We introduce an infinite family of lower triangular matrices $Γ^{(s)}$, where $γ_{n,i}^s$ counts the standard Young tableaux on $n$ cells and with at most $s$ columns on a suitable subset of shapes. We show that the entries of these matrices satisfy a three-term row recurrence and we deduce recursive and asymptotic properties for the total number $τ_s(n)$ of tableaux on $n$ cells and with at most $s$ columns.
We survey some recent works on standard Young tableaux of bounded height. We focus on consequences resulting from numerous bijections to lattice walks in Weyl chambers.
We present a combinatorial model, called \emph{perforated tableaux}, to study $A_{n-1}$ crystals, unifying several previously studied combinatorial models. We identify nodes in the $k$-fold tensor product of the standard crystal with length $k$ words in $[n]= \{ 1, \ldots n\}$. We model this crystal with perforated tableaux (ptableaux), extending this identification isomorphically to biwords, RSK $(P,Q)$ tableaux pairs, and matrix models. In the ptableaux setting, crystal operators are more simply defined and we can identify highest weights visually without computation. We generalize the tenso…
We consider two examples of a fully decodable combinatorial encoding of Bernoulli schemes: the encoding via Weyl simplices and the much more complicated encoding via the RSK (Robinson--Schensted--Knuth) correspondence. In the first case, the decodability is a quite simple fact, while in the second case, this is a nontrivial result obtained by D.~Romik and P.~Śniady and based on the papers~ \cite{KV}, \cite{VK}, and others. We comment on the proofs from the viewpoint of the theory of measurable partitions; another proof, using representation theory and generalized Schur--Weyl duality, will be p…
We derive new combinatorial identities which may be viewed as multivariate analogs of summation formulas for hypergeometric series. As in the previous paper [Re], we start with probability distributions on the space of the infinite Young tableaux. Then we calculate the probability that the entry of a random tableau at a given box equals n=1,2,.... Summing these probabilities over n and equating the result to 1 we get a nontrivial identity. Our choice for the initial distributions is motivated by the recent work on harmonic analysis on the infinite symmetric group and related topics.
We introduce a new family of noncommutative analogues of the Hall-Littlewood symmetric functions. Our construction relies upon Tevlin's bases and simple q-deformations of the classical combinatorial Hopf algebras. We connect our new Hall-Littlewood functions to permutation tableaux, and also give an exact formula for the q-enumeration of permutation tableaux of a fixed shape. This gives an explicit formula for: the steady state probability of each state in the partially asymmetric exclusion process (PASEP); the polynomial enumerating permutations with a fixed set of weak excedances according t…
Chen et al. recently established bijections for $(d+1)$-noncrossing/ nonnesting matchings, oscillating tableaux of bounded height $d$, and oscillating lattice walks in the $d$-dimensional Weyl chamber. Stanley asked what is the total number of such tableaux of length $n$ and of any shape. We find a determinant formula for the exponential generating function. The same idea applies to prove Gessel's remarkable determinant formula for permutations with bounded length of increasing subsequences. We also give short algebraic derivations for some results of the reflection principle.
Tree-like tableaux are objects in bijection with alternative or permutation tableaux. They have been the subject of a fruitful combinatorial study for the past few years. In the present work, we define and study a new subclass of tree-like tableaux enumerated by Baxter numbers. We exhibit simple bijective links between these objects and three other combinatorial classes: (packed or mosaic) floorplans, twisted Baxter permutations and triples of non-intersecting lattice paths. From several (and unrelated) works, these last objects are already known to be enumerated by Baxter numbers, and our mai…
We discuss counting problems linked to finite versions of Cantor's diagonal of infinite tableaux. We extend previous results of [2] by refining an equivalence relation that reduces significantly the exhaustive generation. New enumerative results follow and allow to look at the sub-class of the so- called bi-Cantorian tableaux. We conclude with a correspondence between Cantorian-type tableaux and coloring of hypergraphs having a square number of vertices.
