Hook Length Formula Research Articles

Roots of Descent Polynomials and an Algebraic Inequality on Hook Lengths


 
 
 By reinterpreting the descent polynomial as a function enumerating standard Young tableaux of a ribbon shape, we use Naruse's hook-length formula to express the descent polynomial as a product of two polynomials: one is a trivial part which is a product of linear factors, and the other comes from the excitation factor of Naruse's formula. We expand the excitation factor positively in a Newton basis which arises naturally from Naruse's formula. Under this expansion, each coefficient is the weight of a certain combinatorial object, which we introduce in this paper. We introduce and prove the "Slice and Push Inequality", which compares the weights of such combinatorial objects. As a consequence, we establish a proof of a conjecture by Diaz-Lopez et al. that bounds the roots of descent polynomials.
 
 

Hook length and symplectic content in partitions

The dimension of an irreducible representation of GL(n,C), Sp(2n), or SO(n) is given by the respective hook length and content formulas for the corresponding partition. The first author, inspired by the Nekrasov-Okounkov formula, conjectured combinatorial interpretations of analogous expressions involving hook lengths and symplectic/orthogonal contents. We prove special cases of these conjectures. In the process, we show that partitions of n with all symplectic contents non-zero are equinumerous with partitions of n into distinct even parts. We also present Beck-type companions to this identity. In this context, we give the parity of the number of partitions into distinct parts with odd (respectively, even) rank. We study the connection between the sum of hook lengths and the sum of inversions in the binary representation of a partition. In addition, we introduce a new partition statistic, the x-ray list of a partition, and explore its connection with distinct partitions as well as partitions maximally contained in a given staircase partition.

Naruse Hook Formula for Linear Extensions of Mobile Posets


 
 
 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 a Naruse type hook-length formula to count linear extensions of such posets by proving a major index q-analogue. We also a inversion index q-analogue of the Naruse formula for mobile tree posets.
 
 

The metric space of limit laws for $q$-hook formulas

Billey-Konvalinka-Swanson studied the asymptotic distribution of the coefficients of Stanley's \(q\)-hook length formula, or equivalently the major index on standard tableaux of straight shape and certain skew shapes. We extend those investigations to Stanley's \(q\)-hook-content formula related to semistandard tableaux and \(q\)-hook length formulas of Björner-Wachs related to linear extensions of labeled forests. We show that, while their coefficients are "generically" asymptotically normal, there are uncountably many non-normal limit laws. More precisely, we introduce and completely describe the compact closure of the metric space of distributions of these statistics in several regimes. The additional limit distributions involve generalized uniform sum distributions which are topologically parameterized by certain decreasing sequence spaces with bounded \(2\)-norm. The closure of these distributions in the Lévy metric gives rise to the space of DUSTPAN distributions. As an application, we completely classify the limiting distributions of the size statistic on plane partitions fitting in a box.Mathematics Subject Classifications: 05A16 (Primary), 60C05, 60F05 (Secondary)Keywords: Hook length, \(q\)-analogues, major index, semistandard tableaux, plane partitions, forests, asymptotic normality, limit laws, Irwin-Hall distribution

Multiplication theorems for self-conjugate partitions

In 2011, Han and Ji proved addition-multiplication theorems for integer partitions, from which they derived modular analogues of many classical identities involving hook-length. In the present paper, we prove addition-multiplication theorems for the subset of self-conjugate partitions. Although difficulties arise due to parity questions, we are almost always able to include the BG-rank introduced by Berkovich and Garvan. This gives us as consequences many self-conjugate modular versions of classical hook-lengths identities for partitions. Our tools are mainly based on fine properties of the Littlewood decomposition restricted to self-conjugate partitions.Mathematics Subject Classifications: 05A15, 05A17, 05A19, 05E05, 05E10, 11P81Keywords: Hook-length formulas, BGP-ranks, Integers partitions, Littlewood decomposition, core partitions

