Asymptotic normality of the major index on standard tableaux

TL;DR

This paper investigates the asymptotic distribution of the major index on standard tableaux of various shapes, providing a classification of limit laws and connecting to representation theory of complex reflection groups.

Contribution

It introduces a cumulant-based approach to classify all possible limit laws for the major index on standard tableaux of arbitrary shapes, extending previous results.

Findings

Classifies limit laws using a new auxiliary statistic, aft.

Provides a detailed description of the distribution of irreducible representations in coinvariant algebras.

Suggests conjectures on unimodality, log-concavity, and local limit theorems.

Abstract

We consider the distribution of the major index on standard tableaux of arbitrary straight shape and certain skew shapes. We use cumulants to classify all possible limit laws for any sequence of such shapes in terms of a simple auxiliary statistic, aft, generalizing earlier results of Canfield--Janson--Zeilberger, Chen--Wang--Wang, and others. These results can be interpreted as giving a very precise description of the distribution of irreducible representations in different degrees of coinvariant algebras of certain complex reflection groups. We conclude with some conjectures concerning unimodality, log-concavity, and local limit theorems.

Figures

Tables

Table 1. Table 1 . Summary of some asymptotic normality results for combinatorial statistics. See [ Bón15 , Ch. 3] .

Statistic	Set	Generating Function	References
$\#$ elements	subsets	${\left(1+q\right)}^n$	classical
$\#$ parts	strict partitions	${\prod }_{m=1}^{\infty }\left(1+x{y}^m\right)$	[EL41]
length/inversion number/major index	$S_n$	${\left[n\right]}_q!$	[Fel45], [Gon44]
$\#$ cycles; $\#$ left-to-right minima	$S_n$	${\prod }_{i=0}^{n-1}\left(q+i\right)$	[Fel45], [Gon44]
$\#$ descents	$S_n$	Eulerian   polynomial $A_n\left(q\right)$	[DB62, pp. 150–154]
$\#$ descents	conjugacy classes in $S_n$	[Ful98, Thm. 1]	[Ful98, KL18]
$\#$ blocks	set partitions	${\sum }_{k}S(n,k)q^k$	[Har67]
$\#$ valleys	Dyck paths	$\frac{1}{{\left[n+1\right]}_q}{\left(\frac{2n}{n}\right)}_q$	[CWW08, Cor. 3.3]; [FH85, p. 255]
length/inversion number/major index	$S_n/S_J$, words type $\alpha$	${\left(\frac{n}{\alpha }\right)}_q$	see 3.17
major index	$\mathrm{SYT}\left(\lambda \right)$	$q^{b\left(\lambda \right)}\frac{{\left[n\right]}_q!}{{\prod }_{c\in \lambda }{\left[h_c\right]}_q}$	1.3

Equations

$${\mathcal{X}}^{\ast }:=\frac{\mathcal{X}-\mu }{\sigma }$$

$$\underset{n\to \infty }{\lim }\mathbb{P}[{\mathcal{X}}_n[\mathrm{maj}{]}^{\ast }\le t]=\mathbb{P}[\mathcal{N}\le t]$$

$$\mathrm{aft}(\lambda ):=|\lambda |-\max \{{\lambda }_1,{\lambda }_1^{\prime }\}.$$

$$\mathrm{SYT}(\lambda {)}^{\mathrm{maj}}(q):=\sum _{T\in \mathrm{SYT}(\lambda )}{q}^{\mathrm{maj}(T)}=q^{b(\lambda )}\frac{[n{]}_q!}{{\prod }_{c\in \lambda }[h_c{]}_q},$$

$${\kappa }_d^{\lambda }=\frac{B_d}{d}\left[\sum _{j=1}^n{j}^d-\sum _{c\in \lambda }{h}_c^d\right]$$

$$O_g(t):=\sum _{d\ge 0}{g}_d{t}^d$$

$$E_g(t):=\sum _{d\ge 0}{g}_d\frac{t^d}{d!}.$$

$$F(t)=\sum _{d\ge 0}{f}_d{t}^d=\sum _{d\ge 0}d!f_d\frac{t^d}{d!}$$

$$h_d=\sum _{\pi \in {\Pi }_d}\prod _{b\in \pi }{f}_{|b|},$$

$$h_d=\sum _{\lambda ⊢d}\frac{d!}{z_{\lambda }}\prod _i\frac{f_{{\lambda }_i}}{({\lambda }_i-1)!}$$

$$h_d=f_d+\sum _{m=1}^{d-1}\left(\genfrac{}{}{0ex}{}{d-1}{m-1}\right)f_m{h}_{d-m}.$$

$$B_{0}=1,\ B_{1}=\frac{1}{2},\ B_{2}=\frac{1}{6},\ B_{3}=0,\ B_{4}=-\frac{1}{30},\ B_{5}=0,\ B_{6}=\frac{1}{42},\$$

$$B_7=0,B_8=-\frac{1}{30},B_9=0,B_{10}=\frac{5}{66},B_{11}=0,B_{12}=-\frac{691}{2730}.$$

$$E_D(t):=\sum _{d\ge 1}\frac{B_d}{d}\frac{t^d}{d!}=\log \left(\frac{e^t-1}{t}\right).$$

$$\sum _{k=1}^n{k}^d=\frac{1}{d+1}\sum _{k=0}^d\left(\genfrac{}{}{0ex}{}{d+1}{k}\right)B_k{n}^{d+1-k}.$$

$$F(t):={\int }_{-\infty }^{t}f(x)dx\text{ or }F(t):=\sum _{k\le t}f(k)$$

$$\mathbb{E}[g(\mathcal{X})]:={\int }_{\mathbb{R}}g(x)f(x)dx\text{ or }\mathbb{E}[g(\mathcal{X})]:=\sum _{k=-\infty }^{\infty }g(k)f(k).$$

$$\mu :=\mathbb{E}[\mathcal{X}]\text{ and }{\sigma }^2:=\mathbb{E}[(\mathcal{X}-\mu {)}^2].$$

$${\mu }_d:=\mathbb{E}[{\mathcal{X}}^d]\text{ and }{\alpha }_d:=\mathbb{E}[(\mathcal{X}-\mu {)}^d].$$

$$M_{\mathcal{X}}(t):=\mathbb{E}[e^{t\mathcal{X}}]=\sum _{d=0}^{\infty }{\mu }_d\frac{t^d}{d!},$$

$${\varphi }_{\mathcal{X}}(t):=\mathbb{E}[e^{it\mathcal{X}}],$$

$$W^{\mathrm{stat}}(q):=\sum _{w\in W}{q}^{\mathrm{stat}(w)}$$

$$\mathbb{E}[q^{\mathcal{X}}]=\frac{1}{\#W}{W}^{\mathrm{stat}}(q):=\frac{1}{\#W}\sum _{w\in W}{q}^{\mathrm{stat}(w)}.$$

$$M_{\mathcal{X}}(t)=\frac{1}{\#W}{W}^{\mathrm{stat}}(e^t)\text{ and }{\varphi }_{\mathcal{X}}(t)=\frac{1}{\#W}{W}^{\mathrm{stat}}(e^{it}).$$

$$K_{\mathcal{X}}(t):=\sum _{d=1}^{\infty }{\kappa }_d\frac{t^d}{d!}:=\log M_{\mathcal{X}}(t)=\log \mathbb{E}[e^{t\mathcal{X}}].$$

$${\mu }_d={\kappa }_d+\sum _{m=1}^{d-1}\left(\genfrac{}{}{0ex}{}{d-1}{m-1}\right){\kappa }_m{\mu }_{d-m}.$$

$${\alpha }_d={\kappa }_d+\sum _{m=2}^{d-2}\left(\genfrac{}{}{0ex}{}{d-1}{m-1}\right){\kappa }_m{\alpha }_{d-m}.$$

$${\mu }_3={\kappa }_3+3{\kappa }_2{\kappa }_1+{\kappa }_1^3.$$

$${\kappa }_d=\{\begin{array}{ll}\mu & d=1,\\ {\sigma }^2& d=2,\\ 0& d\ge 3.\end{array}$$

$${\alpha }_d=\{\begin{array}{ll}0& \text{if d is odd},\\ {\sigma }^d(d-1)!!& \text{if d is even}.\end{array}$$

Full text

The first author was partially supported by the Washington Research Foundation and DMS-1764012 from the National Science Foundation. The second author was partially supported by Research Project BI-US/16-17-042 of the Slovenian Research Agency and research core funding No. P1-0294.

1. 1 Introduction
2. 2 Background on cumulants
3. 3 Combinatorial background
4. 4 Asymptotic normality for $\operatorname{baj}-\operatorname{inv}$ on $S_{n}$
5. 5 Asymptotic normality for $\operatorname{maj}$ on $\operatorname{SYT}(\underline{\lambda})$
6. 6 Uniform sum limits for $\operatorname{maj}$ on $\operatorname{SYT}(\underline{\lambda})$
7. 7 Discrete distributions for $\operatorname{maj}$ on $\operatorname{SYT}(\lambda)$
8. 8 Future work

1. Introduction

The study of permutation and partition statistics is a classic topic in enumerative combinatorics. The major index statistic on permutations was introduced a century ago by Percy MacMahon in his seminal works [Mac13, Mac17]. This statistic, denoted $\operatorname{maj}(w)$, is defined to be the sum of the positions of the descents of the permutation $w=[w_{1},w_{2},\ldots,w_{n}]$ in one-line notation. A descent is any position $i$ such that $w_{i}>w_{i+1}$. At first glance, this function on permutations may be unintuitive, but it has inspired hundreds of papers and many generalizations; for example on Macdonald polynomials [HHL05], posets [ER15], quasisymmetric functions [SW10], cyclic sieving [RSW04, AS18], and bijective combinatorics [Foa68, Car75].

The following central limit theorem for $\operatorname{maj}$ on $S_{n}$ is well known and is an archetype for our results. Given a real-valued random variable $\mathcal{X}$, we let

[TABLE]

denote the corresponding normalized random variable with mean [math] and variance $1$. Briefly, we say $\operatorname{maj}$ on $S_{n}$ is asymptotically normal as $n\to\infty$ based on the following classical result. See Table 1 for further examples.

Theorem 1.1**.**

[Fel45*]**

Let $\mathcal{X}_{n}[\operatorname{maj}]$ denote the major index random variable on $S_{n}$ under the uniform distribution. Then, for all $t\in\mathbb{R}$,*

[TABLE]

where $\mathcal{N}$ is the standard normal random variable.

In this paper, we study the distribution of the major index statistic generalized to standard Young tableaux of straight and skew shapes. The properties we discuss here naturally generalize known properties of the major index distribution on permutations. They also have representation theoretic consequences in terms of coinvariant algebras of complex reflection groups. We will briefly introduce the main results. See Section 2 for more details on the background.