This paper presents a combinatorial study of the super plactic monoid of type A, which is related to the representations of the general linear Lie superalgebra. We introduce the analogue of the Schützenberger's jeu de taquin on the structure of super tableaux over a signed alphabet. We show that this procedure which transforms super skew tableaux into super Young tableaux is compatible with the super plactic congruence and it is confluent. We deduce properties relating the super jeu de taquin to insertion algorithms on super tableaux. Moreover, we introduce the super evacuation procedure as an…
Standard set-valued Young tableaux are a generalization of standard Young tableaux in which cells may contain more than one integer, with the added conditions that every integer at position $(i,j)$ must be smaller than every integer at positions $(i,j+1)$ and $(i+1,j)$. This paper explores the combinatorics of standard set-valued Young tableaux with two-rows, and how those tableaux may be used to provide new combinatorial interpretations of generalized Catalan numbers. New combinatorial interpretations are provided for the two-parameter Fuss-Catalan numbers (Raney numbers), the rational Catala…
Staircase tableaux are combinatorial objects which appear as key tools in the study of the PASEP physical model. The aim of this work is to show how the discovery of a tree structure in staircase tableaux is a significant feature to derive properties on these objects.
We define a number of related combinatorial objects, each of which possesses a surprising symmetry. We include several applications such as a combinatorial explanation for certain fixed points of the involution $ω$ on the ring of symmetric functions, as well as a relationship between certain skew Schur functions and skew $Q$-Schur functions. We give a $t$-deformation of these $Q$-Schur functions, and show that it is Schur positive, including a combinatorial description of the Schur coefficients. A corollary of our results is the equality of skew $Q$-Schur functions: $Q_{λ+δ/μ+ δ}=Q_{λ'+δ/μ' + …
In this work, we introduce new combinatorial objects called Dyck tableaux, which present a natural insertion algorithm. These tools may be useful to describe statistics which are relevant in the study of the physical model named PASEP.
Dumont and Foata introduced in 1976 a three-variable symmetric refinement of Genocchi numbers, which satisfies a simple recurrence relation. A six-variable generalization with many similar properties was later considered by Dumont. They generalize a lot of known integer sequences, and their ordinary generating function can be expanded as a Jacobi continued fraction. We give here a new combinatorial interpretation of the six-variable polynomials in terms of the alternative tableaux introduced by Viennot. A powerful tool to enumerate alternative tableaux is the so-called "matrix Ansatz", and usi…
Tree-like tableaux are combinatorial objects that appear in a combinatorial understanding of the PASEP model from statistical mechanics. In this understanding, the corners of the Southeast border correspond to the locations where a particle may jump to the right. Such corners may be of two types: either empty or occupied. Our main result is the following: on average there is one occupied corner per tree-like tableau. We give two proofs of this, a short one which gives us a polynomial version of the result, and another one using a bijection between tree-like tableaux and permutations which give…
In this paper we study a relatively new combinatorial object called staircase tableaux. Staircase tableaux were introduced by Corteel and Williams in the connection with Asymmetric Exclusion Process and has since found interesting connections with Askey - Wilson polynomials. We develop a probabilistic approach that allows us to analyze several parameters of a randomly chosen staircase tableaux of a given size. In particular, we obtain limiting distributions for statistics associated with appearances of Greek letters in staircase. tableaux.
We study a two-species PASEP, in which there are two types of particles, "heavy" and "light," hopping right and left on a one-dimensional lattice of $n$ cells with open boundaries. In this process, only the "heavy" particles can enter on the left of the lattice and exit from the right of the lattice. In the bulk, any transition where a heavier particle type swaps places with an adjacent lighter particle type is possible. We generalize combinatorial results of Corteel and Williams for the ordinary PASEP by defining a combinatorial object which we call a rhombic alternative tableau that gives a …
Egecioglu and Remmel gave an interpretation for the entries of the inverse Kostka matrix K^{-1} in terms of special rim-hook tableaux. They were able to use this interpretation to give a combinatorial proof that KK^{-1}=I but were unable to do the same for the equation K^{-1}K=I. We define a sign-reversing involution on rooted special rim-hook tableaux which can be used to prove that the last column of this second product is correct. In addition, following a suggestion of Chow we combine our involution with a result of Gasharov to give a combinatorial proof of a special case of the (3+1)-free …
We give a combinatorial proof of the skew Kostka analogue of the K-saturation theorem. More precisely, for any positive integer k, we give an explicit injection from the set of skew semistandard Young tableaux with skew shape kλ/kμ and type kν to the set of skew semistandard Young tableaux of shape λ/μ and type ν. Based on this method, we pose some natural conjectural refinements on related problems.
