The $k$-Young lattice $Y^k$ is a partial order on partitions with no part larger than $k$. This weak subposet of the Young lattice originated from the study of the $k$-Schur functions(atoms) $s_λ^{(k)}$, symmetric functions that form a natural basis of the space spanned by homogeneous functions indexed by $k$-bounded partitions. The chains in the $k$-Young lattice are induced by a Pieri-type rule experimentally satisfied by the $k$-Schur functions. Here, using a natural bijection between $k$-bounded partitions and $k+1$-cores, we establish an algorithm for identifying chains in the $k$-Young lattice with certain tableaux on $k+1$ cores. This algorithm reveals that the $k$-Young lattice is isomorphic to the weak order on the quotient of the affine symmetric group $\tilde S_{k+1}$ by a maximal parabolic subgroup. From this, the conjectured $k$-Pieri rule implies that the $k$-Kostka matrix connecting the homogeneous basis $\{h_\la\}_{\la\in\CY^k}$ to $\{s_\la^{(k)}\}_{\la\in\CY^k}$ may now be obtained by counting appropriate classes of tableaux on $k+1$-cores. This suggests that the conjecturally positive $k$-Schur expansion coefficients for Macdonald polynomials (reducing to $q,t$-Kostka polynomials for large $k$) could be described by a $q,t$-statistic on these tableaux, or equivalently on reduced words for affine permutations.
We introduce the quasi-key basis of the polynomial ring. We prove this basis contains the quasi-Schur polynomials of of Haglund, Luoto, Mason and van Willigenburg and that stable limits of quasi-key polynomials are quasi-Schur functions, thus giving a lifting of the quasi-Schur basis of quasisymmetric polynomials to the full polynomial ring. We introduce the combinatorial model of Kohnert tableaux, and use this model to prove that key polynomials expand positively in quasi-key polynomials which in turn expand positively in the fundamental slide polynomials introduced earlier by the authors. We…
We define crystal operators on semistandard shifted tableaux, giving a new proof that Schur $P$-functions are Schur positive. We define a queer crystal operator to construct a connected queer crystal on semistandard shifted tableaux of a given shape, providing a new proof that products of Schur $P$-functions are Schur $P$-positive. We also give a rectification map from shifted tableaux to Young tableaux that commutes with the crystal operators and provides a dual algorithm to shifted insertion.
Many important statistics of signed permutations are realized in the corresponding permutation tableaux or bare tableaux of type $B$: Alignments, crossings and inversions of signed permutations are realized in the corresponding permutation tableaux of type $B$, and the cycles of signed permutations are understood in the corresponding bare tableaux of type $B$. This leads us to relate the number of alignments and crossings with other statistics of signed permutations and also to characterize the covering relation in weak Bruhat order on Coxeter system of type $B$ in terms of permutation tableau…
We study Schur Q-polynomials evaluated on a geometric progression, or equivalently q-enumeration of marked shifted tableaux, seeking explicit formulas that remain regular at q=1. We obtain several such expressions as multiple basic hypergeometric series, and as determinants and pfaffians of continuous q-ultraspherical or continuous q-Jacobi polynomials. As special cases, we obtain simple closed formulas for staircase-type partitions.
We consider the relation between various permutation statistics and properties of permutation tableaux. We answer some of the questions of Steingrimsson and Williams (math.CO/0507149), in particular, on the distribution of the bistatistic of numbers of rows and essential ones in permutation tableaux. We also consider and enumerate sets of permutation tableaux related to some pattern restrictions on permutations.
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' + …
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.
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…
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 …
In this paper we introduce and study a class of tableaux which we call permutation tableaux; these tableaux are naturally in bijection with permutations, and they are a distinguished subset of the Le-diagrams of Alex Postnikov. The structure of these tableaux is in some ways more transparent than the structure of permutations; therefore we believe that permutation tableaux will be useful in furthering the understanding of permutations. We give two bijections from permutation tableaux to permutations. The first bijection carries tableaux statistics to permutation statistics based on relative si…
We give a solution to a problem posed by Corteel and Nadeau concerning permutation tableaux of length n and the number of occurrences of the dashed pattern 32--1 in permutations on [n]. We introduce the inversion number of a permutation tableau. For a permutation tableau T and the permutation $π$ obtained from T by the bijection of Corteel and Nadeau, we show that the inversion number of T equals the number of occurrences of the dashed pattern 32--1 in the reverse complement of $π$. We also show that permutation tableaux without inversions coincide with L-Bell tableaux introduced by Corteel an…
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…
We use the 0-1 tableaux as a tool for enumerating permutations and partitions with restricted minima. The method used is extended for permutation pairs and partition pairs generated by a bipartite 0-1 tableaux.