Let $\operatorname{SYT}(\lambda)$ denote the set of all standard Young tableaux of partition shape $\lambda$. We say $i$ is a descent in a standard tableau $T$ if $i+1$ comes before $i$ in the row reading word of $T$, read from bottom to top along rows in English notation. Equivalently, $i$ is a descent in $T$ if $i+1$ appears in a lower row in $T$. Let $\operatorname{maj}(T)$ denote the major index statistic on $\operatorname{SYT}(\lambda)$, which is again defined to be the sum of the descents of $T$. Figure 1 shows some sample distributions for the major index on standard tableaux for three particular partition shapes. Note that Gaussian approximations fit the data well.

In 1.1, we simply let $n\to\infty$. For partitions, the shape $\lambda$ may “go to infinity” in many different ways. The following statistic on partitions overcomes this difficulty.

Definition 1.2**.**

Suppose $\lambda$ is a partition. Let the aft of $\lambda$ be

[TABLE]

Intuitively, if the first row of $\lambda$ is at least as long as the first column, then $\operatorname{aft}(\lambda)$ is the number of cells not in the first row. This definition is strongly reminiscent of a representation stability result of Church and Farb [CF13, Thm. 7.1], which is proved with an analysis of the major index on standard tableaux.

Our first main result gives the analogue of 1.1 for $\operatorname{maj}$ on $\operatorname{SYT}(\lambda)$. In particular, it completely classifies which sequences of partition shapes give rise to asymptotically normal sequences of $\operatorname{maj}$ statistics on standard tableaux.

Theorem 1.3**.**

Suppose $\lambda^{(1)},\lambda^{(2)},\ldots$ is a sequence of partitions, and let $\mathcal{X}_{N}=\mathcal{X}_{\lambda^{(N)}}[\operatorname{maj}]$ be the corresponding random variables for the $\operatorname{maj}$ statistic on $\operatorname{SYT}(\lambda^{(N)})$. Then, the sequence $\mathcal{X}_{1},\mathcal{X}_{2},\ldots$ is asymptotically normal if and only if $\operatorname{aft}(\lambda^{(N)})\to\infty$ as $N\to\infty$.

Remark 1.4**.**

In Section 5, we more generally consider $\operatorname{maj}$ on $\operatorname{SYT}(\underline{\lambda})$ where $\underline{\lambda}$ is a block diagonal skew partition. See [BKS18, §2] for further representation-theoretic motivation and [BKS18, Thm. 6.3] for the classification of the support of $\operatorname{maj}$ on $\operatorname{SYT}(\underline{\lambda})$.

The generalization of 1.3 to $\operatorname{SYT}(\underline{\lambda})$ is 5.8. Special cases of 5.8 include Canfield–Janson–Zeilberger’s main result in [CJZ11] classifying asymptotic normality for $\operatorname{inv}$ or $\operatorname{maj}$ on words (though see [CJZ12] for earlier, essentially equivalent results due to Diaconis [Dia88]). The case of words generalizes 1.1. The $\lambda^{(N)}=(N,N)$ case of 1.3 also recovers the main result of Chen–Wang–Wang [CWW08], giving asymptotic normality for $q$-Catalan coefficients.

Our proof of 1.3 relies on the method of moments, which requires useful descriptions of the moments of $\mathcal{X}_{\lambda}[\operatorname{maj}]$. Adin–Roichman [AR01] gave exact formulas for the mean and variance of $\mathcal{X}_{\lambda}[\operatorname{maj}]$ in terms of the hook lengths of $\lambda$. Their argument leverages the following $q$-analogue of the celebrated Frame–Robinson–Thrall Hook Length Formula [FRT54, Thm. 1] (obtained by setting $q=1$):

[TABLE]

where $h_{c}$ denotes the hook length of a cell $c$ in $\lambda$ and $b(\lambda)\coloneqq\sum_{i\geq 1}(i-1)\lambda_{i}$. Equation (1) is due to Stanley [Sta99, Cor. 7.21.5] and is strongly related to the stable principal specialization of Schur functions by the identity $s_{\lambda}(1,q,q^{2},\ldots)=\operatorname{SYT}(\lambda)^{\operatorname{maj}}(q)/\prod_{i=1}^{|\lambda|}(1-q^{i})$ [Sta99, Prop. 7.19.11].

In fact, formulas for the $d$th moment $\mu_{d}^{\lambda}$, $d$th central moment $\alpha_{d}^{\lambda}$, and $d$th cumulant $\kappa_{d}^{\lambda}$ of $\operatorname{maj}$ on $\operatorname{SYT}(\lambda)$ may be derived from (1). The most elegant of these formulas is for the cumulants, from which the moments and central moments are all easy to compute.

Theorem 1.5**.**

Let $\lambda\vdash n$ and $d\in\mathbb{Z}_{>1}$. We have

[TABLE]

where $B_{0},B_{1},B_{2},\ldots=1,\frac{1}{2},\frac{1}{6},0,-\frac{1}{30},0,\frac{1}{42},0,\ldots$ are the Bernoulli numbers.

See 2.9 for a generalization of (2) along with exact formulas for the moments and central moments. See 2.10 for the some of the history of this formula.

Remark 1.6**.**

For “most” partition shapes, one expects the term $\sum_{j=1}^{n}j^{d}$ in (2) to dominate $\sum_{c\in\lambda}h_{c}^{d}$, in which case asymptotic normality is quite straightforward. However, for some shapes there is a very large amount of cancellation in (2) and determining the limit law can be quite subtle.

While $\mathcal{X}_{\lambda}[\operatorname{maj}]$ can be written as the sum of scaled indicator random variables $D_{1},2D_{2},3D_{3},\ldots,$ $(n-1)D_{n-1}$ where $D_{i}$ determines if there is a descent at position $i$, the $D_{i}$ are not at all independent, so one may not simply apply standard central limit theorems. Interestingly, the $D_{i}$ are identically distributed [Sta99, Prop. 7.19.9]. The lack of independence of the $D_{i}$’s likewise complicates related work by Fulman [Ful98] and Kim–Lee [KL18] considering the limiting distribution of descents in certain classes of permutations.

The non-normal continuous limit laws for $\operatorname{maj}$ on $\operatorname{SYT}(\lambda)$ turn out to be the Irwin–Hall distributions $\mathcal{IH}_{M}\coloneqq\sum_{k=1}^{M}\mathcal{U}[0,1]$, which are the sum of $M$ i.i.d. continuous $[0,1]$ random variables. The following result completely classifies all possible limit laws for $\operatorname{maj}$ on $\operatorname{SYT}(\lambda)$ for any sequence of partition shapes. See 6.3 for the generalization to block diagonal skew shapes.

Theorem 1.7**.**

Let $\lambda^{(1)},\lambda^{(2)},\ldots$ be a sequence of partitions. Then $(\mathcal{X}_{\lambda^{(N)}}[\operatorname{maj}]^{*})$ converges in distribution if and only if

1. (i)

$\operatorname{aft}(\lambda^{(N)})\to\infty$; or 2. (ii)

$|\lambda^{(N)}|\to\infty$* and $\operatorname{aft}(\lambda^{(N)})\to M<\infty$; or* 3. (iii)

the distribution of $\mathcal{X}_{\lambda^{(N)}}^{*}[\operatorname{maj}]$ is eventually constant.

The limit law is $\mathcal{N}$ in case (i), $\mathcal{IH}_{M}^{*}$ in case (ii), and discrete in case (iii).

Case (iii) naturally leads to the question, when does $\mathcal{X}_{\lambda}^{*}[\operatorname{maj}]=\mathcal{X}_{\mu}^{*}[\operatorname{maj}]$? Such a description in terms of hook lengths is given in 7.1. 1.7 naturally raises several open questions and conjectures concerning unimodality, log-concavity, and local limit theorems, which are described in Section 8.

Example 1.8**.**

We illustrate each possible limit in 1.7. For (i), let $\lambda^{(N)}\coloneqq(N,\lfloor\ln N\rfloor)$, so that $\operatorname{aft}(\lambda^{(N)})=\lfloor\ln N\rfloor\to\infty$ and the distributions are asymptotically normal. For (ii), fix $M\in\mathbb{Z}_{\geq 0}$ and let $\lambda^{(N)}\coloneqq(N+M,M)$, so that $\operatorname{aft}(\lambda^{(N)})=M$ is constant and the distributions converge to $\Sigma_{M}^{*}$. For (iii), let $\lambda^{(2N)}\coloneqq(12,12,3,3,3,2,2,1,1)$ and $\lambda^{(2N+1)}\coloneqq(15,6,6,6,4,2)$, which have the same multisets of hook lengths despite not being transposes of each other, and consequently the same normalized $\operatorname{maj}$ distributions.

The rest of the paper is organized as follows. In Section 2, we give background focused on cumulants aimed at the combinatorial audience. In Section 3, we collect combinatorial background on permutations, tableaux, etc, aimed more at the probabilistic audience. In Section 4, we analyze $\operatorname{baj}-\operatorname{inv}$ on $S_{n}$ as an introductory example. In Section 5, we classify when $\operatorname{maj}$ on $\operatorname{SYT}(\underline{\lambda})$ is asymptotically normal. In Section 6, we determine the remaining continuous limit laws for $\operatorname{maj}$ on $\operatorname{SYT}(\underline{\lambda})$. In Section 7, we characterize the possible discrete distributions for $\operatorname{maj}$ on $\operatorname{SYT}(\lambda)$ in terms of hook lengths. Finally, Section 8 lists conjectures concerning unimodality, log-concavity, and local limit theorems.

2. Background on cumulants

In this section, we review some standard terminology and results on generating functions, random variables, and asymptotic normality, with a focus on cumulants. An excellent source for many further details in this area can be found in Canfield’s Chapter 3 of [Bón15].

2.1. Exponential generating functions

We now introduce our notation for exponential generating functions and the Bernoulli numbers, which will be used with cumulants shortly.

Definition 2.1**.**

Given a rational sequence $(g_{d})_{d=0}^{\infty}=(g_{0},g_{1},\ldots)$, the corresponding ordinary generating function is

[TABLE]

and the corresponding exponential generating function is

[TABLE]

Conversely, any rational power series

[TABLE]

is the ordinary generating function of the sequence $(f_{d})_{d=0}^{\infty}=(f_{0},f_{1},\ldots)$ and the exponential generating function of the sequence $(d!f_{d})_{d=0}^{\infty}$. The exponential generating functions we will encounter will all have a positive radius of convergence.

