Chalykh, Veselov and Feigin introduced the notions of quasiinvariants for Coxeter groups, which is a generalization of invariants. In [2], Bandlow and Musiker showed that for the symmetric group $S_n$ of order $n$, the space of quasiinvariants has a decomposition indexed by standard tableaux. They gave a description of basis for the components indexed by standard tableaux of shape $(n-1,1)$. In this paper, we generalize their results to a description of basis for the components indexed by standard tableaux of arbitrary hook shape.
In 2002, Feigin and Veselov defined the space of m-quasiinvariants for any Coxeter group, building on earlier work of Chalykh and Veselov. While many properties of those spaces were proven from this definition, an explicit computation of a basis was only done in certain cases. In particular, Feigin and Veselov computed bases for the m-quasiinvariants of dihedral groups, including S_3, and Felder and Veselov computed the non-symmetric m-quasiinvariants of lowest degree for general S_n. In this paper, we provide a new characterization of the m-quasiinvariants of S_n, and use this to provide a ba…
A few years ago, Naruse presented a beautiful cancellation-free hook-length formula for skew shapes, both straight and shifted. The formula involves a sum over objects called \emph{excited diagrams}, and the term corresponding to each excited diagram has hook lengths in the denominator, like the classical hook-length formula due to Frame, Robinson and Thrall. Recently, the formula for skew straight shapes was proved via a simple bumping algorithm. The aim of this paper is to extend this result to skew shifted shapes. Since straight skew shapes are special cases of skew shifted shapes, this is …
Recently, Naruse presented a beautiful cancellation-free hook-length formula for skew shapes. The formula involves a sum over objects called excited diagrams, and the term corresponding to each excited diagram has hook lengths in the denominator, like the classical hook-length formula due to Frame, Robinson and Thrall. In this paper, we present a simple bijection that proves an equivalent recursive version of Naruse's result, in the same way that the celebrated hook-walk proof due to Green, Nijenhuis and Wilf gives a bijective (or probabilistic) proof of the hook-length formula for ordinary sh…
The generating function of reverse plane partitions of a fixed shape factors into a product featuring the hook-lengths of this shape. This result, which was first obtained by Stanley, can be explained bijectively using the Hillman-Grassl correspondence between reverse plane partitions and tableaux weighted by hook-lengths. In this extended abstract an alternative bijection between the same families of objects is presented. This construction is best perceived as a set of rules for building reverse plane partitions, viewed as arrangements of stacks of cubes, using bricks in the shape of rim-hook…
We present a new family of hook-length formulas for the number of standard increasing tableaux which arise in the study of factorial Grothendieck polynomials. In the case of straight shapes our formulas generalize the classical hook-length formula and Stanley's formula. For skew shapes, our formulas generalize the Naruse hook-length formula and its $q$-analogues, which were studied in previous papers of the series.
It is known that a lattice path matroid polytope can be associated with two given noncrossing lattice paths on $\mathbb{Z}\times\mathbb{Z}$ with the same end points. In this short note we give explicit formulae for the $f$-vector, toric $f$- and $g$-polynomials of a lattice path matroid polytope when two boundary paths enclose a hook shape.