For a $(3+1)$-free poset $P$, we define a hybrid of $P$-tableaux and cylindric tableaux called cylindric $P$-tableaux. We introduce $P$-analogs of cylindric Schur functions, defined by a determinantal formula, and prove that they are the weight generating functions of cylindric $P$-tableaux. We deduce that certain sums of the $e$-expansion coefficients of the chromatic symmetric function $X_{inc(P)}$ are positive. This improves on Gasharov's theorem on the Schur positivity of $X_{inc(P)}$ and gives further evidence for the Stanley-Stembridge conjecture.
In this paper, we give bijections between the set of 4123-avoiding down-up alternating permutations of length $2n$ and the set of standard Young tableaux of shape $(n,n,n)$, and between the set of 4123-avoiding down-up alternating permutations of length $2n-1$ and the set of shifted standard Young tableaux of shape $(n+1, n, n-1)$ via an intermediate structure of Yamanouchi words. Moreover, we get the enumeration of 4123-avoiding up-down alternating permutations of even and odd length by presenting bijections between 4123-avoiding up-down alternating permutations and standard Young tableaux.
We give another bijective proof of a result of Corteel and Nadeau. We find a generating function related to unrestricted columns of permutation tableaux. As a consequence, we obtain a sign-imbalance formula for permutation tableaux. We extend the first bijection of Corteel and Nadeau between permutations and permutation tableaux to type $B$ objects. Using this type $B$ bijection, we generalize a result of Lam and Williams. We prove that the bijection of Corteel and Nadeau and our type $B$ bijection can be expressed as zigzag maps on the alternative representation.
Quasi-Yamanouchi tableaux are a subset of semistandard Young tableaux and refine standard Young tableaux. They are closely tied to the descent set of standard Young tableaux and were introduced by Assaf and Searles to tighten Gessel's fundamental quasisymmetric expansion of Schur functions. The descent set and descent statistic of standard Young tableaux repeatedly prove themselves useful to consider, and as a result, quasi-Yamanouchi tableaux make appearances in many ways outside of their original purpose. Some examples, which we present in this paper, include the Schur expansion of Jack poly…
In this paper, we confirm a conjecture of Laborde-Zubieta on the enumeration of corners in tree-like tableaux. Our proof is based on Aval, Boussicault and Nadeau's bijection between tree-like tableaux and permutation tableaux, and Corteel and Nadeau's bijection between permutation tableaux and permutations. This last bijection sends a corner in permutation tableaux to an ascent followed by a descent in permutations, this enables us to enumerate the number of corners in permutation tableaux, and thus to completely solve L.-Z.'s conjecture. Moreover, we give a bijection between corners and runs …
This paper contains a partial answer to the open problem 3.11 of \cite{[H2008]}. That is to find an explicit bijection on Schröder paths that inverts the statistics area and bounce. This paper started as an attempt to write the sum over $m$-Schröder paths with a fix number of diagonal steps into Schur functions in the variables $q$ and $t$. Some results have been generalized to parking functions, and some bijections were made with standard Young tableaux giving way to partial combinatorial formulas in the basis $s_μ(q,t)s_λ(X)$ for $\nabla(e_n)$ (respectively, $\nabla^m(e_n)$), when $μ$ and $λ…
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…
Young's lattice, the lattice of all Young diagrams, has the Robinson-Schensted-Knuth correspondence, the correspondence between certain matrices and pairs of semi-standard Young tableaux with the same shape. Fomin introduced generalized Schur operators to generalize the Robinson-Schensted-Knuth correspondence. In this sense, generalized Schur operators are generalizations of semi-standard Young tableaux. We define a generalization of Schur polynomials as expansion coefficients of generalized Schur operators. We show that the commutating relation of generalized Schur operators implies Pieri's f…
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…
We present a family of analogs of the Hall-Littlewood symmetric functions in the $Q$-function algebra. The change of basis coefficients between this family and Schur's $Q$-functions are $q$-analogs of numbers of marked shifted tableaux. These coefficients exhibit many parallel properties to the Kostka-Foulkes polynomials.
We prove that graded $k$-Schur functions are $G$-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We expose a new miraculous shift invariance property of the graded $k$-Schur functions and resolve the Schur positivity and $k$-branching conjectures in the strongest possible terms by providing direct combinatorial formulas using strong marked tableaux.