It is easy to describe products, quotients and compositions of generating functions. We recall in particular a formula for compositions of exponential generating functions for later use. Given two rational sequences $f=(f_{d})_{d=0}^{\infty}$, $g=(g_{d})_{d=0}^{\infty}$ such that $f_{0}=0$ and $g_{0}=1$, the composition of their exponential generating functions $E_{g}\circ E_{f}$ is again an exponential generating function for a rational sequence $h$, say $E_{h}(t)=E_{g}(E_{f}(t))$. For example, if $E_{f}(t)=\sum f_{d}t^{d}/d!$ and $E_{g}(t)=e^{t}$, so $g_{i}=1$ for all $i$, then by [Sta99, Cor. 5.1.6], the corresponding sequence $(h_{d})_{d=0}^{\infty}$ is given by $h_{0}=1$ and, for $d\geq 1$,

[TABLE]

where $\Pi_{d}$ is the collection of all set partitions $\pi=\{b_{1},b_{2},\ldots,b_{k}\}$ of $\{1,2,\ldots,d\}$. Collecting together $S_{d}$-orbits of $\Pi_{d}$ in (3) quickly gives

[TABLE]

where if $\lambda$ has $m_{i}$ parts of length $i$, then $z_{\lambda}\coloneqq 1^{m_{1}}2^{m_{2}}\cdots m_{1}!m_{2}!\cdots$. A more computationally efficient, recursive approach to (3) is the formula [Sta99, Prop. 5.1.7]

[TABLE]

Example 2.2**.**

The Bernoulli numbers $(B_{d})_{d=0}^{\infty}$ are rational numbers determined by the exponential generating function $E_{B}(t)\coloneqq t/(1-e^{-t})$. The first few terms in the sequence are

[TABLE]

[TABLE]

The divided Bernoulli numbers are given by $\frac{B_{d}}{d}$ for $d\geq 1$. Their exponential generating function $E_{D}(t)$ satisfies $1+t\frac{d}{dt}E_{D}(t)=E_{B}(t)$, from which it follows that

[TABLE]

We caution that a common alternate convention for Bernoulli numbers uses $B_{1}=-\frac{1}{2}$ with all other entries the same, corresponding with the exponential generating function $t/(e^{t}-1)$.

The Bernoulli numbers have many interesting properties; see [Maz08, Wik17] and [GKP89, Section 6.5]. For example, they appear in the polynomial expansion of the sums of $d$th powers,

[TABLE]

Compare the formula for sums of $d$th powers to the Riemann zeta function $\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}$ which can be evaluated at complex values $s\neq 1$ by analytic continuation. The divided Bernoulli numbers which appear in our formula (2) satisfy $\frac{B_{d}}{d}=-\zeta(1-d)$.

2.2. Probabilistic generating functions

We next review basic vocabulary and notation for moments and cumulants of random variables. All random variables we encounter will have moments of all orders. See [Bil95] for more details.

Definition 2.3**.**

Let $\mathcal{X}$ be a real-valued random variable where either $\mathcal{X}$ is continuous with probability density function $f\colon\mathbb{R}\to\mathbb{R}_{\geq 0}$ or $\mathcal{X}$ is discrete with probability mass function $f\colon\mathbb{Z}\to\mathbb{R}_{\geq 0}$. The cumulative distribution function (CDF) of $\mathcal{X}$ is given by

[TABLE]

depending on whether $\mathcal{X}$ is continuous or discrete. For any continuous real-valued function $g$, there is an associated random variable $g(\mathcal{X})$. The expectation of $g(\mathcal{X})$ is given by

[TABLE]

The mean and variance of $\mathcal{X}$ are, respectively,

[TABLE]

For $d\in\mathbb{Z}_{\geq 0}$, the $d$th moment and $d$th central moment of $\mathcal{X}$ are, respectively,

[TABLE]

The moment-generating function of $\mathcal{X}$ is

[TABLE]

which for us will always have a positive radius of convergence. The characteristic function of $\mathcal{X}$ is

[TABLE]

which exists for all $t\in\mathbb{R}$ and which is the Fourier transform of $f$, the density or mass function associated to $\mathcal{X}$.

Example 2.4**.**

Let $W$ be a finite set with an integer statistic $\operatorname{stat}\colon W\to\mathbb{Z}_{\geq 0}$. We will use the notation

[TABLE]

for the corresponding polynomial generating function. If $W^{\operatorname{stat}}(q)=\sum c_{k}q^{k}$, define a random variable $\mathcal{X}$ associated with $\operatorname{stat}\colon W\to\mathbb{Z}_{\geq 0}$ sampled uniformly on $W$ by $\mathbb{P}(\mathcal{X}=k)=c_{k}/\#W.$ The probability generating function for $\mathcal{X}$ is

[TABLE]

Letting $q=e^{t}$, an easy computation shows that the moment-generating function and characteristic function of $\mathcal{X}$ are

[TABLE]

These expressions reveal an intimate connection between the study of generating functions of combinatorial statistics evaluated on the unit circle and the underlying probability distribution via the Laplace and Fourier transforms. In particular, the distribution determines the characteristic function and the moment-generating function, and conversely each of these determines the distribution.

Definition 2.5**.**

The cumulants $\kappa_{1},\kappa_{2},\ldots$ of $\mathcal{X}$ are defined to be the coefficients of the exponential generating function

[TABLE]

While cumulants of random variables may initially be less intuitive than moments, they lead to nicer formulas in many cases, including 1.5, and they often have more useful properties. See [NS11] for some history and applications. We will use the following properties of cumulants. The proofs are straightforward from the definitions.

(Familiar Values) The first three cumulants are $\kappa_{1}=\mu$, $\kappa_{2}=\sigma^{2}$, and $\kappa_{3}=\alpha_{3}$. The higher cumulants typically differ from the moments and central moments. 2. 2.

(Shift Invariance) The second and higher cumulants of $\mathcal{X}$ agree with those for $\mathcal{X}-c$ for $c\in\mathbb{R}$. 3. 3.

(Homogeneity) The $d$th cumulant of $c\mathcal{X}$ is $c^{d}\kappa_{d}$ for $c\in\mathbb{R}$. 4. 4.

(Additivity) The cumulants of the sum of independent random variables are the sums of the cumulants. 5. 5.

(Polynomial Equivalence) The cumulants, moments, and central moments are determined by polynomials in any one of these three sequences.

The polynomial equivalence property can be made explicit by the results in Section 2.1. Equation (5) allows us to express the $d$th moment of $\mathcal{X}$ as a polynomial function of the first $d$ cumulants of $\mathcal{X}$ and vice versa via the recurrence

[TABLE]

Using the shift invariance property of cumulants, the corresponding formula for the central moments in terms of the cumulants can be obtained from (7) by setting $\kappa_{1}=0$ and leaving the other cumulants alone. This gives, for $d>1$,

[TABLE]

For instance, at $d=3$ we have

[TABLE]

Setting $\kappa_{1}=0$ yields $\alpha_{3}=\kappa_{3}$ as mentioned above.

2.3. Cumulant formulas

Next we describe the cumulants of some well-known distributions and use one of them to deduce a result of Hwang–Zacharovas, which immediately yields 1.5 as a corollary.

Example 2.6**.**

Let $\mathcal{X}=\mathcal{N}(\mu,\sigma^{2})$ be the normal random variable with mean $\mu$ and variance $\sigma^{2}$. The density function of $\mathcal{X}$ is $f(x;\mu,\sigma^{2})=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(x-\mu)^{2}}{2\sigma^{2}}\right)$. Taking the Fourier transform gives the characteristic function $\mathbb{E}[e^{it\mathcal{X}}]=\exp\left(i\mu t-\frac{1}{2}\sigma^{2}t^{2}\right)$, so the moment-generating function is $\mathbb{E}[e^{t\mathcal{X}}]=\exp\left(\mu t+\frac{1}{2}\sigma^{2}t^{2}\right)$ and the cumulants are

[TABLE]

Using (4) to compute the central moments of $\mathcal{X}$ from (9), we effectively set $\kappa_{1}=0$ and note that only $\lambda=(2,2,\ldots,2)=(2^{d/2})$ contributes, in which case $\alpha_{d}=\kappa_{2}^{d/2}d!/(2^{d/2}(d/2)!)$. It follows that

[TABLE]

Example 2.7**.**

Let $\mathcal{U}=\mathcal{U}[0,1]$ be the continuous uniform random variable whose density takes the value $1$ on the interval $[0,1]$ and [math] otherwise. Then the moment generating function is $M_{\mathcal{U}}(t)=\int_{0}^{1}e^{tx}dx=(e^{t}-1)/t$, so the cumulant generating function $\log M_{\mathcal{U}}(t)$ coincides with the exponential generating function for the divided Bernoulli numbers from Section 2.1. That is, $\kappa_{d}^{\mathcal{U}}=B_{d}/d$ for $d\geq 1$.

Recall from Section 1, $\mathcal{IH}_{m}$ is the Irwin–Hall distribution obtained by adding $m$ independent, identically distributed $\mathcal{U}[0,1]$ random variables. By Additivity, the $d$th cumulant of $\mathcal{IH}_{m}$ is $mB_{d}/d$. More generally, let $\mathcal{S}\coloneqq\sum_{k=1}^{m}\mathcal{U}[\alpha_{k},\beta_{k}]$ be the sum of $m$ independent uniform continuous random variables. Then the $d$th cumulant of $\mathcal{S}$ for $d\geq 2$ is

[TABLE]

by the Homogeneity and Additivity Properties of cumulants.

Example 2.8**.**

Let $\mathcal{U}_{n}$ be the discrete uniform random variable supported on $\{0,1,\ldots,n-1\}$. The probability generating function for $\mathcal{U}_{n}$ is $[n]_{q}/n\coloneqq(q^{n}-1)/(n(q-1))$, so the cumulant generating function is

[TABLE]

It follows that for $d\geq 1$, the divided Bernoulli numbers arise again in this context,

[TABLE]

Product formulas for polynomials such as Stanley’s formula (1) give rise to explicit formulas for cumulants and moments according to the following theorem. The proof is immediate from 2.8 and the exponential generating function identity (4).

Theorem 2.9**.**

Suppose $\{a_{1},\ldots,a_{m}\}$ and $\{b_{1},\ldots,b_{m}\}$ are multisets of positive integers such that

[TABLE]

so in particular each $c_{k}\in\mathbb{Z}_{\geq 0}$. Let $\mathcal{X}$ be a discrete random variable with $\mathbb{P}[\mathcal{X}=k]=c_{k}/P(1)$. Then the $d$th cumulant of $\mathcal{X}$ is

[TABLE]

where $B_{d}$ is the $d$th Bernoulli number (with $B_{1}=\frac{1}{2}$). Moreover, the $d$th central moment of $\mathcal{X}$ is

[TABLE]

and the $d$th moment of $\mathcal{X}$ is

[TABLE]

Remark 2.10**.**

Equation (12) appeared explicitly in the work of Hwang–Zacharovas [HZ15, §4.1] building on the work of Chen–Wang–Wang [CWW08, Thm. 3.1], who in turn used an argument going back at least to Sachkov [Sac97, §1.3.1]. It was rediscovered experimentally through (14) by the present authors, and by Thiel–Williams [TW18].