A notion of cyclic descents on standard Young tableaux (SYT) of rectangular shape was introduced by Rhoades, and extended to certain skew shapes by the last two authors. The cyclic descent set restricts to the usual descent set when the largest value is ignored, and has the property that the number of SYT of a given shape with a given cyclic descent set $D$ is invariant under cyclic shifts of the entries of $D$. Following these results, the existence of cyclic descent sets for standard Young tableaux of any skew shape other than a ribbon was conjectured by the authors, and recently proved by A…
In a recent article (arXiv:1507.03499) (joint with Alon Regev) we studied sums of squares of characters Chi(L,M) of the Symmetric Group over shapes L that are two-rowed, and shapes L that are hook shapes, and M is an arbitrary shape that mostly consists of ones, and designed algorithms for closed-form evaluations of each of these. We noted (and proved) that when M is the shape with n cells consisting of 3 followed by n-3 ones, the former sum equals one half time the analogous sum over hook shapes with n+2 cells and M is the partition consisting of 3,2, followed by n-3 ones. Here we show that t…
A new algorithm for inserting rim-hooks into reverse plane partitions is presented. The insertion is used to define a bijection between reverse plane partitions of a fixed shape and multi-sets of rim-hooks. In turn this yields a bijective proof of the fact that the generating function for reverse plane partitions of a fixed shape, which was first obtained by R. Stanley, factors into a product featuring the hook-lengths of this shape. Our bijection turns out to be equivalent to a map defined by I. Pak by different means, and can be related to the Hillman-Grassl correspondence and the Robinson-S…
Linear extensions of posets are important objects in enumerative and algebraic combinatorics that are difficult to count in general. Families of posets like Young diagrams of straight shapes and $d$-complete posets have hook-length product formulas to count linear extensions, whereas families like Young diagrams of skew shapes have determinant or positive sum formulas like the Naruse hook-length formula from 2014. In 2020, Garver et. al. gave determinant formulas to count linear extensions of a family of posets called mobile posets that refine $d$-complete posets and border strip skew shapes. …
We give new product formulas for the number of standard Young tableaux of certain skew shapes and for the principal evaluation of the certain Schubert polynomials. These are proved by utilizing symmetries for evaluations of factorial Schur functions, extensively studied in the first two papers in the series "Hook formulas for skew shapes" [arxiv:1512.08348, arxiv:1610.04744]. We also apply our technology to obtain determinantal and product formulas for the partition function of certain weighted lozenge tilings, and give various probabilistic and asymptotic applications.
The Naruse hook-length formula is a recent general formula for the number of standard Young tableaux of skew shapes, given as a positive sum over excited diagrams of products of hook-lengths. In 2015 we gave two different $q$-analogues of Naruse's formula: for the skew Schur functions, and for counting reverse plane partitions of skew shapes. In this paper we give an elementary proof of Naruse's formula based on the case of border strips. For special border strips, we obtain curious new formulas for the Euler and $q$-Euler numbers in terms of certain Dyck path summations.
Another bijective proof of Stanley's hook-content formula for the generating function for semistandard tableaux of a given shape is given that does not involve the involution principle of Garsia and Milne. It is the result of a merge of the modified jeu de taquin idea from the author's previous bijective proof (``An involution principle-free bijective proof of Stanley's hook-content formula", Discrete Math. Theoret. Computer Science, to appear) and the Novelli-Pak-Stoyanovskii bijection (Discrete Math. Theoret. Computer Science 1 (1997), 53-67) for the hook formula for standard Young tableaux …
Recently Blasiak gave a combinatorial rule for the Kronecker coefficient $g_{λμν}$ when $μ$ is a hook shape by defining a set of colored Yamanouchi tableaux with cardinality $g_{λμν}$ in terms of a process called conversion. We give a characterization of colored Yamanouchi tableaux that does not rely on conversion, which leads to a simpler formulation and proof of the Kronecker rule for one hook shape.
We present a bijective proof of the hook-length formula for shifted standard tableaux of a fixed shape based on a modified jeu de taquin and the ideas of the bijective proof of the hook-length formula for ordinary standard tableaux by Novelli, Pak and Stoyanovskii. In their proof Novelli, Pak and Stoyanovskii define a bijection between arbitrary fillings of the Ferrers diagram with the integers $1,2,...,n$ and pairs of standard tableaux and hook tabloids. In our shifted version of their algorithm the map from the set of arbitrary fillings of the shifted Ferrers diagram onto the set of shifted …
Several bases of the Garsia-Haiman modules for hook shapes are given, as well as combinatorial decomposition rules for these modules. These bases and rules extend the classical ones for the coinvariant algebra of type $A$. We also give a decomposition of the Garsia-Haiman modules into descent representations.
