ASYMPTOTICS OF THE MAJOR INDEX OF A RANDOM STANDARD TABLEAU PIERRE-LOÏC MÉLIOT AND ASHKAN NIKEGHBALI

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ASYMPTOTICS OF THE MAJOR INDEX

OF A RANDOM STANDARD TABLEAU

PIERRE-LOÏC MÉLIOT AND ASHKAN NIKEGHBALI

Abstract. In this article, we establish the mod-φconvergence of the major index of a uniform

random standard tableau whose shape converges in the Thoma simplex. This implies various

probabilistic estimates, in particular speed of convergence estimates of Berry–Esseen type, and

strong large deviation principles.

Contents

1. Major index of a random standard tableau 2

1.1. Integer partitions and standard tableaux 2

1.2. Major index and Schur functions 3

1.3. Cumulants of the major index 4

1.4. Growing partitions and the Thoma simplex 7

1.5. Main results and outline of the paper 9

2. Observables of integer partitions 13

2.1. Cumulants and the Kerov–Olshanski algebra 13

2.2. Descents and Frobenius moments 16

2.3. Symmetric functions of the contents 18

2.4. Asymptotics of the cumulants 24

3. Asymptotics of the distribution of maj(T)25

3.1. Asymptotics of the log-Laplace transform 25

3.2. Upper bounds on cumulants and Berry–Esseen estimates 27

3.3. Exponential tilting of measures and control of the tilted Fourier transforms 29

References 35

Date: October 25, 2022.

arXiv:2210.13022v1 [math.PR] 24 Oct 2022

2 PIERRE-LOÏC MÉLIOT AND ASHKAN NIKEGHBALI

1. Major index of a random standard tableau

1.1. Integer partitions and standard tableaux. If λ= (λ1≥λ2≥ ··· ≥ λ`(λ))is a non-

increasing sequence of positive integers, we recall that it is called integer partition of the integer

n=|λ|=P`(λ)

i=1 λi, and that it is represented by a Young diagram, which is the array of n

boxes with λ1boxes on the ﬁrst row, λ2boxes on the second row, and so on. For instance, the

integer partition λ= (4,2,2,1) has size |λ|= 9, and it is represented by the Young diagram:

We denote Y(n)the set of integer partitions with size n, and Y=Fn∈NY(n)the set of all

integer partitions. Given λ∈Y(n), a standard tableau with shape λis a numbering of the cells

of the Young diagram of λby the integers in [[1, n]] which is bijective and strictly increasing

along the rows and columns. For instance,

4 7

3 5

1269

is a standard tableau with shape (4,2,2,1). The set of standard tableaux with shape λwill

be denoted ST(λ), and the cardinality of this set is given by the Frame–Robinson–Thrall

hook length formula [FRT54]. Given a cell @of a Young diagram λ, its hook length h(@)is the

number of cells of the hook which is included in the Young diagram λand which contains @, the

cells above @and the cells on the right of @. For instance, the integer partition λ= (4,2,2,1)

yields the following hook lengths:

3 1

4 2

7521

Then, for any integer partition λ∈Y(n),

|ST(λ)|=n!

Q@∈λh(@);

see for instance [Mac95, Section I.4, Example 3]. In this article, we shall be interested in the

distribution of a certain statistics X: ST(λ)→N, the set ST(λ)being endowed with the

uniform law. Before presenting this statistics, let us associate to the cells of a Young diagram

λanother integer: if the cell @is in i-th row and the j-th column of the Young diagram λ, its

content is c(@) = j−i. For instance, λ= (4,2,2,1) admits the following contents:

2 1

1 0

0123

The contents are involved in numerous combinatorial formulæ in the theory of integer partitions.

For instance, the cardinality of the set SST(λ, m)of semistandard tableaux with shape λand

entries in [[1, m]] is given by the product Q@∈λ

m+c(@)

h(@); see [Sta99, Corollary 7.21.4]. The contents

of the cells of the Young diagrams shall also play an important role in our work.

ASYMPTOTICS OF THE MAJOR INDEX OF A RANDOM STANDARD TABLEAU 3

1.2. Major index and Schur functions. Adescent of a standard tableau of size nis an

integer i∈[[1, n −1]] such that i+ 1 appears in a row strictly above the row containing i. The

major index of a standard tableau Tis the sum of its descents:

maj(T) = X

i∈Desc(T)

For instance, if λ= (4,2,2,1) and Tis the standard tableau given as an example in the previous

paragraph, then the set of descents of Tis {2,3,6,7}, and the major index is 2 +3+6 +7 = 18.

The deﬁnitions of descent and major index for a standard tableau are closely related to the

analogue deﬁnitions for a permutation. Given σ∈S(n), a descent of the permutation σis an

index i∈[[1, n −1]] such that σ(i)> σ(i+ 1). For instance, the descent set of the permutation

σ= 592138647 is {2,3,6,7}. The Robinson–Schensted–Knuth algorithm yields a bijection

between S(n)and the set Fλ∈Y(n)ST(λ)×ST(λ)of pairs of standard tableaux with the same

shape. For instance, 592138647 ∈S(9) corresponds to the pair of standard tableaux

5 8

2 6

1347

;Q=

4 7

3 5

1269

This bijection preserves the set of descents: if RSK(σ) = (P, Q), then Desc(σ) = Desc(Q); see

for instance [Loe11, Theorem 10.117].

