Roland Bacher, Laurent Manivel
This paper shows that the number of hooks of length k contained in all partitions of n equals k times the number of parts of length k in all partitions of n. It contains also formulas for the moments (under uniform distribution) of k-th parts in partitions of n.
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Inequalities and asymptotics for hook lengths in $\ell$-regular partitions and $\ell$-distinct partitions
本文通过推导ℓ-正则分拆与ℓ-互异分拆中钩长为1、2、3的钩的总数的渐近公式，建立了这两类分拆之间及其内部的钩长不等式，并证明相应钩数比值趋于依赖ℓ和t的常数。
We exactly determine the maximum possible hook length of $(s,t)$-core partitions with $d$-distinct parts when there are finitely many such partitions. Moreover, we provide an algorithm to construct a $d$-distinct $(s,t)$-core partition with this maximum possible hook length.
Hook length biases for self-conjugate partitions and partitions with distinct odd parts
本文建立了自共轭分拆与具有不同奇数部分分拆间的钩长偏置，推导出固定长度钩总数的渐近公式，并解决了相关猜想。
Inequalities and asymptotics for hook numbers in restricted partitions
本文研究了受限划分中钩数的渐进行为，通过渐近公式证明了奇数部分划分在平均意义上比不同部分划分有更多钩数。
On the distribution of $t$-hooks of doubled distinct partitions
本文研究双重相异分拆中 $t$-hooks 与严格分拆中 $t$-shifted hooks 的计数分布，通过建立生成函数证明二者均渐近正态分布。
Let $p_t(a,b;n)$ denote the number of partitions of $n$ such that the number of $t$ hooks is congruent to $a \bmod{b}$. For $t\in \{2, 3\}$, arithmetic progressions $r_1 \bmod{m_1}$ and $r_2 \bmod{m_2}$ on which $p_t(r_1,m_1; m_2 n + r_2)$ vanishes were established in recent work by Bringmann, Craig, Males, and Ono using the theory of modular forms. Here we offer a direct combinatorial proof of this result using abaci and the theory of $t$-cores and $t$-quotients.
Inequalities for the number of $t$-hooks in two partition classes arising from sum-product identities
本文研究了由求和-乘积恒等式产生的两个划分类中t-hooks数量的不等式，通过生成函数和渐进行式证明了t=1,2时的不等式。
Hook-Length Biases in $t$-regular partitions
本文通过生成函数方法证明了t-正则划分中钩长度相关结果，并揭示了钩长度与不同部分划分之间的新联系。
Hook Length Biases in $t$-Core Partitions
本研究将钩长度偏倚理论扩展到t-核心分区，发现特定长度的钩数量存在偏倚关系。
Motivated by the many roles that hook lengths play in mathematics, we study the distribution of the number of $t$-hooks in the partitions of $n$. We prove that the limiting distribution is normal with mean $μ_t(n)\sim \frac{\sqrt{6n}}π-\frac{t}{2}$ and variance $σ_t^2(n)\sim \frac{(π^2-6)\sqrt{6n}}{2π^3}.$ Furthermore, we prove that the distribution of the number of hook lengths that are multiples of a fixed $t\geq 4$ in partitions of $n$ converge to a shifted Gamma distribution with parameter $k=(t-1)/2$ and scale $θ=\sqrt{2/(t-1)}.$
Finite analogs of partition bias related to hook length two and a variant of Sylvester's map
本文研究带最大部分上界的奇划分与不同划分中长度为二的钩子数量的偏差，利用西尔维斯特映射的变体推广已有不等式，并提出并部分证明加权计数的有限偏差猜想。
Let $α_k(λ)$ denote the number of $k$-hooks in a partition $λ$ and let $b(n,k)$ be the maximum value of $α_k(λ)$ among partitions of $n$. Amdeberhan posed a conjecture on the generating function of $b(n,1)$. We give a proof of this conjecture. In general, we obtain a formula that can be used to determine $b(n,k)$. This leads to a generating function formula for $b(n,k)$. We introduce the notion of nearly $k$-triangular partitions. We show that for any $n$, there is a nearly $k$-triangular partition which can be transformed into a partition of $n$ that attains the maximum number of $k$-hooks. T…
Hook length biases and general linear partition inequalities
该研究对比奇分拆与互异分拆的固定长度钩数量偏差，证明2、3长钩的数量差异并提出一般猜想，同时建立互异分拆数的通用线性不等式，方法横跨解析与组合领域。
Recently, the first author generalized a formula of Nekrasov and Okounkov which gives a combinatorial formula, in terms of hook lengths of partitions, for the coefficients of certain power series. In the course of this investigation, he conjectured that $A000731(n)=0$ if and only if $A033687(n)=0$. The numbers $A000731(n)$ are given in terms of hook lengths of partitions, while $A033687(n)$ equals the number of 3-core partitions of $n$. Here we prove this conjecture.