Baryshnikov and Romik derived the combinatorial identities for the numbers of the $m$-strip tableaux. This generalized the classical André's theorem for the number of up-down permutations. They asked for a bijective proof for the enumeration of $3$-strip tableaux. In this paper we will provide such a bijective proof. First we count the $3$-strip tableaux by decomposition. Secondly we will apply this "decomposition" idea on the up-down permutations and down-up permutations to enumerate the $3$-strip tableaux bijectively.
We introduce the notion of "type" of a tableau, that allows us to define new families of tableaux including both balanced and standard Young tableaux. We use these new objects to describe the set of reduced decompositions of any permutation. We then generalize the work of Fomin \emph{et al.} by giving, among other things, a new proof of the fact that balanced and standard tableaux are equinumerous, and by exhibiting many new families of tableaux having similar combinatorial properties to those of balanced tableaux.
Permutation tableaux were introduced by Steingr\'ımsson and Williams. Corteel and Kim defined the sign of a permutation tableau in terms of the number of unrestricted columns. The sign-imbalance of permutation tableaux of length $n$ is the sum of signs over permutation tableaux of length $n$. They have btained a formula for the sign-imbalance of permutation tableaux of length $n$ by using generating functions and asked for a combinatorial proof. Moreover, they raised the question of finding a sign-imbalance formula for type $B$ permutation tableaux introduced by Lam and Williams. We define a s…
A combinatorial description of the crystal B(infinity) for finite-dimensional simple Lie algebras in terms of certain Young tableaux was developed by J. Hong and H. Lee. We establish an explicit bijection between these Young tableaux and canonical bases indexed by Lusztig's parametrization, and obtain a combinatorial rule for expressing the Gindikin-Karpelevich formula as a sum over the set of Young tableaux.
We derive combinatorial identities, involving the Bernoulli and Euler numbers, for the numbers of standard Young tableaux of certain skew shapes. This generalizes the classical formulas of D. Andre on the number of up-down permutations. The analysis uses a transfer operator approach extending the method of Elkies, combined with an identity expressing the volume of a certain polytope in terms of a Schur function.
Cylindric Young tableaux are combinatorial objects that first appeared in the 1990s. A natural extension of the classical notion of a Young tableau, they have since been used several times, most notably by Gessel and Krattenthaler and by Alexander Postnikov. Despite this, relatively little is known about cylindric Young tableaux. This paper is an investigation of the properties of this object. In this paper, we extend the Robinson-Schensted-Knuth Correspondence, a well-known and very useful bijection concerning regular Young tableaux, to be a correspondence between pairs of cylindric tableaux.…
We explain how the moments of the (weight function of the) Askey Wilson polynomials are related to the enumeration of the staircase tableaux introduced by the first and fourth authors. This gives us a direct combinatorial formula for these moments. Then we use techniques developed by Ismail and the third author to give explicit formulae for these moments and for the enumeration of staircase tableaux. Finally we study the enumeration of staircase tableaux at various specializations of the parameterizations; for example, we obtain the Catalan numbers, Fibonacci numbers, Eulerian numbers, the num…
In work with A. Yong, the author introduced genomic tableaux to prove the first positive combinatorial rule for the Littlewood-Richardson coefficients in torus-equivariant $K$-theory of Grassmannians. We then studied the genomic Schur function $U_λ$, a generating function for such tableaux, showing that it is non-trivially a symmetric function, although generally not Schur-positive. Here we show that $U_λ$ is, however, positive in the basis of fundamental quasisymmetric functions. We give a positive combinatorial formula for this expansion in terms of gapless increasing tableaux; this is, more…