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…
We recently generalised the lattice permutation condition for Young tableaux to Kronecker tableaux and hence calculated a large new class of stable Kronecker coefficients labelled by co-Pieri triples. In this extended abstract we discuss important families of co-Pieri triples for which our combinatorics simplifies drastically.
Type-B permutation tableaux are combinatorial objects introduced by Lam and Williams that have an interesting connection with the partially asymmetric simple exclusion process (PASEP). In this paper, we compute the expected value of several statistics on these tableaux. Some of these computations are motivated by a similar paper on permutation tableaux. Others are motivated by the PASEP. In particular, we compute the expected number of rows, unrestricted rows, diagonal ones, adjacent south steps, and adjacent west steps.
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.
In this paper we classify all Schur functions and skew Schur functions that are multiplicity free when expanded in the basis of fundamental quasisymmetric functions, termed F-multiplicity free. Combinatorially, this is equivalent to classifying all skew shapes whose standard Young tableaux have distinct descent sets. We then generalize our setting, and classify all F-multiplicity free quasisymmetric Schur functions with one or two terms in the expansion, or one or two parts in the indexing composition. This identifies composition shapes such that all standard composition tableaux of that shape…
In this paper, we introduce the rhombic alternative tableaux, whose weight generating functions provide combinatorial formulae to compute the steady state probabilities of the two-species ASEP. In the ASEP, there are two species of particles, one heavy and one light, hopping right and left on a one-dimensional finite lattice with open boundaries. Parameters $α$, $β$, and $q$ describe the hopping probabilities. The rhombic alternative tableaux are enumerated by the Lah numbers, which also enumerate certain assemblées of permutations. We describe a bijection between the rhombic alternative table…
For each fully commutative permutation, we construct a "boolean core," which is the maximal boolean permutation in its principal order ideal under the right weak order. We partition the set of fully commutative permutations into the recently defined crowded and uncrowded elements, distinguished by whether or not their RSK insertion tableaux satisfy a sparsity condition. We show that a fully commutative element is uncrowded exactly when it shares the RSK insertion tableau with its boolean core. We present the dynamics of the right weak order on fully commutative permutations, with particular in…
The k-Schur functions were first introduced by Lapointe, Lascoux and Morse (2003) in the hopes of refining the expansion of Macdonald polynomials into Schur functions. Recently, an alternative definition for k-Schur functions was given by Lam, Lapointe, Morse, and Shimozono (2010) as the weighted generating function of starred strong tableaux which correspond with labeled saturated chains in the Bruhat order on the affine symmetric group modulo the symmetric group. This definition has been shown to correspond to the Schubert basis for the affine Grassmannian of type A by Lam (2008), and, at t …
Haglund, Luoto, Mason, and van Willigenburg introduced a basis for quasisymmetric functions called the quasisymmetric Schur function basis, generated combinatorially through fillings of composition diagrams in much the same way as Schur functions are generated through reverse column-strict tableaux. We introduce a new basis for quasisymmetric functions called the row-strict quasisymmetric Schur function basis, generated combinatorially through fillings of composition diagrams in much the same way as Schur functions are generated through row-strict tableaux. We describe the relationship between…
By considering the specialisation $s_λ(1,q,q^2,...,q^{n-1})$ of the Schur function, Stanley was able to describe a formula for the number of semistandard Young tableaux of shape $λ$ in terms of two properties of the boxes in the diagram for $λ$. Using specialisations of symplectic and orthogonal Schur functions, we derive corresponding formulae, first given by El Samra and King, for the number of semistandard symplectic and orthogonal $λ$-tableaux.
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.