One frequently encounters polynomials of the form $q^{\beta}P(q)$ for some $\beta\in\mathbb{Z}_{\geq 0}$, as in (1). The formulas in 2.9 remain valid in this case except that one must add $\beta$ to the expression for $\kappa_{1}$ and add $\beta$ to each factor in the product in (14) for which $\lambda_{i}=1$.

Remark 2.11**.**

The generating function machinery used to construct the cumulants in (12) works whether or not the function $P(q)$ is polynomial. The corresponding $\kappa_{d}$’s are called formal cumulants in the literature.

2.4. Asymptotic normality

Asymptotic normality is a very old topic lying at the intersection of probability and combinatorics. For an introduction, we recommend Canfield’s Chapter 3 in [Bón15].

Definition 2.12**.**

Let $\mathcal{X}_{1},\mathcal{X}_{2},\ldots$ and $\mathcal{X}$ be real-valued random variables with cumulative distribution functions $F_{1},F_{2},\ldots$ and $F$, respectively. We say $\mathcal{X}_{1},\mathcal{X}_{2},\ldots$ converges in distribution to $\mathcal{X}$, written $\mathcal{X}_{n}\Rightarrow\mathcal{X}$, if for all $t\in\mathbb{R}$ at which $F$ is continuous we have

[TABLE]

Recall from the introduction that for a real-valued random variable $\mathcal{X}$ with mean $\mu$ and variance $\sigma^{2}>0$, the corresponding normalized random variable is

[TABLE]

Observe that $\mathcal{X}^{*}$ has mean $\mu^{*}=0$ and variance ${\sigma^{*}}^{2}=1$. The moments and central moments of $\mathcal{X}^{*}$ agree for $d\geq 2$ and are given by

[TABLE]

Similarly, the cumulants of $\mathcal{X}^{*}$ are given by $\kappa_{1}^{*}=0$, $\kappa_{2}^{*}=1$, and $\kappa_{d}^{*}=\kappa_{d}/\sigma^{d}$ for $d\geq 2$.

Definition 2.13**.**

Let $\mathcal{X}_{1},\mathcal{X}_{2},\ldots$ be a sequence of real-valued random variables. We say the sequence is asymptotically normal if $\mathcal{X}_{n}^{*}\Rightarrow\mathcal{N}(0,1)$.

The “original” asymptotic normality result is as follows. Let $2^{[n]}$ be the set of all subsets of $[n]\coloneqq\{1,2,\ldots,n\}$. Let $\mathcal{X}_{2^{[n]}}[\operatorname{size}]$ denote the random variable given by the cardinality, where $2^{[n]}$ is given the uniform distribution. This has the same distribution as the number of heads after $n$ fair coin flips, so the probability generating function up to normalization is $(1+q)^{n}$. The following result is credited to de Moivre and Laplace; see [Bón15, Theorem 3.2.1] for further discussion.

Theorem 2.14** (de Moivre–Laplace).**

The sequence $\mathcal{X}_{2^{[n]}}[\operatorname{size}]$ is asymptotically normal.

Asymptotic normality results for combinatorial statistics are plentiful. See Table 1 for more examples and further references.

2.5. The method of moments

We next describe two standard criteria for establishing asymptotic normality or more generally convergence in distribution of a sequence of random variables.

Theorem 2.15** (Lévy’s Continuity Theorem, [Bil95, Theorem 26.3]).**

A sequence $\mathcal{X}_{1},\mathcal{X}_{2},\ldots$ of real-valued random variables converges in distribution to a real-valued random variable $\mathcal{X}$ if and only if, for all $t\in\mathbb{R}$,

[TABLE]

Theorem 2.16** **(Frechét–Shohat Theorem,

[Bil95, Theorem 30.2]).

Let $\mathcal{X}_{1},\mathcal{X}_{2},\ldots$ be a sequence of real-valued random variables, and let $\mathcal{X}$ be a real-valued random variable. Suppose the moments of $\mathcal{X}_{n}$ and $\mathcal{X}$ all exist and the moment generating functions all have a positive radius of convergence. If

[TABLE]

then $\mathcal{X}_{1},\mathcal{X}_{2},\ldots$ converges in distribution to $\mathcal{X}$.

By 2.15, we may test for asymptotic normality by checking if the normalized characteristic functions tend point-wise to the characteristic function of the standard normal. Likewise by 2.16 we may instead perform the check on the level of individual normalized moments, which is often referred to as the method of moments. By (7) we may further replace the moment condition (15) with the cumulant condition

[TABLE]

For instance, we have the following explicit criterion.

Corollary 2.17**.**

A sequence $\mathcal{X}_{1},\mathcal{X}_{2},\ldots$ of real-valued random variables on finite sets is asymptotically normal if for all $d\geq 3$ we have

[TABLE]

In fact, one may show a converse of the Frechét–Shohat theorem holds for quotients as in 2.9, though we will not have need of it here.

2.6. Local limit theorems

Asymptotic normality concerns cumulative distribution functions, so it gives estimates for the number of combinatorial objects with a large range of statistics. However, our original motivation was to count combinatorial objects with a given statistic. Estimates of this latter form are frequently referred to as local limit theorems. Here we review two motivating examples.

The present work was partly inspired by the following local limit theorem due to the third author with a uniform rather than normal limit law. For $\lambda\vdash n$, let $\operatorname{maj}_{n}\colon\operatorname{SYT}(\lambda)\to[n]$ be $\operatorname{maj}$ modulo $n$.

Theorem 2.18**.**

[Swa18, Theorem 1.9]** For $\lambda\vdash n$, let $X_{\lambda}[\operatorname{maj}_{n}]$ denote the random variable $\operatorname{maj}_{n}$ on $\operatorname{SYT}(\lambda)$. Suppose $\#\operatorname{SYT}(\lambda)\geq n^{5}$. Then, for all $k\in[n]$,

[TABLE]

Further motivation was provided by the following analogue of 3.16.

Theorem 2.19**.**

[CJZ11, Theorem 4.5]** There exists a positive constant $c$ such that for every $C$, the following is true. Uniformly for all compositions $\alpha=(\alpha_{1},\ldots,\alpha_{m})$ such that $\max_{i}\alpha_{i}\leq Ce^{cs(\alpha)}$ and all integers $k$,

[TABLE]

where $X_{\alpha}$ denotes inversions on words of type $\alpha$.

3. Combinatorial background

3.1. Combinatorial background for $\operatorname{baj}-\operatorname{inv}$ on $S_{n}$

Here we introduce the two most well-known permutation statistics, $\operatorname{inv}$ and $\operatorname{maj}$, as well as one unusual permutation statistic, $\operatorname{baj}$.

Definition 3.1**.**

Let $\sigma\in S_{n}$ be a permutation of $\{1,\ldots,n\}$. Set

[TABLE]

Following Zabrocki [Zab03] for the nomenclature, we also set

[TABLE]

The equidistribution of $\operatorname{inv}$ and $\operatorname{maj}$ on $S_{n}$ is due to MacMahon, who also first introduced $\operatorname{maj}$. His proof gave the following generating function expression for both statistics.

Theorem 3.2** ([Mac13, Art. 6]).**

We have

[TABLE]

The statistic $\operatorname{baj}-\operatorname{inv}$ appeared in the context of extended affine Weyl groups and Hecke algebras in the work of Iwahori and Matsumoto in 1965 [IM65]. It is the Coxeter length function restricted to coset representatives of the extended affine Weyl group of type $A_{n-1}$ mod translations by coroots. Stembridge and Waugh [SW98, Remarks 1.5 and 2.3] give a careful overview of this topic and further results. In particular, they prove the following factorization formula for the generating function associated to $\operatorname{baj}-\operatorname{inv}$ on $S_{n}$. From this factorization, the corresponding cumulants can be read off from 2.9.

Theorem 3.3**.**

[IM65, SW98]** We have

[TABLE]

Corollary 3.4**.**

The $d$th cumulant $\kappa_{d}^{n}$ for $\operatorname{baj}-\operatorname{inv}$ on $S_{n}$ is

[TABLE]

Remark 3.5**.**

Indeed, (18) holds with $S_{n}$ replaced by $\{\sigma\in S_{n}:\sigma(n)=k\}$ for any fixed $k=1,\ldots,n$ if the factor of $n$ is deleted from the right-hand side. See [Zab03] for a bijective proof of this generalization. In addition, [SW98, Thm. 1.1] gives another generalization of the product formula (18) to all crystallographic Coxeter groups.

3.2. Combinatorial background for $\operatorname{maj}$ on $\operatorname{W}_{\alpha}$

and $\operatorname{SYT}(\underline{\lambda})$

Here we review standard combinatorial notions related to words, tableaux, and their major index generating functions.

Definition 3.6**.**

Given a word $w=w_{1}w_{2}\cdots w_{n}$ with letters $w_{i}\in\mathbb{Z}_{\geq 1}$, the type of $w$ is the sequence $\alpha=(\alpha_{1},\alpha_{2},\ldots)$ where $\alpha_{i}$ is the number of times $i$ appears in $w$. Such a sequence $\alpha$ is a (weak) composition of $n$, written as $\alpha\vDash n$. Trailing [math]’s are often omitted when writing weak compositions, so $\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{m})$ for some $m$. Note that a word of type $(1,1,\ldots,1)\vDash n$ is a permutation in the symmetric group $S_{n}$ written in one-line notation. Just as for permutations, the inversion number of $w$ is

[TABLE]

The descent set of $w$ is

[TABLE]

and the major index of $w$ is

[TABLE]

Definition 3.7**.**

Let $\alpha=(\alpha_{1},\ldots,\alpha_{m})\vDash n$. We use the following standard $q$-analogues:

[TABLE]

Example 3.8**.**

The identity statistic on the set $W=\{0,\ldots,n-1\}$ has generating function $[n]_{q}$. The “sum” statistic on $W=\prod_{k=1}^{n}\{0,\ldots,k-1\}$ has generating function $[n]_{q}!$.

For $\alpha\vDash n$, let $\operatorname{W}_{\alpha}$ denote the words of type $\alpha$. MacMahon’s classic result generalizing 3.2 in fact shows that $\operatorname{maj}$ and $\operatorname{inv}$ have the same distribution on $\operatorname{W}_{\alpha}$.