A poset on a certain class of partitions known as k-shapes was recently introduced to provide a combinatorial rule for the expansion of a (k-1)-Schur functions into k-Schur functions at t=1. The main ingredient in this construction was a bijection, which we call the weak bijection, that associates to a k-tableau a pair made out of a (k-1)-tableau and a path in the poset of k-shapes. We define here a concept of charge on k-tableaux (which conjecturally gives a combinatorial interpretation for the expansion coefficients of Hall-Littlewood polynomials into k-Schur functions), and show that it is …
The goal of this paper is to provide a combinatorial expression for the steady state probabilities of the two-species PASEP. In this model, there are two species of particles, one "heavy" and one "light", on a one-dimensional finite lattice with open boundaries. Both particles can swap places with adjacent holes to the right and left at rates 1 and $q$. Moreover, when the heavy and light particles are adjacent to each other, they can swap places as if the light particle were a hole. Additionally, the heavy particle can hop in and out at the boundary of the lattice. Our main result is a combina…
Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1 whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We present a unifying perspective on ASMs and other combinatorial objects by studying a certain tetrahedral poset and its subposets. We prove the order ideals of these subposets are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self-complementary plane partitions (TSSCPPs), staircase shaped semistandard Young tableaux, Catalan objects, tournaments, and totally symmetric plane partitions. …
Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, we study six combinatorial Hopf algebras. These Hopf algebras can be thought of as K-theoretic analogues of the by now classical ``square'' of Hopf algebras consisting of symmetric functions, quasisymmetric functions, noncommutative symmetric functions and the Malvenuto-Reutenauer Hopf algebra of permutations. In addition, we develop a theory of set-valued P-partitions and study three new families of symmetric functions which are weight generating functions of reverse plane partitions, weak set-va…
We show that combinatorial objects called row-strict composition tableaux, introduced by Mason and Remmel in 2014 and closely related to the quasi-symmetric Schur functions of Haglund-Luoto-Mason-van Willigenburg, form a basis for Schur functors of finite free modules over arbitrary commutative rings. When the ring is the complex numbers, this produces a new basis for the irreducible polynomial representations of $\operatorname{GL}_n(\mathbb{C})$. Moreover, in this case it also produces new basis for the Plücker algebra, a subalgebra of the polynomial ring over $\mathbb{C}$ in $n^2$ variables…
Schur-Weyl duality is a fundamental framework in combinatorial representation theory. It intimately relates the irreducible representations of a group to the irreducible representations of its centralizer algebra. We investigate the analog of Schur-Weyl duality for the group of unipotent upper triangular matrices over a finite field. In this case, the character theory of these upper triangular matrices is "wild" or unattainable. Thus we employ a generalization, known as supercharacter theory, that creates a striking variation on the character theory of the symmetric group with combinatorics bu…
Permutation tableaux are combinatorial objects related with permutations and various statistics on them. They appeared in connection with total positivity in Grassmannians, and stationary probabilities in a PASEP model. In particular they gave rise to an interesting q-analog of Eulerian numbers. The purpose of this article is to study some combinatorial properties of type B permutation tableaux, defined by Lam and Williams, and links with signed permutation statistics. We show that many of the tools used for permutation tableaux generalize in this case, including: the Matrix Ansatz (a method o…
We introduce a new combinatorial object called tower diagrams and prove fundamental properties of these objects. We also introduce an algorithm that allows us to slide words to tower diagrams. We show that the algorithm is well-defined only for reduced words which makes the algorithm a test for reducibility. Using the algorithm, a bijection between tower diagrams and finite permutations is obtained and it is shown that this bijection specializes to a bijection between certain labellings of a given tower diagram and reduced expressions of the corresponding permutation.