Congruences for hook lengths of partitions
本文通过加法定理推导并推广了整数划分中钩数的同余关系，扩展至z-非对称划分，并回顾了相关历史结果。
The inequality on the number of $1$-hooks, $2$-hooks and $3$-hooks in $t$-regular partitions
本文研究了$t$-正则分拆中钩子数量的关系，为$b_{t,2}(n)\geq b_{t,1}(n)$给出了显式界$N=O(t^5)$，并证明对所有$n\ge 4$均有$b_{t,2}(n) \geq b_{t,3}(n)$。
Distribution of hooks in self-conjugate partitions
本文证明了无限制整数分拆中 t-钩的分布可下降到自共轭分拆，并给出大小为 n 的自共轭分拆中长度为 t 的钩数渐近正态分布的均值与方差的精确公式。
Using only a symmetric p-core partition and p-quotient, we give an explicit formula for the set of diagonal hook lengths of the associated symmetric partition.
We introduce the difference operator for functions defined on strict partitions and prove a polynomiality property for a summation involving the hook length and content statistics. As an application, several new hook-content formulas for strict partitions are derived.
It is well-known that two conjugate partitions have the same hook multiset. But two different partitions with the same hook multiset may not be conjugate to each other. In $1977$, Herman and Chung proposed the following question: What are the necessary and sufficient conditions for partitions to be determined by their hook multisets up to conjugation? In this paper, we will answer this question by giving a criterion to determine whether two different partitions with the same hook multiset are conjugate to each other.
We give relations between the joint distributions of multiple hook lengths and of frequencies and part sizes in partitions, extending prior work in this area. These results are discovered by investigating truncations of the Han/Nekrasov-Okounkov hooklength formula and of (k,j)-colored partitions, a unification of k-colored partitions and overpartitions. We establish the observed relations at the constant and linear terms for all n, and for j=2 in their quadratic term, with the associated hook/frequency identities. Further results of this type seem likely.
Hook length biases in ordinary and $t$-regular partitions
本文研究普通分拆与 $t$-正则分拆的钩长偏差，证明普通分拆中长度 $k$ 的钩数不少于长度 $k+1$ 的钩数（除 $n=k+1$ 外），并发现 $2$-正则分拆中某些偏差与普通分拆相反，同时给出生成函数和猜想。
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…
Recent works at the interface of algebraic combinatorics, algebraic geometry, number theory, and topology have provided new integer-valued invariants on integer partitions. It is natural to consider the distribution of partitions when sorted by these invariants in congruence classes. We consider the prominent situations which arise from extensions of the Nekrasov-Okounkov hook product formula, and from Betti numbers of various Hilbert schemes of $n$ points on $\mathbb{C}^2.$ For the Hilbert schemes, we prove that homology is equidistributed as $n\to \infty.$ For $t$-hooks, we prove distributio…
Some conjectures on partition hook lengths, recently stated by the author, have been proved and generalized by Stanley, who also needed a formula by Andrews, Goulden and Jackson on symmetric functions to complete his derivation. Another identity on symmetric functions can be used instead. The purpose of this note is to prove it.