On the Okounkov-Olshanski formula for standard tableaux of skew shapes

The classical hook length formula counts the number of standard tableaux of straight shapes. In 1996, Okounkov and Olshanski found a positive formula for the number of standard Young tableaux of a skew shape. We prove various properties of this formula, including three determinantal formulas for the number of nonzero terms, an equivalence between the Okounkov-Olshanski formula and another skew tableaux formula involving Knutson-Tao puzzles, and two $q$-analogues for reverse plane partitions, which complement work by Stanley and Chen for semistandard tableaux. We also give several reformulations of the formula, including two in terms of the excited diagrams appearing in a more recent skew tableaux formula by Naruse. Lastly, for thick zigzag shapes we show that the number of nonzero terms is given by a determinant of the Genocchi numbers and improve on known upper bounds by Morales-Pak-Panova on the number of standard tableaux of these shapes.Mathematics Subject Classifications: 05A15, 05A15, 05A19, 05A05

Hook Formulas for Skew Shapes IV. Increasing Tableaux and Factorial Grothendieck Polynomials

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.

A generalization of balanced tableaux and marriage problems with unique solutions

We consider families of finite sets that we call shellable and that have been characterized by Chang and by Hirst and Hughes as being the families of sets that admit unique solutions to Hall's marriage problem. In this paper, we introduce a natural generalization of Edelman and Greene's balanced tableaux that involves families of sets that satisfy Hall's marriage Condition and certain words in $[n]^m$, then prove that shellable families can be characterized by a strong existence condition relating to this generalization. As a consequence of this characterization, we show that the average number of such generalized tableaux is given by a generalization of the hook-length formula.

On the growth and zeros of polynomials attached to arithmetic functions

In this paper we investigate growth properties and the zero distribution of polynomials attached to arithmetic functions g and h, where g is normalized, of moderate growth, and 0<h(n) le h(n+1). We put P_0^{g,h}(x)=1 and Png,h(x):=xh(n)∑k=1ng(k)Pn-kg,h(x).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} P_n^{g,h}(x) := \\frac{x}{h(n)} \\sum _{k=1}^{n} g(k) \\, P_{n-k}^{g,h}(x). \\end{aligned}$$\\end{document}As an application we obtain the best known result on the domain of the non-vanishing of the Fourier coefficients of powers of the Dedekind eta -function. Here, g is the sum of divisors and h the identity function. Kostant’s result on the representation of simple complex Lie algebras and Han’s results on the Nekrasov–Okounkov hook length formula are extended. The polynomials are related to reciprocals of Eisenstein series, Klein’s j-invariant, and Chebyshev polynomials of the second kind.

Randomized Schützenberger’s Jeu De Taquin and Approximate Calculation of the Cotransition Probabilities of a Central Markov Process on the 3D Young Graph

There exists a well-known hook-length formula for calculating the dimensions of two-dimensional Young diagrams. Unfortunately, an analogous formula for the three-dimensional case is unknown. We introduce an approach to calculating estimated dimensions of three-dimensional Young diagrams, also known as plane partitions. The most difficult part of this task is the calculation of the cotransition probabilities for a central Markov process. We suggest an algorithm for an approximate calculation of these probabilities. It generates numerous random paths to a given diagram. In the case where the generated paths are uniformly distributed, the proportion of paths passing through a certain edge gives us an approximate value of the cotransition probability. As our numerical experiments show, a random generator based on the randomized version of the Schutzenberger transformation allows one to obtain accurate values of cotransition probabilities. Also, a method for constructing three-dimensional Young diagrams with large dimensions is suggested. Bibliography: 13 titles.

Weyl-Kac character formula for affine Lie algebras in Deligne's categories

We study the characters of simple modules in the parabolic BGG category of the affine Lie algebra in Deligne's category. More specifically, we take the limit of the Weyl-Kac formula to compute the character of the irreducible quotient L(X,k) of the parabolic Verma module M(X,k) of level k, where X is an indecomposable object of Deligne's category Re_p(GLt), Re_p(Ot), or Re_p(Spt), under conditions that the highest weight of X plus the level gives a fundamental weight, t is transcendental, and the base field k has characteristic 0. We compare our result to the partial result in [7, Problem 6.2], and evaluate the characters to the categorical dimensions to get a categorical interpretation of the Nekrasov-Okounkov hook length formula, [12, Formula (6.12)].