Theorem 3.9** ([Mac13, Art. 6]).**

For each $\alpha\vDash n$,

[TABLE]

Definition 3.10**.**

A composition $\lambda\vDash n$ such that $\lambda_{1}\geq\lambda_{2}\geq\ldots$ is called a partition of $n$, written as $\lambda\vdash n$. The size of $\lambda$ is $|\lambda|\coloneqq n$ and the length $\ell(\lambda)$ of $\lambda$ is the number of non-zero entries. The Young diagram of $\lambda$ is the upper-left justified arrangement of unit squares called cells where the $i$th row from the top has $\lambda_{i}$ cells following the English notation; see Figure 2(a). The hook length of a cell $c\in\lambda$ is the number $h_{c}$ of cells in $\lambda$ in the same row as $c$ to the right of $c$ and in the same column as $c$ and below $c$, including $c$ itself; see Figure 2(b). A corner of $\lambda$ is any cell with hook length $1$. A bijective filling of $\lambda$ is any labeling of the cells of $\lambda$ by the numbers $[n]=\{1,2,\ldots,n\}$.

Definition 3.11**.**

A skew partition $\lambda/\nu$ is a pair of partitions $(\nu,\lambda)$ such that the Young diagram of $\nu$ is contained in the Young diagram of $\lambda$. The cells of $\lambda/\nu$ are the cells in the diagram of $\lambda$ which are not in the diagram of $\nu$, written $c\in\lambda/\nu$. We identify straight partitions $\lambda$ with skew partitions $\lambda/\varnothing$ where $\varnothing=(0,0,\ldots)$ is the empty partition. The size of $\lambda/\nu$ is $|\lambda/\nu|\coloneqq|\lambda|-|\nu|$. The notions of bijective filling, hook lengths, and corners naturally extend to skew partitions as well.

Definition 3.12**.**

Given a sequence of partitions $\underline{\lambda}=(\lambda^{(1)},\ldots,\lambda^{(m)})$, we identify the sequence with the block diagonal skew partition obtained by translating the Young diagrams of the $\lambda^{(i)}$ so that the rows and columns occupied by these components are disjoint, form a valid skew shape, and appear in order from top to bottom as depicted in Figure 3.

Definition 3.13**.**

A standard Young tableau of shape $\lambda/\nu$ is a bijective filling of the cells of $\lambda/\nu$ such that labels increase to the right in rows and down columns; see Figure 4. The set of standard Young tableaux of shape $\lambda/\nu$ is denoted $\operatorname{SYT}(\lambda/\nu)$. The descent set of $T\in\operatorname{SYT}(\lambda/\nu)$ is the set $\operatorname{Des}(T)$ of all labels $i$ in $T$ such that $i+1$ is in a strictly lower row than $i$. The major index of $T$ is

[TABLE]

Remark 3.14**.**

The block diagonal skew partitions $\underline{\lambda}$ allow us to simultaneously consider words and tableaux as follows. Recall that $\operatorname{W}_{\alpha}$ is set of all words with type $\alpha=(\alpha_{1},\ldots,\alpha_{k})$. Letting $\underline{\lambda}=((\alpha_{k}),\ldots,(\alpha_{1}))$, we have a bijection

[TABLE]

which sends a tableau $T$ to the word whose $i$th letter is the row number in which $i$ appears in $T$, counting from the bottom up rather than top down. For example, using the skew tableau $T$ on the right of Figure 4, we have $\phi(T)=1312231\in\operatorname{W}_{(3,2,2)}$. It is easy to see that $\operatorname{Des}(\phi(T))=\operatorname{Des}(T)$, so that $\operatorname{maj}(\phi(T))=\operatorname{maj}(T)$. Hence $\operatorname{SYT}((\alpha_{1}),\ldots,(\alpha_{k}))^{\operatorname{maj}}(q)=\operatorname{W}_{\alpha}^{\operatorname{maj}}(q)=\binom{n}{\alpha}_{q}$.

Remark 3.15**.**

We also recover $q$-integers, $q$-binomials, $q$-multinomials, and $q$-Catalan numbers up to $q$-shifts as special cases of the major index generating function for tableaux given in (1):

[TABLE]

Many combinatorial statistics arise from sets indexed by more complicated objects than the positive integers, in which case one can “let $n\to\infty$” in many different ways. The following result due to Canfield, Janson, and Zeilberger illustrates a more interesting limit. Their result is characterized by the statistic $s(\alpha)\coloneqq n-m$ where $\alpha=(\alpha_{1},\ldots,\alpha_{\ell})\vDash n$ with $\max\{\alpha_{i}\}=m$.

Theorem 3.16**.**

[CJZ11, Theorem 1.2]** Let $\alpha^{(1)},\alpha^{(2)},\ldots$ be a sequence of compositions, possibly of differing lengths. Let $\mathcal{X}_{n}$ be the inversion (or major index) statistic on words of type $\alpha^{(n)}$. Then $\mathcal{X}_{1},\mathcal{X}_{2},\ldots$ is asymptotically normal if and only if

[TABLE]

Remark 3.17**.**

Explorations equivalent to 3.16 appeared significantly earlier than [CJZ11] in other contexts, for instance [Dia88, p. 127-128] and (in the two-letter case) [MW47]. See [CJZ12] for further discussion and references.

The cumulant formula for $\mathcal{X}_{\lambda}[\operatorname{maj}]$, 1.5, follows immediately from 2.9 and Stanley’s formula (1). Adin and Roichman [AR01] had previously used (1) to compute the mean and variance of $\mathcal{X}_{\lambda}[\operatorname{maj}]$ as

[TABLE]

and

[TABLE]

The following common generalization of Stanley’s formula (1) and MacMahon’s formula, 3.9, is well known (e.g. see [Ste89, (5.6)]). See [BKS18, Thm. 2.15] for other applications.

Theorem 3.18**.**

Let $\underline{\lambda}=(\lambda^{(1)},\ldots,\lambda^{(m)})$ where $\lambda^{(i)}\vdash\alpha_{i}$ and $n=\alpha_{1}+\cdots+\alpha_{m}$. Then

[TABLE]

Corollary 3.19**.**

Let $\kappa_{d}^{\underline{\lambda}}$ be the $d$th cumulant of $\operatorname{maj}$ on $\operatorname{SYT}(\underline{\lambda})$ for $d>1$. Then

[TABLE]

For general skew shapes, $\operatorname{SYT}(\lambda/\nu)^{\operatorname{maj}}(q)$ does not factor as a product of cyclotomic polynomials times $q$ to a power. A “$q$-Naruse” formula due to Morales–Pak–Panova, [MPP18, (3.4)], gives an analogue of (1) involving a sum over “excited diagrams,” though the resulting sum has a single term precisely for the block diagonal skew partitions $\underline{\lambda}$.

4. Asymptotic normality for $\operatorname{baj}-\operatorname{inv}$ on $S_{n}$

We give with a straightforward example which serves as a warmup and establishes some notation. See Section 3.1 for background. Asymptotic normality of $\operatorname{baj}-\operatorname{inv}$ on $S_{n}$ follows from the cumulant formula in 3.4 by the following routine calculations. Recall that $a_{n}\sim b_{n}$ means that $\lim_{n\to\infty}a_{n}/b_{n}=1$.

Lemma 4.1**.**

Fix $d\geq 1$. Then, as $n\to\infty$,

[TABLE]

Proof.

We have

[TABLE]

∎

Remark 4.2**.**

The value of the integral in 4.1 is well known:

[TABLE]

See [OEI17, A002457] for a surprisingly large number of interpretations of the reciprocals of these values. Equation (23) is also a very special case of the Selberg integral formula [Sel44], which has many interesting connections to algebraic combinatorics such as those in [KO17].

Corollary 4.3**.**

Fix $d\in\{1,2,4,6,\ldots\}$. Let $\kappa_{d}^{n}$ be the $d$th cumulant of $\operatorname{baj}-\operatorname{inv}$ on $S_{n}$, and let ${\kappa_{d}^{n}}^{*}$ be the $d$th cumulant of the corresponding normalized random variable with mean [math] and variance $1$. Then, uniformly for all $n$, we have

[TABLE]

That is, there are constants $c,C>0$ depending only on $d$ such that

[TABLE]

Proof.

It follows immediately from 3.4 and 4.1 that $|\kappa_{d}^{n}|=\Theta(n^{2d+1})$. Hence

[TABLE]

∎

Theorem 4.4**.**

Let $\mathcal{X}_{n}=\mathcal{X}_{S_{n}}[\operatorname{baj}-\operatorname{inv}]$ be the random variable for the $\operatorname{baj}-\operatorname{inv}$ statistic taken uniformly at random from $S_{n}$. Then, $\mathcal{X}_{1},\mathcal{X}_{2},\ldots$ is asymptotically normal.

Proof.

For fixed $d>2$ even, we have $1-d/2<0$, so by 4.3, ${\kappa_{d}^{n}}^{*}\to 0$ as $n\to\infty$. The odd cumulants for $d>2$ vanish since the odd Bernoulli numbers are [math]. The result now follows from 2.17. ∎

Remark 4.5**.**

A key step in the above argument was to show that the variance $\sigma_{n}^{2}$ of $\operatorname{baj}-\operatorname{inv}$ on $S_{n}$ satisfies $\sigma_{n}^{2}=\Theta(n^{5})$. Indeed, the argument gives $\sigma_{n}^{2}\sim n^{5}/360$. The weaker observation that $\sum_{i=1}^{n-1}[i(n-i)]^{2}$ is the dominant contribution to $\sigma_{n}^{2}$ is essentially enough to deduce asymptotic normality in this case. Our analysis of $\operatorname{maj}$ on standard tableaux includes non-normal limits, so more precise estimates like the above will become absolutely necessary. A straightforward modification of the above argument together with 3.2 also proves 1.1.

5. Asymptotic normality for $\operatorname{maj}$ on $\operatorname{SYT}(\underline{\lambda})$

The main result of this section, 5.8, classifies the sequences of block diagonal skew partitions for which $\operatorname{maj}$ is asymptotically normal. We begin with a series of estimates for the differences $\sum_{k=1}^{|\lambda/\nu|}k^{d}-\sum_{c\in\lambda/\nu}h_{c}^{d}$, culminating in 5.7.

Definition 5.1**.**

A reverse standard Young tableau of shape $\lambda/\nu$ is a bijective filling of $\lambda/\nu$ which strictly decreases along rows and columns. The set of reverse standard Young tableaux of shape $\lambda/\nu$ is denoted $\operatorname{RSYT}(\lambda/\nu)$.

Lemma 5.2**.**

Let $\lambda/\nu\vdash n$ and $T\in\operatorname{RSYT}(\lambda/\nu)$. Then for all $c\in\lambda/\nu$,

[TABLE]

Furthermore, for any positive integer $d$,

[TABLE]

where ${\mathbf{h}}_{d-1}$ denotes the complete homogeneous symmetric function.

Proof.