Partitions, the partition function $p(n)$, and the hook lengths of their Ferrers-Young diagrams are important objects in combinatorics, number theory and representation theory. For positive integers $n$ and $t$, we study $p_t^e(n)$ (resp. $p_t^o(n)$), the number of partitions of $n$ with an even (resp. odd) number of $t$-hooks. We study the limiting behavior of the ratio $p_t^e(n)/p(n)$, which also gives $p_t^o(n)/p(n)$ since $p_t^e(n) + p_t^0(n) = p(n)$. For even $t$, we show that $$\lim\limits_{n \to \infty} \dfrac{p_t^e(n)}{p(n)} = \dfrac{1}{2},$$ and for odd $t$ we establish the non-unifor…
A {\em pointed partition} of $n$ is a pair $(λ, v)$ where $λ\vdash n$ and $v$ is a cell in its Ferrers diagram. We construct an involution on pointed partitions of $n$ exchanging "hook length" and "part length". This gives a bijective proof of a recent result of Bessenrodt and Han.
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…
Fixed hooks in arbitrary columns
本文将分拆第一列的不动点概念推广到任意列的固定钩，建立其与彩色分拆的组合联系，并给出多种分拆限制下按固定度分类的生成函数。
In 2009, the first author proved the Nekrasov-Okounkov formula on hook lengths for integer partitions by using an identity of Macdonald in the framework of type $\widetilde A$ affine root systems, and conjectured that some summations over the set of all partitions of size $n$ are always polynomials in $n$. This conjecture was generalized and proved by Stanley. Recently, Pétréolle derived two Nekrasov-Okounkov type formulas for $\widetilde C$ and $\widetilde C\,\check{}$ which involve doubled distinct and self-conjugate partitions. Inspired by all those previous works, we establish the polynomi…
The dimension of an irreducible representation of $GL(n,\mathbb{C})$, $Sp(2n)$, 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 compani…
On hook length biases in $t$-regular partitions
针对 $t$-正则分拆中长度为 2 的钩数单调性猜想，本文证明了 $t=3$ 时 $b_{4,2}(n)\geq b_{3,2}(n)$ 对所有 $n\geq0$ 成立。
Motivated by a recent paper of Straub, we study the distribution of integer partitions according to the length of their largest hook, instead of the usual statistic, namely the size of the partitions. We refine Straub's analogue of Euler's Odd-Distinct partition theorem, derive a generalization in the spirit of Alder's conjecture, as well as a curious analogue of the first Rogers-Ramanujan identity. Moreover, we obtain a partition theorem that is the counterpart of Euler's pentagonal number theory in this setting, and connect it with the Rogers-Fine identity. We concludes with some congruence …
This paper proves a generalization of a conjecture of Guoniu Han, inspired originally by an identity of Nekrasov and Okounkov. The main result states that certain sums over partitions p of n, involving symmetric functions of the squares of the hook lengths of p, are polynomial functions of n. A similar result is obtained for symmetric functions of the contents and shifted parts of n.
Strongly refining results by Regev, Regev and Zeilberger, we prove surprising coincidences between characters to 2-part partitions of size n and characters to hooks of size n+2 on two related families obtained by extending 2-regular conjugacy classes.
Hook length inequalities for $t$-regular partitions in the $t$-aspect
本文研究$t$-正则分拆的钩长计数问题，证明了固定钩长下不同$t$值对应的钩数不等式，并提出未来研究方向。
Jordan types commuting with a hook partition
本文完全分类了与钩子划分Jordan型幂零矩阵交换的幂零矩阵的Jordan型，并证明具有相同通用交换Jordan型的两个划分不一定相互交换。
In this paper, we present, for any integer d, a description of the set of hooks in a d-symbol. We then introduce generalized hook length functions for a d-symbol, and prove a general result about them, involving the core and quotient of the symbol. We list some applications, for example to the well-known hook lengths in integer partitions. This leads in particular to a generalization of a relative hook formula for the degree of characters of the symmetric group discovered by G. Malle and G. Navarro in [3].