Minimal factorizations of a cycle: a multivariate generating function

It is known that the number of minimal factorizations of the long cycle in the symmetric group into a product of k cycles of given lengths has a very simple formula: it is nk−1 where n is the rank of the underlying symmetric group and k is the number of factors. In particular, this is nn−2 for transposition factorizations. The goal of this work is to prove a multivariate generalization of this result. As a byproduct, we get a multivariate analog of Postnikov's hook length formula for trees, and a refined enumeration of final chains of noncrossing partitions.

Hook formulas for skew shapes

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 two q-analogues of Naruse's formula for the skew Schur functions and for counting reverse plane partitions of skew shapes. We also apply our results to border strip shapes and their generalizations.

A bijective proof of the hook-length formula for skew shapes

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 Greene, Nijenhuis and Wilf gives a bijective (or probabilistic) proof of the hook-length formula for ordinary shapes.In particular, we also give a new bijective proof of the classical hook-length formula, quite different from the known proofs.

Hook, line and sinker: A bijective proof of the skew shifted hook-length formula

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 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 by the author 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 a bijection that proves the whole family of hook-length formulas, and is also the simplest known bijective proof for shifted (non-skew) shapes. The complexity of the algorithm is studied, and a weighted generalization of Naruse’s formula is also presented.

The Dedekind eta function and D’Arcais-type polynomials

D’Arcais-type polynomials encode growth and non-vanishing properties of the coefficients of powers of the Dedekind eta function. They also include associated Laguerre polynomials. We prove growth conditions and apply them to the representation theory of complex simple Lie algebras and to the theory of partitions, in the direction of the Nekrasov–Okounkov hook length formula. We generalize and extend results of Kostant and Han.

Position Sequences and a $q$-Analogue for the Modular Hook Length Formula

We prove a $q$-analogue of the modular hook length formula using position sequences. These position sequences, which correspond to moving the beads in a mathematical abacus, provide a new combinatorial interpretation for the characters of the irreducible representations of the symmetric group.

Product formulas for certain skew tableaux

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,aa−1,1). They also conjectured a product formula for the number of standard Young tableaux of a certain skew shifted shape. In this paper we prove their conjectures using Selberg-type integrals. We also give a generalization of MacMahon’s box theorem and a product formula for the trace generating function for a certain skew shape, which is a generalization of a recent result of Morales, Pak and Panova.

Reverse plane partitions of skew staircase shapes and q-Euler numbers

Recently, Naruse discovered a hook length formula for the number of standard Young tableaux of a skew shape. Morales, Pak and Panova found two q-analogs of Naruse's hook length formula over semistandard Young tableaux (SSYTs) and reverse plane partitions (RPPs). As an application of their formula, they expressed certain q-Euler numbers, which are generating functions for SSYTs and RPPs of a zigzag border strip, in terms of weighted Dyck paths. They found a determinantal formula for the generating function for SSYTs of a skew staircase shape and proposed two conjectures related to RPPs of the same shape. One conjecture is a determinantal formula for the number of pleasant diagrams in terms of Schröder paths and the other conjecture is a determinantal formula for the generating function for RPPs of a skew staircase shape in terms of q-Euler numbers.In this paper, we show that the results of Morales, Pak and Panova on the q-Euler numbers can be derived from previously known results due to Prodinger by manipulating continued fractions. These q-Euler numbers are naturally expressed as generating functions for alternating permutations with certain statistics involving maj. It has been proved by Huber and Yee that these q-Euler numbers are generating functions for alternating permutations with certain statistics involving inv. By modifying Foata's bijection we construct a bijection on alternating permutations which sends the statistics involving maj to the statistic involving inv. We also prove the aforementioned two conjectures of Morales, Pak and Panova.

Combinatorics of patience sorting monoids

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 and pairs of tableaux.We also exhibit formulas to count both the number of each type of tableaux with given evaluations (that is, containing a given number of each symbol). Observing that for any natural number n, the nth Bell number is given by the number of standard tableaux containing n symbols, we restrict the previous formulas to standard words and extract a formula for the Bell numbers. Finally, we present a ‘hook length formula’ that gives the number of standard tableaux of a given shape and deduce some consequences.