Motivated by the question of finding a type B analogue of the bijection between oscillating tableaux and matchings, we find a correspondence between oscillating m-rim hook tableaux and m-colored matchings, where m is a positive integer. An oscillating m-rim hook tableau is defined as a sequence $(λ^0,λ^1,...,λ^{2n})$ of Young diagrams starting with the empty shape and ending with the empty shape such that $λ^{i}$ is obtained from $λ^{i-1}$ by adding an m-rim hook or by deleting an m-rim hook. Our bijection relies on the generalized Schensted algorithm due to White. An oscillating 2-rim hook ta…
The Kronecker product of two Schur functions $s_μ$ and $s_ν$, denoted by $s_μ*s_ν$, is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the partitions $μ$ and $ν$. The coefficient of $s_λ$ in this product is denoted by $γ^λ_{μν}$, and corresponds to the multiplicity of the irreducible character $χ^λ$ in $χ^μχ^ν.$ We use Sergeev's Formula for a Schur function of a difference of two alphabets and the comultiplication expansion for $s_λ[XY]$ to find closed formulas for the Kronecker coefficients $γ^λ_{μν}$ when $λ$ is an…
A well-known representation-theoretic model for the transformed Macdonald polynomial $\widetilde{H}_μ(Z;t,q)$, where $μ$ is an integer partition, is given by the Garsia-Haiman module $\mathcal{H}_μ$. We study the $\frac{n!}{k}$ conjecture of Bergeron and Garsia, which concerns the behavior of certain $k$-tuples of Garsia-Haiman modules under intersection. In the special case that $μ$ has hook shape, we use a basis for $\mathcal{H}_μ$ due to Adin, Remmel, and Roichman to resolve the $\frac{n!}{2}$ conjecture by constructing an explicit basis for the intersection of two Garsia-Haiman modules.
The celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give an algebraic and a combinatorial proof of Naruse's formula, by using factorial Schur functions and a generalization of the Hillman--Grassl correspondence, respectively. The main new results are two different $q$-analogues of Naruse's formula: for the skew Schur functions, and for counting rever…
We give a positive combinatorial formula for the Kronecker coefficient g_{lambda mu(d) nu} for any partitions lambda, nu of n and hook shape mu(d) := (n-d,1^d). Our main tool is Haiman's \emph{mixed insertion}. This is a generalization of Schensted insertion to \emph{colored words}, words in the alphabet of barred letters \bar{1},\bar{2},... and unbarred letters 1,2,.... We define the set of \emph{colored Yamanouchi tableaux of content lambda and total color d} (CYT_{lambda, d}) to be the set of mixed insertion tableaux of colored words w with exactly d barred letters and such that w^{blft} is…
We give new bounds and asymptotic estimates on the number of standard Young tableaux of skew shape in a variety of special cases. Our approach is based on Naruse's hook-length formula. We also compare our bounds with the existing bounds on the numbers of linear extensions of the corresponding posets.
We provide simple necessary and sufficient conditions for the existence of a standard Young tableau of a given shape and major index $r$ mod $n$, for all $r$. Our result generalizes the $r=1$ case due essentially to (1974) and proves a recent conjecture due to Sundaram (2016) for the $r=0$ case. A byproduct of the proof is an asymptotic equidistribution result for "almost all" shapes. The proof uses a representation-theoretic formula involving Ramanujan sums and normalized symmetric group character estimates. Further estimates involving "opposite" hook lengths are given which are well-adapted …
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.