Let $λ$ be a partition, viewed as a Young diagram. We define the hook difference of a cell of $λ$ to be the difference of its leg and arm lengths. Define $h_{1,1}(λ)$ to be the number of cells of $λ$ with hook difference one. In the paper of Buryak and Feigin (arXiv:1206.5640), algebraic geometry is used to prove a generating function identity which implies that $h_{1,1}$ is equidistributed with $a_2$, the largest part of a partition that appears at least twice, over the partitions of a given size. In this paper, we propose a refinement of the theorem of Buryak and Feigin and prove some partia…
Fix $t \geq 2$. We first give an asymptotic formula for certain sums of the number of $t$-cores. We then use this result to compute the distribution of the size of the $t$-core of a uniformly random partition of an integer $n$. We show that this converges weakly to a gamma distribution after dividing by $\sqrt{n}$. As a consequence, we find that the size of the $t$-core is of the order of $\sqrt{n}$ in expectation. We then apply this result to show that the probability that $t$ divides the hook length of a uniformly random cell in a uniformly random partition equals $1/t$ in the limit. Finally…
Representation theory of the symmetric group $\mathfrak{S}_n$ has a very distinctive combinatorial flavor. The conjugacy classes as well as the irreducible characters are indexed by integer partitions $λ\vdash n$. We introduce class functions on $\mathfrak{S}_n$ that count the number of certain tilings of Young diagrams. The counting interpretation gives a uniform expression of these class functions in the ring of character polynomials, as defined by \cite{murnaghanfirst}. A modern treatment of character polynomials is given in \cite{orellana-zabrocki}. We prove a relation between these combin…
In this paper we construct for a given arbitrary skew diagram A all partitions nu with maximal principal hook lengths among all partitions with the character [nu] appearing in the skew character [A]. Furthermore we show that these are also partitions with minimal Durfee size. This we use to give the maximal Durfee size for [nu] appearing in [A] for the cases when A decays into two partitions and for some special cases of A. Also this gives conditions for two skew diagrams to represent the same skew character.
We generalize multivariate hook product formulae for $P$-partitions. We use Macdonald symmetric functions to prove a $(q,t)$-deformation of Gansner's hook product formula for the generating functions of reverse (shifted) plane partitions. (The unshifted case has also been proved by Adachi.) For a $d$-complete poset, we present a conjectural $(q,t)$-deformation of Peterson--Proctor's hook product formula.
We introduce the hook length expansion technique and explain how to discover old and new hook length formulas for partitions and plane trees. The new hook length formulas for trees obtained by our method can be proved rather easily, whereas those for partitions are much more difficult and some of them still remain open conjectures. We also develop a Maple package HookExp for computing the hook length expansion. The paper can be seen as a collection of hook length formulas for partitons and plane trees. All examples are illustrated by HookExp and, for many easy cases, expained by well-known com…
Proofs of two conjectures of Berkovich and Dhar on hook lengths
本文证明了 Berkovich 和 Dhar 关于有界划分中钩长系数的两个猜想的更强形式，通过组合和解析方法建立奇数划分与不同划分中钩对数量的不等式。
Partitions with fixed points in the sequence of first-column hook lengths
本文研究整数分拆Young图第一列钩长序列中的不动钩，通过生成函数和组合证明进行计数，并建立其与部分大小等于重数的出现次数及五边形数定理截断等研究的联系。
The concept of $t$-difference operator for functions of partitions is introduced to prove a generalization of Stanley's theorem on polynomiality of Plancherel averages of symmetric functions related to contents and hook lengths. Our extension uses a generalization of the notion of Plancherel measure, based on walks in the Young lattice with steps given by the addition of $t$-hooks. It is well-known that the hook lengths of multiples of $t$ can be characterized by the Littlewood decomposition. Our study gives some further information on the contents and hook lengths of other congruence classes …
On the hook length biases of the $2$- and $3$-regular partitions
本文证实了关于3-正则分拆钩数差的正性猜想，并指出2-正则分拆钩数单调性猜想在奇数情形无穷多次失效，但在k=4和6时成立，同时提出偶数情形的修正猜想。
We show that the distribution of the major index over the set of involutions in S_n that avoid the pattern 321 is given by the q-analogue of the n-th central binomial coefficient. The proof consists of a composition of three non-trivial bijections, one being the Robinson-Schensted correspondence, ultimately mapping those involutions with major index m into partitions of m whose Young diagram fits inside an n/2 by n/2 box. We also obtain a refinement that keeps track of the descent set, and we deduce an analogous result for the comajor index of 123-avoiding involutions.