We take inspiration from the Okounkov-Vershik approach to the representation theory of the symmetric groups to develop a new way of understanding how the Schur-Weyl duality can be used to perform the Quantum Schur Transform. The Quantum Schur Transform is a unitary change of basis transformation between the computational basis of $(\mathbb{C}^d)^{\otimes n}$ and the Schur-Weyl basis of $(\mathbb{C}^d)^{\otimes n}$. We describe a new multigraph, which we call the Schur-Weyl-Young graph, that represents both standard Weyl tableaux and standard Young tableaux in the same diagram. We suggest a maj…
In this work we introduce and study tree-like tableaux, which are certain fillings of Ferrers diagrams in simple bijection with permutation tableaux and alternative tableaux. We exhibit an elementary insertion procedure on our tableaux which gives a clear proof that tree-like tableaux of size n are counted by n!, and which moreover respects most of the well-known statistics studied originally on alternative and permutation tableaux. Our insertion procedure allows to define in particular two simple new bijections between tree-like tableaux and permutations: the first one is conceived specifical…
The problem of finding Schur-positive sets of permutations, originally posed by Gessel and Reutenauer, has seen some recent developments. Schur-positive sets of pattern-avoiding permutations have been found by Sagan et al and a general construction based on geometric operations on grid classes has been given by the authors. In this paper we prove that horizontal rotations of Schur-positive subsets of permutations are always Schur-positive. The proof applies a cyclic action on standard Young tableaux of certain skew shapes and a jeu-de-taquin type straightening algorithm. As a consequence of th…
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…
The notion of set-valued Young tableaux was introduced by Buch in his study of the Littlewood-Richardson rule for stable Grothendieck polynomials. Knutson, Miller and Yong showed that the double Grothendieck polynomials of 2143-avoiding permutations can be generated by set-valued Young tableaux. In this paper, we introduce the structure of set-valued Rothe tableaux of permutations. Given the Rothe diagram $D(w)$ of a permutation $w$, a set-valued Rothe tableau of shape $D(w)$ is a filling of finite nonempty subsets of positive integers into the squares of $D(w)$ such that the rows are weakly d…
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 …
We define a class L_{n, k} of permutations that generalizes alternating (up-down) permutations and give bijective proofs of certain pattern-avoidance results for this class. As a special case of our results, we give two bijections between the set A_{2n}(1234) of alternating permutations of length 2n with no four-term increasing subsequence and standard Young tableaux of shape (3^n), and between the set A_{2n + 1}(1234) and standard Young tableaux of shape (3^{n - 1}, 2, 1). This represents the first enumeration of alternating permutations avoiding a pattern of length four. We also extend previ…
Permutation statistics $\wnm$ and $\rlm$ are both arising from permutation tableaux. $\wnm$ was introduced by Chen and Zhou, which was proved equally distributed with the number of unrestricted rows of a permutation tableau. While $\rlm$ is showed by Nadeau equally distributed with the number of $1$'s in the first row of a permutation tableau. In this paper, we investigate the joint distribution of $\wnm$ and $\rlm$. Statistic $(\rlm,\wnm,\rlmin,\des,(\underline{321}))$ is shown equally distributed with $(\rlm,\rlmin,\wnm,\des,(\underline{321}))$ on $S_n$. Then the generating function of $(\rl…
We investigate pattern avoidance in alternating permutations and generalizations thereof. First, we study pattern avoidance in an alternating analogue of Young diagrams. In particular, we extend Babson-West's notion of shape-Wilf equivalence to apply to alternating permutations and so generalize results of Backelin-West-Xin and Ouchterlony to alternating permutations. Second, we study pattern avoidance in the more general context of permutations with restricted ascents and descents. We consider a question of Lewis regarding permutations that are the reading words of thickened staircase Young t…
By considering type B analogs of permutations and tableaux, we extend abstract dual equivalence to type B in two directions. In one direction, we define involutions on signed permutations and shifted tableaux that give a weak dual equivalence, thereby giving another proof of the Schur positivity of Schur Q- and P-functions. In another direction, we define an abstract shifted dual equivalence parallel to dual equivalence and prove that it can be used to establish Schur P-positivity of a function expressed as a sum of shifted fundamental quasisymmetric functions.
In this paper we use a probabilistic approach to derive the expressions for the characteristic functions of basic statistics defined on permutation tableaux. Since our expressions are exact, we can identify the distributions of basic statistics (like the number of unrestricted rows, the number of rows, and the number of 1s in the first row) exactly. In all three cases the distributions are known to be asymptotically normal after a suitable normalization. We also establish the asymptotic normality of the number of superfluous 1s. The latter result relies on a bijection between permutation table…
In this paper we study alternative tableaux introduced by Viennot. These tableaux are in simple bijection with permutation tableaux, defined previously by Postnikov . We exhibit a simple recursive structure for alternative tableaux. From this decomposition, we can easily deduce a number of enumerative results. We also give bijections between these tableaux and certain classes of labeled trees. Finally, we exhibit a bijection with permutations, and relate it to some other bijections that already appeared in the literature.
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…
Inspired by the spin-inversion statistics of Schilling et al. and Haglund et al., we relate the symmetry of ribbon functions to a result of van Leeuwen, and also describe the multiplication of a domino function by a Schur function.