We introduce a family of tableaux that simultaneously generalizes the tableaux used to characterize Grothendieck polynomials and k-Schur functions. We prove that the polynomials drawn from these tableaux are the affine Grothendieck polynomials and k-K-Schur functions -- Schubert representatives for the K-theory of affine Grassmannians and their dual in the nil Hecke ring. We prove a number of combinatorial properties including Pieri rules.
Using the expansion of the inverse of the Kostka matrix in terms of tabloids as presented by Egecioglu and Remmel, we show that the fusion coefficients can be expressed as an alternating sum over cylindric tableaux. Cylindric tableaux are skew tableaux with a certain cyclic symmetry. When the skew shape of the tableau has a cutting point, meaning that the cylindric skew shape is not connected, or if its weight has at most two parts, we give a positive combinatorial formula for the fusion coefficients. The proof uses a slight modification of a sign-reversing involution introduced by Remmel and …
This paper concerns a relatively new combinatorial structure called staircase tableaux. They were introduced in the context of the asymmetric exclusion process and Askey--Wilson polynomials, however, their purely combinatorial properties have gained considerable interest in the past few years. In this paper we further study combinatorial properties of staircase tableaux. We consider a general model of staircase tableaux in which symbols that appear in staircase tableaux may have arbitrary positive weights. Under this general model we derive a number of results. Some of our results concern the …
Tableau sequences of bounded height have been central to the analysis of k-noncrossing set partitions and matchings. We show here that familes of sequences that end with a row shape are particularly compelling and lead to some interesting connections. First, we prove that hesitating tableaux of height at most two ending with a row shape are counted by Baxter numbers. This permits us to define three new Baxter classes which, remarkably, do not obviously possess the antipodal symmetry of other known Baxter classes. We then conjecture that oscillating tableau of height bounded by k ending in a ro…
Irreducible characters in the symmetric group are of special interest in combinatorics. They can be expressed either combinatorially with ribbon tableaux, or algebraically with contents. In this paper, these two expressions are related in a combinatorial way. We first introduce a fine structure in the famous jeu de taquin called "trace forest", with which we are able to count certain types of ribbon tableaux, leading to a simple bijective proof of a character evaluation formula in terms of contents that dates back to Frobenius (1901). Inspired by this proof, we give an inductive scheme that gi…
Set-valued tableaux play an important role in combinatorial $K$-theory. Separately, semistandard skyline fillings are a combinatorial model for Demazure atoms and key polynomials. We unify these two concepts by defining a set-valued extension of semistandard skyline fillings and then give analogues of results of J. Haglund, K. Luoto, S. Mason, and S. van Willigenberg. Additionally, we give a bijection between set-valued semistandard Young tableaux and C. Lenart's Schur expansion of the Grothendieck polynomial $G_λ$, using the uncrowding operator of V. Reiner, B. Tenner, and A. Yong.
For any polynomial representation of the special linear group, the nodes of the corresponding crystal may be indexed by semi-standard Young tableaux. Under certain conditions, the standard Young tableaux occur, and do so with weight 0. Standard Young tableaux also parametrize the vertices of dual equivalence graphs. Motivated by the underlying representation theory, in this paper, we explainthis connection by giving a combinatorial manifestation of Schur-Weyl duality. In particular, we put a dual equivalence graph structure on the 0-weight space of certain crystal graphs, producing edges combi…
In this article, we consider a generalization of Young tableaux in which we allow some consecutive pairs of cells with decreasing labels. We show that this leads to a rich variety of combinatorial formulas, which suggest that these new objects could be related to deeper structures, similarly to the ubiquitous Young tableaux. Our methods rely on variants of hook-length type formulas, and also on a new efficient generic method (which we call the density method) which allows not only to generate constrained combinatorial objects, but also to enumerate them. We also investigate some repercussions …
The RSK correspondence generalises the Robinson-Schensted correspondence by replacing permutation matrices by matrices with entries in ${\bf N}$, and standard Young tableaux by semistandard ones. For $r>0$, the Robinson-Schensted correspondence can be trivially extended, using the $r$-quotient map, to one between coloured permutations and pairs of standard $r$-ribbon tableaux built on a fixed $r$-core (the Stanton-White correspondence). This correspondence can also be generalised to arbitrary matrices with entries in ${\bf N}^r$ and pairs of semistandard $r$-ribbon tableaux built on a fixed $r…
The main aim of the paper is to present a~combinatorial algorithm that, applying Littlewood-Richardson tableaux with entries equal to $1$, computes generic extensions of semisimple invariant subspaces of nilpotent linear operators. Moreover, we discuss geometric properties of generic extensions and their connections with combinatorics.