The aim of this paper is to describe structural properties of spaces of diagonal rectangular harmonic polynomials in several sets (say $k$) of $n$ variables, both as $GL_k$-modules and $S_n$-modules. We construct explicit such modules associated to any hook shape partitions. For the two sets of variables case, we conjecture that the associated graded Frobenius characteristic corresponds to the effect of the operator Nabla on the corresponding hook-indexed Schur function, up to a usual renormalization. We prove identities that give indirect support to this conjecture, and show that its restrict…
Using equivariant cohomology theory, Naruse obtained a hook length formula for the number of standard Young tableaux of skew shape $λ/μ$. Morales, Pak and Panova found two $q$-analogues of Naruse's formula respectively by counting semistandard Young tableaux of shape $λ/μ$ and reverse plane partitions of shape $λ/μ$. When $λ$ and $μ$ are both staircase shape partitions, Morales, Pak and Panova conjectured that the generating function of reverse plane partitions of shape $λ/ μ$ can be expressed as a determinant whose entries are related to $q$-analogues of the Euler numbers. The objective of th…
The hook length formula gives a product formula for the number of standard Young tableaux of a partition shape. The number of standard Young tableaux of a skew shape does not always have a product formula. However, for some special skew shapes, there is a product formula. Recently, Morales, Pak and Panova joint with Krattenthaler conjectured a product formula for the number of standard Young tableaux of shape $λ/μ$ for $λ=((2a+c)^{c+a},(a+c)^a)$ and $μ=(a+1,a^{a-1},1)$. They also conjectured a product formula for the number of standard Young tableaux of a certain skew shifted shape. In this pa…
For their bijective proof of the hook-length formula for the number of standard tableaux of a fixed shape Novelli, Pak and Stoyanovskii define a modified jeu de taquin which transforms an arbitrary filling of the Ferrers diagram with $1,2,...,n$ (tabloid) into a standard tableau. Their definition relies on a total order of the cells in the Ferrers diagram induced by a special standard tableau, however, this definition also makes sense for the total order induced by any other standard tableau. Given two standard tableaux $P,Q$ of the same shape we show that the number of tabloids which result i…
Let $S_n$ denote the symmetric group on $n$ letters. The $k$-point fixing graph $\mathcal{F}(n,k)$ is defined to be the graph with vertex set $S_n$ and two vertices $g,h$ of $\mathcal{F}(n,k)$ are joined by an edge, if and only if $gh^{-1}$ fixes exactly $k$ points. Ku, Lau and Wong [Cayley graph on symmetric group generated by elements fixing $k$ points, Linear Algebra Appl. 471 (2015) 405-426] obtained a recursive formula for the eigenvalues of $\mathcal{F}(n,k)$. In this paper, we use objects called excited diagrams defined as certain generalizations of skew shapes and derive an explicit fo…
The enumeration of standard Young tableaux (SYTs) of shape λ can be easily computed by the hook-length formula. In 1981, Amitai Regev proved that the number of SYTs having at most three rows with n entries equals the nth Motzkin number M_n. In 2006, Regev conjectured that the total number of SYTs of skew shape λ/(2, 1) over all partitions λ having at most three parts with n entries is the difference of two Motzkin numbers, M_{n-1} - M_{n-3}. Ekhad and Zeilberger proved Regev's conjecture using a computer program. In 2009, S.-P. Eu found a bijection between Motzkin paths and SYTs of skew shape …
The number of standard Young tableaux of a fixed shape is famously given by the hook-length formula due to Frame, Robinson and Thrall. A bijective proof of Novelli, Pak and Stoyanovskii relies on a sorting algorithm akin to jeu-de-taquin which transforms an arbitrary filling of a partition into a standard Young tableau by exchanging adjacent entries. Recently, Krattenthaler and Müller defined the complexity of this algorithm as the average number of performed exchanges, and Neumann and the author proved it fulfils some nice symmetry properties. In this paper we recall and extend the previous r…
We present a randomized algorithm for generating standard set-valued tableaux by extending the Green-Nijenhuis-Wilf hook walk algorithm. In the case of asymptotically rank two partitions, we use this algorithm to give a fully polynomial almost uniform sampler (FPAUS) for standard set-valued tableaux. This FPAUS is then used to construct a fully polynomial randomized approximation scheme (FPRAS) for counting the number of standard set-valued tableaux for such shapes. We also construct a FPAUS and FPRAS for standard set-valued tableaux when either the size of the partition or the difference betw…
This paper makes a combinatorial study of the two monoids and the two types of tableaux that arise from the two possible generalizations of the Patience Sorting algorithm from permutations (or standard words) to words. For both types of tableaux, we present Robinson--Schensted--Knuth-type correspondences (that is, bijective correspondences between word arrays and certain pairs of semistandard tableaux of the same shape), generalizing two known correspondences: a bijective correspondence between standard words and certain pairs of standard tableaux, and an injective correspondence between words…
