilting modules are recovered under respective assumptions., Comment: 11 pages, minor changes

An ordered set partition of $\{1,2,\ldots,n\}$ is a partition with an ordering on the parts. Let $\mathcal{OP}_{n,k}$ be the set of ordered set partitions of $[n]$ with $k$ blocks. Godbole, Goyt, Herdan and Pudwell defined $\mathcal{OP}_{n,k}(\sigma)$ to be the set of ordered set partitions in $\mathcal{OP}_{n,k}$ avoiding a permutation pattern $\sigma$ and obtained the formula for $|\mathcal{OP}_{n,k}(\sigma)|$ when the pattern $\sigma$ is of length $2$. Later, Chen, Dai and Zhou found a formula algebraically for $|\mathcal{OP}_{n,k}(\sigma)|$ when the pattern $\sigma$ is of length $3$. In this paper, we define a new pattern avoidance for the set $\mathcal{OP}_{n,k}$, called $\mathcal{WOP}_{n,k}(\sigma)$, which includes the questions proposed by Godbole, Goyt, Herdan and Pudwell. We obtain formulas for $|\mathcal{WOP}_{n,k}(\sigma)|$ combinatorially for any $\sigma$ of length $ 3$. We also define 3 kinds of descent statistics on ordered set partitions and study the distribution of the descent statistics on $\mathcal{WOP}_{n,k}(\sigma)$ for $\sigma$ of length $3$., Comment: 42 pages, 16 figures

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 , … , b n ) -core partitions where { b 1 , b 2 , … , 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.

We deduce decompositions of natural representations of general linear groups and symmetric groups from combinatorial bijections involving tableaux. These include some of Howe's dualities, Gelfand models, the Schur-Weyl decomposition of tensor space, and multiplicity-free decompositions indexed by threshold partitions.