We study k-Schur functions characterized by k-tableaux, proving combinatorial properties such as a k-Pieri rule and a k-conjugation. This new approach relies on developing the theory of k-tableaux, and includes the introduction of a weight-permuting involution on these tableaux that generalizes the Bender-Knuth involution. This work lays the groundwork needed to prove that the set of k-Schur Littlewood-Richardson coefficients contains the 3-point Gromov-Witten invariants; structure constants for the quantum cohomology ring.
We introduce the new combinatorial approach of plethystic type of tableaux, as a method to understand coefficients of Schur functions appearing in plethysms $s_ν[h_λ]$ and $s_ν[e_λ]$, for any partitions $λ$ and $ν$. We first give general results about this approach, then use results on tableaux, ribbon tableaux and integer points in polytopes to understand the case where $ν$ is a partition of $3$ and $λ$ has one part. We then use a \textit{Kronecker map} to extend these results to any partition $λ$.
A tableau inversion is a pair of entries in row-standard tableau $T$ that lie in the same column of $T$ yet lack the appropriate relative ordering to make $T$ column-standard. An $i$-inverted Young tableau is a row-standard tableau along with a precisely $i$ inversion pairs. Tableau inversions were originally introduced by Fresse to calculate the Betti numbers of Springer fibers in Type A, with the number of $i$-inverted tableaux that standardize to a fixed standard Young tableau corresponding to a specific Betti number of the associated fiber. In this paper we approach the topic of tableau in…
We introduce and study some affine Hecke algebras of type ADE, generalising the affine Hecke algebras of GL. We construct irreducible calibrated representations and describe the calibrated spectrum. This is done in terms of new families of combinatorial objects equipped with actions of the corresponding Weyl groups. These objects are built from and generalise the usual standard Young tableaux, and are controlled by the considered affine Hecke algebras. By restriction and limiting procedure, we obtain several combinatorial models for representations of finite Hecke algebras and Weyl groups of t…
The expectation of the descent number of a random Young tableau of a fixed shape is given, and concentration around the mean is shown. This result is generalized to the major index and to other descent functions. The proof combines probabilistic arguments together with combinatorial character theory. Connections with Hecke algebras are mentioned.
We introduce a new combinatorial object, semistandard increasing decomposition tableau and study its relation to a semistandard decomposition tableau introduced by Kraśkiewicz and developed by Lam and Serrano. We also introduce generalized Littlewood--Richardson coefficients for products of Schur's symmetric functions and give combinatorial descriptions in terms of tableau words. The insertion algorithms play central roles for proofs. A new description of shifted Littlewood--Richardson coefficients is given in terms of semistandard increasing decomposition tableaux. We show that a "big" Schur …
Bouvel and Pergola introduced the notion of minimal permutations in the study of the whole genome duplication-random loss model for genome rearrangements. Let $\mathcal{F}_d(n)$ denote the set of minimal permutations of length $n$ with $d$ descents, and let $f_d(n)= |\mathcal{F}_d(n)|$. They derived that $f_{n-2}(n)=2^{n}-(n-1)n-2$ and $f_n(2n)=C_n$, where $C_n$ is the $n$-th Catalan number. Mansour and Yan proved that $f_{n+1}(2n+1)=2^{n-2}nC_{n+1}$. In this paper, we consider the problem of counting minimal permutations in $\mathcal{F}_d(n)$ with a prescribed set of ascents. We show that suc…