For (25), equality holds at the outer corner $c$ where $T_{c}=1$. Removing $c$ and subtracting $1$ from each remaining entry in $T$ allows us to induct. Equation (26) follows immediately by rearranging the terms and factoring $(T_{c}^{d}-h_{c}^{d})=(T_{c}-h_{c})\sum_{k=0}^{d-1}T_{c}^{d-1-k}h_{c}^{k}$. ∎

Lemma 5.3**.**

Let $\lambda/\nu\vdash n$ such that $\max_{c\in\lambda/\nu}h_{c}<0.8n$. Let $d$ be any positive integer. Then

[TABLE]

Proof.

Using Riemmann sums for $\int_{0}^{n}x^{d}dx$, we obtain the bounds

[TABLE]

for all positive integers $d,n$. The upper bound in the lemma now follows immediately.

For the lower bound, label the cells of $\lambda/\nu$ by some $T\in\operatorname{RSYT}(\lambda/\nu)$. By (25), $h_{c}\leq T_{c}$, and by assumption we have $h_{c}<0.8n$ for all $c\in\lambda/\nu$. Considering the tighter of these two bounds on each summand and using (27) again, we have

[TABLE]

Consequently,

[TABLE]

It is easy to check that the coefficient on $n^{d+1}$ is bounded below by $\frac{1}{26(d+1)}$ for all positive integers $d$. The result follows. ∎

Definition 5.4**.**

Given any partition $\lambda/\nu\vdash n$, let the aft of $\lambda/\nu$ be the statistic

[TABLE]

where $\operatorname{arm}(c)$ is the number of cells in the same row as $c$ to the right of $c$, including $c$ itself, and $\operatorname{leg}(c)$ is the number of cells in the same column as $c$ below $c$, including $c$. When $\nu=\varnothing$, we have $\operatorname{aft}(\lambda)=n-\max\{\lambda_{1},\lambda_{1}^{\prime}\}$ as above. When $\lambda/\nu=\underline{\lambda}$, we have $\operatorname{aft}(\underline{\lambda})=n-\max_{i}\{\lambda_{1}^{(i)},{\lambda^{(i)}}^{\prime}_{1}\}$. Note that $h_{c}=\operatorname{arm}(c)+\operatorname{leg}(c)-1$.

Lemma 5.5**.**

Let $\lambda/\nu\vdash n$ such that $\max_{c\in\lambda/\nu}h_{c}\geq 0.8n$, and let $d$ be any positive integer. Furthermore, suppose $n\geq 10$. Then,

[TABLE]

Proof.

The result holds trivially if $\operatorname{aft}(\lambda/\nu)=0$ since in that case $\lambda/\nu$ is a single row or column, so assume $\operatorname{aft}(\lambda/\nu)>0$. Let $m\in\lambda/\nu$ have $h_{m}\geq 0.8n$, where we may assume $m$ is the first cell in its row and column. For convenience, we may further assume by symmetry that $\operatorname{arm}(m)\geq\operatorname{leg}(m)$. Since $h_{m}\geq 0.8n$, it also follows that $\operatorname{aft}(\lambda/\nu)=n-\operatorname{arm}(m)$.

Now let $R$ be the set of cells in the row of $m$, not including $m$ itself, which are the only cells of $\lambda/\nu$ in their columns. Since $\lambda/\nu$ is a skew partition, $R$ is connected. We claim that $\#R\geq 0.1n$. To prove the claim, we first observe that the hypothesis $h_{m}\geq 0.8n$ implies there are at most $n-h_{m}\leq 0.2n$ cells of $\lambda/\nu$ which could possibly be in the columns of the cells of the row of $m$ not including $m$. Since $\operatorname{arm}(m)\geq\operatorname{leg}(m)$ and $\operatorname{arm}(m)+\operatorname{leg}(m)-1=h_{m}\geq 0.8n$, we have $\operatorname{arm}(m)\geq 0.4n$. Hence no more than $0.2n$ of the $0.4n-1$ cells in the row of $m$ not including $m$ can be excluded from $R$, so $\#R\geq 0.4n-1-0.2n\geq 0.1n$ for $n\geq 10$.

Construct $T\in\operatorname{RSYT}(\lambda/\nu)$ iteratively as follows; see Figure 5 for an example. At each step of the iteration, we will first increment all existing labels by $1$ and then label a new outer cell with $1$. Begin by adding the cells of the row of $m$ from left to right until the last cell of $R$ has been added. Now add the remaining cells of $\lambda/\nu$ row by row starting at the topmost row and going from left to right. It is easy to see that the result respects the decreasing row and column conditions, so $T\in\operatorname{RSYT}(\lambda/\nu)$.

By 5.2, we have inequalities $T_{c}\geq h_{c}$. At every step of the iteration, a labeled cell has $T_{c}$ increase by $1$, while $h_{c}$ increases by $1$ if and only if the newly labeled cell is in the hook of $c$. That is, for the final filling $T$, $T_{c}-h_{c}$ counts the number of times after cell $c$ was filled that the new cell was not in the same row or column as $c$. For each $c\in R$, it follows that $T_{c}-h_{c}=n-\operatorname{arm}(m)=\operatorname{aft}(\lambda/\nu)$.

For the lower bound, we now find

[TABLE]

where the first inequality uses the fact that $\{h_{c}:c\in R\}$ has pointwise lower bounds of $\{1,2,\ldots,\#R\}$ and the last inequality uses (27).

For the upper bound, we construct a new $T\in\operatorname{RSYT}(\lambda/\nu)$ as follows; see Figure 6 for an example. First, for each cell $c$ in the row of $m$ taken from left to right, add the topmost cell in the column of $c$. Now add the remaining cells of $\lambda/\nu$ exactly as before. Again consider the final differences $T_{c}-h_{c}$. For cells added in the second stage, $T_{c}-h_{c}$ could increase no more than $n-\operatorname{arm}(m)=\operatorname{aft}(\lambda/\nu)$ times, so $T_{c}-h_{c}\leq\operatorname{aft}(\lambda/\nu)$ for such $c$. For cells added in the first stage, we claim that $T_{c}-h_{c}\leq 2\operatorname{aft}(\lambda/\nu)$. For the claim, it suffices to show that after the first stage, for cells added in the first stage, $T_{c}-h_{c}\leq\operatorname{aft}(\lambda/\nu)$. During the first stage, the differences $T_{c}-h_{c}$ are zero while cells of row $m$ are being added. Afterwards during the first phase, cells not in row $m$ are added, of which there are no more than $n-\operatorname{arm}(m)=\operatorname{aft}(\lambda/\nu)$, so the differences $T_{c}-h_{c}$ can increase no more than $\operatorname{aft}(\lambda/\nu)$ many times during the first phase, completing the claim.

Having established that $T_{c}-h_{c}\leq 2\operatorname{aft}(\lambda/\nu)$, we now find by (26) and (27),

[TABLE]

∎

Corollary 5.6**.**

For fixed $d\in\mathbb{Z}_{\geq 1}$, uniformly for all skew shapes $\lambda/\nu$,

[TABLE]

Proof.

Let $n=|\lambda/\nu|$. When $\max_{c\in\lambda/\nu}h_{c}\geq 0.8n$, the result follows from 5.5. On the other hand, when $\max_{c\in\lambda/\nu}h_{c}<0.8n$, then $n\geq\operatorname{aft}(\lambda/\nu)\geq 0.2n$, and the result follows from 5.3. ∎

Corollary 5.7**.**

Fix $d$ to be an even positive integer. Uniformly for all block diagonal skew shapes $\underline{\lambda}$, the absolute value of the normalized cumulant $|{\kappa_{d}^{\underline{\lambda}}}^{*}|$ of $\mathcal{X}_{\underline{\lambda}}[\operatorname{maj}]$ is $\Theta(\operatorname{aft}(\underline{\lambda})^{1-d/2})$.

Proof.

For $d$ even, by (22) and 5.6, we have

[TABLE]

where $n=|\underline{\lambda}|$. Consequently by the homogeneity of cumulants, we have

[TABLE]

∎

We now state and prove the generalization of 1.3 for the block diagonal skew shapes $\underline{\lambda}$ from Section 3.2.

Theorem 5.8**.**

Suppose $\underline{\lambda}^{(1)},\underline{\lambda}^{(2)},\ldots$ is a sequence of block diagonal skew partitions, and let $\mathcal{X}_{N}\coloneqq\mathcal{X}_{\underline{\lambda}^{(N)}}[\operatorname{maj}]$ be the corresponding random variables for the $\operatorname{maj}$ statistic. Then, the sequence $\mathcal{X}_{1},\mathcal{X}_{2},\ldots$ is asymptotically normal if and only if $\operatorname{aft}(\underline{\lambda}^{(N)})\to\infty$ as $N\to\infty$.

Proof.

If $\operatorname{aft}(\underline{\lambda}^{(N)})\to\infty$, the result follows immediately from 2.17, 5.7, and the fact that the odd cumulants vanish. On the other hand, if $\operatorname{aft}(\underline{\lambda}^{(N)})\not\to\infty$, in the next section we will show that $\mathcal{X}_{1}^{*},\mathcal{X}_{2}^{*},\ldots$ has a subsequence which converges to either a discrete or uniform-sum distribution, which in either case is non-normal. ∎

Remark 5.9**.**

Using work of Hwang–Zacharovas [HZ15, Thm. 1.1], considering just the $d=4$ case is sufficient to prove both directions of 5.8. However, the estimates we’ve given for $\kappa_{d}^{\underline{\lambda}}$ are strong enough to bound all the normalized cumulants simultaneously, and restricting to $d=4$ (or even $d=2$) does not simplify the argument.

6. Uniform sum limits for $\operatorname{maj}$ on $\operatorname{SYT}(\underline{\lambda})$

The estimates from Section 5 apply when $\operatorname{aft}\to\infty$. We next give an analogous estimate handling the case when $\operatorname{aft}$ is bounded, resulting in 6.2. We may then deduce 1.7 from the introduction and its generalization to block diagonal skew shapes, 6.3. Recall from Section 1 and 2.7 that $\mathcal{IH}_{M}$ is the Irwin–Hall distribution obtained by adding $M$ i.i.d. $\mathcal{U}[0,1]$ random variables.

Lemma 6.1**.**

Suppose $\lambda^{(N)}/\nu^{(N)}\vdash n_{N}$ is a sequence of skew partitions such that $\lim_{N\to\infty}n_{N}=\infty$ and

[TABLE]

Then for each fixed $d\in\mathbb{Z}_{\geq 1}$, we have

[TABLE]

Proof.

Take $N$ large enough so that $\operatorname{aft}(\lambda^{(N)}/\nu^{(N)})=M$ and $n_{N}\gg M$. Let $m\in\lambda^{(N)}/\nu^{(N)}$ be such that $\operatorname{aft}(\lambda^{(N)}/\nu^{(N)})=M=n_{N}-\operatorname{arm}(m)$ so $m$ is the first cell in its row and column, as in the proof of 5.5. Consider three regions of $\lambda^{(N)}/\nu^{(N)}$:

1. (i)

The rightmost $\operatorname{arm}(m)-M=n_{N}-2M$ cells in the row of $m$. 2. (ii)

The remaining leftmost $M$ cells in the row of $m$. 3. (iii)

The remaining $M$ cells in $\lambda^{(N)}/\nu^{(N)}$.