We use recent work of Jonah Blasiak (2012) to prove a stability result for the coefficients in the Kronecker product of two Schur functions: one indexed by a hook partition and one indexed by a rectangle partition. We also give bounds for the size of the partition starting with which the Kronecker coefficients are stable. Moreover, we show that once the bound is reached, no new Schur functions appear in the decomposition of Kronecker product, thus allowing one to recover the decomposition from the smallest case in which the stability holds.
Hook lengths in self-conjugate partitions
本文针对自共轭分拆，推导了偶数大小钩子数量的生成函数及简单计数公式，并证实了相关猜想。
On the smallest partition associated to a numerical semigroup
研究给定钩长集的划分数量，分析最小划分大小分布，探讨数值半群与钩长集的关系。
The Nekrasov--Okounkov hook length formula provides a fundamental link between the theory of partitions and the coefficients of powers of the Dedekind eta function. In this paper we examine three conjectures presented by Amdeberhan. The first conjecture is a refined Nekrasov--Okounkov formula involving hooks with trivial legs. We prove the conjecture. The second conjecture is on properties of the roots of the underlying D'Arcais polynomials. We give a counterexample and present a new conjecture. The third conjecture is on the unimodality of the coefficients of the involved polynomials. We conf…
In this article we consider hook removal operators on odd partitions, i.e., partitions labelling odd-degree irreducible characters of finite symmetric groups. In particular we complete the discussion, started by Isaacs, Navarro, Olsson and Tiep in 2016, concerning the commutativity of such operators.
We use the hook lengths of a partition to define two rectangular tableaux. We prove these tableaux have equal multisets of entries, first by elementary combinatorial arguments, and then using Stanley's Hook Content Formula and symmetric polynomials.
Peterson and Proctor obtained a formula which expresses the multivariate generating function for $P$-partitions on a $d$-complete poset $P$ as a product in terms of hooks in $P$. In this paper, we give a skew generalization of Peterson--Proctor's hook formula, i.e., a formula for the generating function for $(P \setminus F)$-partitions for a $d$-complete poset $P$ and its order filter $F$, by using the notion of excited diagrams. Our proof uses the Billey-type formula and the Chevalley-type formula in the equivariant $K$-theory of Kac--Moody partial flag varieties. This generalization provides…
We discover an explicit expansion formula for the powers $s$ of the Euler Product (or Dedekind $η$-function) in terms of hook lengths of partitions, where the exponent $s$ is any complex number. Several classical formulas have been derived for certain integers $s$ by Euler, Jacobi, Klein, Fricke, Atkin, Winquist, Dyson and Macdonald. In particular, Macdonald obtained expansion formulas for the integer exponents $s$ for which there exists a semi-simple Lie algebra of dimension $s$. For the type $A_l^{(a)}$ he has expressed the $(t^2-1)$-st power of the Euler Product as a sum of weighted integer…
Motivated by the Nekrasov-Okounkov formula on hook lengths, the first author conjectured that the Plancherel average of the $2k$-th power sum of hook lengths of partitions with size $n$ is always a polynomial of $n$ for any $k\in \mathbb{N}$. This conjecture was generalized and proved by Stanley (Ramanujan J., 23(1--3): 91--105, 2010). In this paper, inspired by the work of Stanley and Olshanski on the differential poset of Young lattice, we study the properties of two kinds of difference operators $D$ and $D^-$ defined on functions of partitions. Even though the calculations for higher orders…
On the nonnegativity of monomial immanants for hook partitions
本文解决了 Stembridge 提出的关于单项式隐式量非负性的开放问题，证明了对于所有钩子分区，当矩阵的所有低阶子式非负且分区长度受限时，隐式量非负。
The hook length formula for $d$-complete posets states that the $P$-partition generating function for them is given by a product in terms of hook lengths. We give a new proof of the hook length formula using $q$-integrals. The proof is done by a case-by-case analysis consisting of two steps. First, we express the $P$-partition generating function for each case as a $q$-integral and then we evaluate the $q$-integrals. Several $q$-integrals are evaluated using partial fraction expansion identities and others are verified by computer.
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