Construct $T\in\operatorname{RSYT}(\lambda^{(N)}/\nu^{(N)})$ iteratively as in the proof of 5.5 as follows. First add cells in region (iii) row by row starting at the topmost row proceeding from left to right, stopping just before inserting the row of $m$. Next add the cells from region (ii) from left to right. Now add the remaining cells in region (iii) row by row starting at the row immediately below the row of $m$ proceeding from left to right. Finally insert the cells from region (i) from left to right. It is easy to see that the cells in region (i) are the lowest cells in their column, from which it follows that $T$ indeed satisfies the column and row decreasing conditions.

We now consider the contributions of regions (i)-(iii) to the quotient

[TABLE]

Recall that $T_{c}-h_{c}$ can be interpreted as the number of times a cell inserted after cell $c$ was not inserted in the same hook as $c$. It follows that $T_{c}-h_{c}=0$ for region (i), leaving only contributions from the $2M$ cells in regions (ii) and (iii), a bounded sum. For region (ii), we have $T_{c}-h_{c}\leq M$, so that

[TABLE]

Dividing by $Mn_{N}^{d}$, cells in region (ii) contribute [math] to the sum in the limit. Finally, for region (iii), we find $1\leq h_{c}\leq M+1$ and $n_{N}-2M+1\leq T_{c}\leq n_{N}$, so that for each of the $M$ cells $c$ in region (iii),

[TABLE]

Dividing by $n_{N}^{d}$, both bounds are asymptotic to $1$ as $n_{N}\to\infty$. Adding up all $M$ such contributions, the result follows. ∎

Theorem 6.2**.**

Suppose that $\underline{\lambda}^{(1)},\underline{\lambda}^{(2)},\ldots$ is a sequence of block diagonal skew partitions such that $\lim_{N\to\infty}|\underline{\lambda}^{(N)}|=\infty$ and $\operatorname{aft}(\underline{\lambda}^{(N)})=M$ is constant. Let $\mathcal{X}_{N}\coloneqq\mathcal{X}_{\underline{\lambda}^{(N)}}[\operatorname{maj}]$ be the corresponding random variable for the $\operatorname{maj}$ statistic. Then $\mathcal{X}_{1}^{*},\mathcal{X}_{2}^{*},\ldots$ converges in distribution to $\mathcal{IH}_{M}^{*}$.

Proof.

Using Equation 22 and 6.1, we have for $d\geq 2$ that

[TABLE]

From 2.7 and the homogeneity and additivity properties of cumulants, we have

[TABLE]

The result now follows from 2.16 after converting moments to cumulants. ∎

Theorem 6.3**.**

Let $\underline{\lambda}^{(1)},\underline{\lambda}^{(2)},\ldots$ be a sequence of block diagonal skew partitions. Then the sequence $(\mathcal{X}_{\underline{\lambda}^{(N)}}[\operatorname{maj}]^{*})$ converges in distribution if and only if

1. (i)

$\operatorname{aft}(\underline{\lambda}^{(N)})\to\infty$; or 2. (ii)

$|\underline{\lambda}^{(N)}|\to\infty$* and $\operatorname{aft}(\underline{\lambda}^{(N)})\to M<\infty$; or* 3. (iii)

the distribution of $\mathcal{X}_{\underline{\lambda}^{(N)}}[\operatorname{maj}]$ is eventually constant.

The limit law is $\mathcal{N}$ in case (i), $\mathcal{IH}_{M}^{*}$ in case (ii), and discrete in case (iii).

Proof.

The backwards direction follows from 5.8 and 6.2. In the forwards direction, let $\underline{\lambda}^{(N)}$ be such a sequence where $(\mathcal{X}_{\underline{\lambda}^{(N)}}[\operatorname{maj}]^{*})$ converges in distribution. If $|\underline{\lambda}^{(N)}|$ is bounded, then there are only finitely many distinct $\underline{\lambda}^{(N)}$, forcing case (iii). If $|\underline{\lambda}^{(N)}|$ is unbounded, then we have subsequences satisfying either (i) or (ii) since the sequence converges in distribution, which from 5.8 and 6.2 gives convergence in distribution to $\mathcal{N}$ or $\mathcal{IH}_{M}^{*}$, which are continuous, distinct distributions. The result follows. ∎

From the Central Limit Theorem, we know the Irwin–Hall distribution $\mathcal{IH}_{M}^{*}$ for $M$ large closely resembles a normal distribution, so it will be quite rare for a plot of the coefficients of $\operatorname{SYT}(\lambda)^{\operatorname{maj}}(q)$ to look anything but normal. Since Irwin–Hall distributions are finitely supported, the difference between the two distributions is mainly in the tails. We note that even for $M=5$, there is a close resemblance. See the plot in Figure 7.

7. Discrete distributions for $\operatorname{maj}$ on $\operatorname{SYT}(\lambda)$

We conclude by analyzing more carefully the discrete case of the limit law classification for $\operatorname{maj}$ on $\operatorname{SYT}(\lambda)$, 1.7. The result is 7.1, which lists several families of pairs of shapes $\lambda$ and $\nu$ of differing sizes for which we nonetheless have $\#\operatorname{SYT}(\lambda)=\#\operatorname{SYT}(\nu)$.

A well-known corollary of (1) is that for partitions $\lambda$ and $\nu$ of $n$, $\operatorname{maj}$ is equidistributed on $\operatorname{SYT}(\lambda)$ and $\operatorname{SYT}(\nu)$ if and only if $b(\lambda)=b(\nu)$ and the multisets $\{h_{c}:c\in\lambda\}$ and $\{h_{d}:d\in\nu\}$ are equal. These hook multisets do not entirely characterize the partition—see [HC78]. The following theorem gives a similar result even if we consider the corresponding standardized random variables $\mathcal{X}_{\lambda}[\operatorname{maj}]$ and $\mathcal{X}_{\nu}[\operatorname{maj}]$.

Theorem 7.1**.**

Let $\lambda$ and $\nu$ be partitions. Then $\mathcal{X}_{\lambda}[\operatorname{maj}]^{*}$ and $\mathcal{X}_{\nu}[\operatorname{maj}]^{*}$ have the same distribution if and only if

1. (i)

the multisets of hook lengths $\{h_{c}:c\in\lambda\}$ and $\{h_{d}:d\in\nu\}$ are equal; or 2. (ii)

the multisets $\{h_{c}:c\in\lambda\}$ and $\{|\lambda|\}\sqcup\{h_{d}:d\in\nu\}$ are equal; or 3. (iii)

$\lambda$* and $\nu$ are each either a single row or column; or* 4. (iv)

$\lambda,\nu\in\{(2,1),(2,2)\}$.

Moreover, case (ii) occurs if and only if, up to transposing,

1. (a)

$\lambda=(n)$* and $\nu=(n-1)$ for $n\geq 2$; or* 2. (b)

$\lambda=(r+1,1^{2r+2})$* and $\nu=(2^{r+1},1^{r})$ for $r\geq 1$; or* 3. (c)

$\lambda=(s,1^{s+2})$* and $\nu=(s,s,1)$ for $s\geq 4$; or* 4. (d)

$\lambda=(3,1^{5})$* and $\nu=(3^{2},1)$, or $\lambda=(4,1^{6})$ and $\nu=(3^{3},1)$.*

Proof.

Let $n\coloneqq|\lambda|$ and $m\coloneqq|\nu|$. Let $f^{\lambda}(q)=\frac{[n]_{q}!}{\prod_{c\in\lambda}[h_{c}]}$, which is a polynomial by (1) with constant coefficient $1$. Let $f^{\lambda}=f^{\lambda}(1)=|\operatorname{SYT}(\lambda)|$. Let $f^{\nu}$ and $f^{\nu}(q)$ be defined similarly.

In the backwards direction, if (i) holds, then $n=m$, both variances agree by 1.5, and $f^{\lambda}(q)=f^{\nu}(q)$, so $\mathcal{X}_{\lambda}[\operatorname{maj}]^{*}$ and $\mathcal{X}_{\nu}[\operatorname{maj}]^{*}$ have the same distribution. Similarly if (ii) holds $f^{\lambda}(q)=f^{\nu}(q)$, both variances agree, and $\mathcal{X}_{\lambda}[\operatorname{maj}]^{*}$ and $\mathcal{X}_{\nu}[\operatorname{maj}]^{*}$ have the same distribution again. Condition (iii) holds if and only if the distributions are concentrated at a single point. For (iv), we have $f^{(2,1)}(q)=1+q$ and $f^{(2,2)}(q)=1+q^{2}$, so the normalized distributions are clearly equal.

In the forwards direction, suppose $\mathcal{X}_{\lambda}[\operatorname{maj}]^{*}$ and $\mathcal{X}_{\nu}[\operatorname{maj}]^{*}$ have the same distribution. Since $f^{\lambda}(q)$ has constant coefficient $1$, $\mathcal{X}_{\lambda}[\operatorname{maj}]$ is concentrated at a single point if and only if $f^{\lambda}=1$, which occurs if and only if $\lambda$ is a single row or column which is covered by case (iii). It is easy to see that $f^{\lambda}=2$ if and only if $\lambda\in\{(2,1),(2,2)\}$ which is covered by case (iv).

Assume $f^{\lambda},f^{\nu}>2$. By [BKS18, Thm. 1.1], it follows that $f^{\lambda}(q)$ and $f^{\nu}(q)$ each have two adjacent non-zero coefficients. Since $f^{\lambda}(q)$ and $f^{\nu}(q)$ each have constant term 1 and two adjacent non-zero coefficients, then it follows from the assumption $\mathcal{X}_{\lambda}[\operatorname{maj}]^{*}$ and $\mathcal{X}_{\nu}[\operatorname{maj}]^{*}$ have the same distribution that

[TABLE]

Without loss of generality, we can assume $n\geq m$. If $n=m$, we have $\prod_{c\in\lambda}[h_{c}]_{q}=\prod_{d\in\nu}[h_{d}]_{q}$, from which it follows that the multisets of hook lengths are equal by considering multiplicities of zeros at all primitive roots of unity as in case (i).

From here on, assume $n>m$. The multiplicity of a zero of a primitive $n$th root of unity in (32) is [math] on the right, so from the left $\lambda$ must have a hook of length $n$ so it itself is a hook shape partition. Since $\lambda$ is not a single row or column by the assumption $f^{\lambda}>2$, we know $\lambda$ does not have a cell with hook length $n-1$. Consequently, the multiplicity of a zero at a primitive $(n-1)$th root of unity in (32) is $1$ on the left, forcing $m=n-1$ on the right. Thus (32) becomes

[TABLE]

and as before the multiset condition (ii) must hold. This completes the proof of the first statement in the theorem.

For the second statement, suppose (ii) holds, so the multisets $\{h_{c}:c\in\lambda\}$ and $\{|\lambda|\}\sqcup\{h_{d}:d\in\nu\}$ are equal. Then, $m=n-1$ and $\lambda$ has a cell with hook length $|\lambda|$, so $\lambda$ is a hook shape partition $(n-k,1^{k})$ for some $0\leq k\leq n$, and

[TABLE]

By transposing if necessary, we may assume $k\geq m-k$ is the maximum hook length in $\nu$. If $\lambda$ has one cell with hook length $1$, then (a) holds. Otherwise, both $\lambda$ and $\nu$ have precisely two cells with hook length $1$, so $\nu$ is the union of two rectangles and not itself a rectangle. If $\nu$ were a hook, then it would have a hook length equal to $m$ which would imply $\lambda$ has a cell of hook length $m=n-1$ contradicting the fact that $\lambda$ has two outer corners. Thus $\nu$ is not itself a hook.

Transposing $\nu$ if necessary, we can assume its first two rows are equal, say $\nu_{1}=\nu_{2}=s$. If $\nu_{1}^{\prime}=\nu_{2}^{\prime}$, one may check that the cell furthest from the origin in the intersection of the two rectangles forming $\nu$ would be the only cell of its hook length, and that moreover its two neighbors in the intersection would each have one larger hook length, contrary to (34). It follows that $\nu=(s^{t},1^{r})$ where $r\geq 1$, $s\geq 2$, and $t\geq 2$. We now have several cases.

• •

If $s=2$, the hook lengths of $\nu$ are $\{1,\ldots,r,r+2,\ldots,r+t+1,1,\ldots,t\}$. The “gap” between $r$ and $r+2$ together with (34) forces $t=r+1$, so that $\nu=(2^{r+1},1^{r})$ with $r\geq 1$. Here $k=r+t+1=2r+2$, resulting in case (b).

• •

If $s\geq 3$, the last two columns of $\nu$ already contain two cells with hook length $2$. If $r>1$, the first column would also have a cell with hook length $2$, contradicting (34), so $r=1$.

• –

If $s=3$, the hook lengths of $\nu$ are $\{1,\ldots,t,2,\ldots,t+1,1,4,5,\ldots,t+3\}$. Because of the “gap” between $t+1$ and $t+3$, this is of the form in (34) if and only if $t=2$ or $t=3$, resulting in case (d).

• –

Suppose $s>3$. If $t\geq 3$, then the final three columns of $\nu$ contain three cells with hook length $3$, contradicting (34), so $t=2$. The hook lengths of $\nu$ are then $\{1,1,2,\ldots,s-1,s+1,2,3,\ldots,s,s+2\}$, which is already of the form (34), resulting in case (c).

The reverse implications from (a)-(d) to (ii) were verified in the course of the above argument. ∎

Remark 7.2**.**

The proof of 7.1 applies more generally to arbitrary scaling factors and translations of the distributions of $\mathcal{X}_{\lambda}[\operatorname{maj}]$ and $X_{\nu}[\operatorname{maj}]$, and not just those coming from means and variances.

8. Future work

We conjecture that almost all of the polynomials of the form $\operatorname{SYT}(\lambda)^{\operatorname{maj}}(q)$ are unimodal and log-concave. In this section, we discuss the deviations of each of these properties. In the rare cases where unimodality or log-concavity fails, it only seems to happen at the very beginning and end of the sequence of coefficients or near the middle coefficient.

Recall that a polynomial $P(q)=\sum_{i=0}^{n}c_{i}q^{i}$ is unimodal if

[TABLE]

for some $j$, and $P(q)$ is log-concave if $c_{i}^{2}\geq c_{i-1}c_{i+1}$ for all integers $0<i<n$. A polynomial with nonnegative coefficients which is log-concave and has no internal zero coefficients is necessarily unimodal [Sta89]. By [BKS18], we know exactly where internal zeros occur so log-concavity would imply unimodality in these cases.

We say $P(q)$ is nearly unimodal if instead

[TABLE]

for some $j$ and $P(q)$ has symmetric coefficients. Also, a symmetric polynomial $P(q)$ is nearly log-concave if $c_{i}^{2}\geq c_{i-1}c_{i+1}$ for all $1<i<\lfloor\frac{n}{2}\rfloor$.

Conjecture 8.1**.**

The polynomial $\operatorname{SYT}(\lambda)^{\operatorname{maj}}(q)$ is unimodal if $\lambda$ has at least $4$ corners. If $\lambda$ has $3$ corners or fewer, then $\operatorname{SYT}(\lambda)^{\operatorname{maj}}(q)$ is unimodal except when $\lambda$ or $\lambda^{\prime}$ is among the following partitions:

1. (1)

Any partition of rectangle shape that has more than one row and column. 2. (2)

Any partition of the form $(k,2)$ with $k\geq 4$ and $k$ even. 3. (3)

Any partition of the form $(k,4)$ with $k\geq 6$ and $k$ even. 4. (4)

Any partition of the form $(k,2,1,1)$ with $k\geq 2$ and $k$ even. 5. (5)

Any partition of the form $(k,2,2)$ with $k\geq 6$. 6. (6)

Any partition on the list of 40 special exceptions:

[TABLE]

8.1 was checked for all partitions up to size $n=50$. Each of the families $(k,2)$, $(k,4)$, or $(k,2,1,1)$ have a relatively simple set of hook lengths so explicit formulas can be derived for the coefficients of $\operatorname{SYT}(\lambda)^{\operatorname{maj}}(q)$. We have found explicit proofs of near unimodality for each of these cases. They are related to known integer sequences [OEI17, A266755] and [OEI17, A008642] with nice generating functions. Furthermore, these families are all nearly unimodal as well as 20 of the special exceptions. All rectangles with at least 2 rows and columns are nearly unimodal for $30\leq n\leq 100$. The only deviation occurs at $i=1$ up to symmetry. We conjecture this trend also continues, hence the claim that all coefficients in $\operatorname{SYT}(\lambda)^{\operatorname{maj}}(q)$ are close to unimodal. The family $(k,2,2)$ is a bit further from being unimodal. The proof of the following result is omitted, but follows directly from a careful analysis of the hook lengths.

Proposition 8.2**.**

If $\lambda=(k,2,2)$ for any positive integer $k\geq 3$, then the maximal coefficient of $f^{\lambda}(q)$, say $c_{j}$, satisfies the equation $c_{j}=c_{j+1}+\mathrm{floor}(k/6)+I(4=(k\ \mathrm{mod}\ 6))$ and $c_{0}\leq c_{1}\leq\cdots\leq c_{j}$ and $j+1$ is the median nonzero coefficient. Here $I$ is an indicator function which is 1 if true and 0 if false.

Conjecture 8.3**.**

The polynomials $\operatorname{SYT}(\lambda)^{\operatorname{maj}}(q)$ are “nearly unimodal but not unimodal” for partitions $\lambda$ or $\lambda^{\prime}$ in the following cases:

1. (1)

Any partition of rectangle shape that has more than one row and column with more than 30 cells. 2. (2)

Any partition of the form $(k,2)$ with $k\geq 4$ and $k$ even. 3. (3)

Any partition of the form $(k,4)$ with $k\geq 6$ and $k$ even. 4. (4)

Any partition of the form $(k,2,1,1)$ with $k\geq 2$ and $k$ even.

8.3 was checked for all paritions of size up to $n=100$. It also holds for the following 14 special exceptions:

[TABLE]

[TABLE]

Log-concavity for the polynomials $\operatorname{SYT}_{\lambda}^{\operatorname{maj}}(q)$ appears to be harder to characterize. There are examples of partitions with even 5 corners which are not log-concave. For example $f^{\lambda}(q)$ for $\lambda=(9,9,7,7,5,5,3,3,2)$ is nearly log-concave but $c_{1}^{2}=4^{2}=16<17=c_{0}c_{2}$. The only deviation occurs at $i=1$ up to symmetry. Thus, we summarize what we have observed in the following conjecture.

Conjecture 8.4**.**

The polynomials $\operatorname{SYT}(\lambda)^{\operatorname{maj}}(q)$ are almost always log-concave for partitions $\lambda\vdash n$ for large $n$.

This conjecture is based on the fact that the normal distribution is log-concave and the following evidence. The approximate probability that a uniformly chosen partition of $n$ has the log-concave property $\mathbb{P}(\mathrm{LC})$ and the corresponding probability for the nearly log-concave property $\mathbb{P}(\mathrm{NLC})$ is given in the following table:

By 1.3 and the conjectured claim that the coefficients of $\operatorname{SYT}(\lambda)^{\operatorname{maj}}(q)$ are unimodal or almost unimodal for large $\lambda$, one might hope that we could approximate the number of $T\in\operatorname{SYT}(\lambda)$ with $\operatorname{maj}(T)=k$ by the density function $f(k;\kappa_{1}^{\lambda},\kappa_{2}^{\lambda})$ for the normal distribution with mean $\kappa_{1}^{\lambda}$ and variance $\kappa_{2}^{\lambda}$. We have the following conjectured bounds on such an approximation.

Conjecture 8.5**.**

Let $\lambda\vdash n$ be any partition. Uniformly for all $n$, for all integers $k$, we have

[TABLE]

The conjecture has been verified for $25<n\leq 50$ and $\operatorname{aft}(\lambda)>1$ with a constant of $1/9$, which is tight up to reasonable limits on computation in the sense that if it is changed to $1/10$ with the other constraints the same, it fails at $n=50$.

Conjecture 8.6**.**

Asymptotic normality for general skew shapes and not just block diagonal skew shapes holds if and only if $\operatorname{aft}(\lambda/\nu^{(N)})\to\infty$ as $N\to\infty$, generalizing the result in 5.8.

The argument in Section 5 proves that the “formal cumulants” associated with

[TABLE]

exhibit asymptotic normality when $\operatorname{aft}(\lambda/\mu)\to\infty$. However, this is only the first term in the general $q$-Naruse formula for $\operatorname{SYT}(\lambda/\mu)^{\operatorname{maj}}(q)$. One approach to 8.6 would be to show the remaining terms are “appropriately negligible.”

Acknowledgments

We would like to thank Krzysztof Burdzy, Rodney Canfield, Persi Diaconis, Sergey Fomin, Pavel Galashin, Svante Janson, William McGovern, Andrew Ohana, Greta Panova, Mihael Perman, Martin Raič, Richard Stanley, Sheila Sundaram, Vasu Tewari, Lauren Williams, and Alex Woo for helpful discussions related to this work.