The purpose of this article is to study the distribution of maj(T)when Tis taken uniformly at

random in the set ST(λ)of standard tableaux with shape λ, and λ=λ(n)is an integer partition

with size n,ngoing to inﬁnity. Equivalently, this amounts to consider the distribution of maj(σ)

when σis a permutation taken uniformly at random in a RSK class of growing size. The case of

a uniform random permutation in S(n)can essentially be recovered as a particular case of our

results; see the discussion after the statement of Theorems Aand B. The following result due to

Billey–Konvalinka–Swanson [BKS20] ensures that under mild hypotheses and after appropriate

scaling, this distribution of maj(T)with T∼ U(ST(λ)) is asymptotically normal:

Theorem 1 (Billey–Konvalinka–Swanson).We denote λ0the conjugate of an integer partition

λ, which is obtained by symmetrizing the Young diagram with respect to the diagonal. Consider

a sequence of integer partitions (λ(n))n≥1such that

|λ(n)|=n;n−λ(n)

1→+∞;n−λ(n)0

1→+∞.

Then, with T(n)taken uniformly at random in ST(λ(n)),

maj(T(n))−E[maj(T(n))]

pvar(maj(T(n))) *n→+∞N(0,1).

The conditions n−λ(n)

1→+∞and n−λ(n)0

1→+∞are actually necessary: otherwise, the

limiting distribution exists but is not Gaussian, see [BKS20, Theorem 1.3]. We shall see in the

sequel that if |λ|=nand T∼ U(ST(λ)), then

`(λ)

i=1

(i−1)λi≤maj(T)≤n

2−

`(λ)

i=1 λi

2

and the variance of maj(T)is of order O(n3). Thus, in most cases, the values of maj(T)are

of order O(n2), and their ﬂuctuations around the mean are of order O(n3

2)and asymptotically

normal. This raises the question of the large deviations of maj(T)of order O(n2): given y > 0,

what are the asymptotics of

P[maj(T(n))−E[maj(T(n))] ≥yn2] ?

4 PIERRE-LOÏC MÉLIOT AND ASHKAN NIKEGHBALI

We shall explain in this article how to estimate these probabilities if one knows the limit shape

of the partitions λ(n). On the other hand, Theorem 1ensures that

dKol maj(T(n))−E[maj(T(n))]

pvar(maj(T(n))) ,N(0,1)!

= sup

s∈RPmaj(T(n))−E[maj(T(n))] ≤sqvar(maj(T(n)))−Zs

−∞

e−x2

2dx

√2π

goes to 0when ngoes to inﬁnity, but it does not give the speed of convergence, related to the

quality of the Gaussian approximation. One of our main results is a uniform upper bound for

this Kolmogorov distance, which is of order O(n−1

2).

The major index of standard tableaux is related to the Schur functions and to the so-called

q-hook length formula. For any family of variables (x1, . . . , xN)with N≥`(λ), set

sλ(x1, . . . , xN) = det((xi)λj+N−j)1≤i,j≤N

det((xi)N−j)1≤i,j≤N

with by convention λj= 0 if j > `(λ). The specialisation xN+1 = 0 from R[x1, . . . , xN+1]to

R[x1, . . . , xN]stabilises these Schur polynomials:

∀N≥`(λ), sλ(x1, . . . , xN,0) = sλ(x1, . . . , xN).

In the projective limit in the category of graded algebras R[X] = lim

←−N→∞ R[x1, . . . , xN], there

exists a unique element sλ(X)whose projections are the symmetric polynomials deﬁned above.

This is the Schur function sλ, see for instance [Mac95, Section I.3]. An important result in

the theory of these symmetric functions is the Stanley q-hook length formula [Sta99, Corollary

7.21.3]: if |q|<1, then

sλ(1, q, q2, . . . , qn, . . .) = qb(λ)Y

@∈λ1

1−qh(@),

with b(λ) = P`(λ)

i=1 (i−1)λi. Moreover, up to a combinatorial factor, the principal specialisation

sλ(1, q, q2, . . .)is the generating series of the major indices of the standard tableaux with shape

λ:

sλ(1, q, q2, . . .) = 

|λ|

i=1

1−qi

X

T∈ST(λ)

qmaj(T),

see [Sta99, Proposition 7.19.11].

1.3. Cumulants of the major index. The starting point of the argument of [BKS20] is the

following important remark, which is based on earlier works by Chen–Wang–Wang [CWW08]

and by Hwang–Zachavoras [HZ15]. Given T∈ST(λ), set X(T) = maj(T)−b(λ), and consider

the generating series of this statistics:

T∈ST(λ)

qX(T)=Q|λ|

i=1(1 −qi)

Q@∈λ(1 −qh(@)).

We obtain a product of ratios of q-integers [a]q=1−qa

1−q. This leads to simple formulas for the

cumulants of the random variable X(T)with T∼ U(ST(λ)). Given a random variable whose

Laplace transform E[ezX ]is convergent on a disk D(0,R)={z∈C| |z|< R}with R > 0, we

recall that the cumulants of Xare the coeﬃcients κ(r)(X)of the log-Laplace transform:

log E[ezX ] = ∞

r=1

κ(r)(X)

r!zrin a neighborhood of 0.

ASYMPTOTICS OF THE MAJOR INDEX OF A RANDOM STANDARD TABLEAU 5

The cumulants of Xare related to its moments by a Möbius inversion formula with respect to

the lattice of set partitions:

κ(r)(X) = X

π∈P(r)

(−1)`(π)−1(`(π)−1)! 



`(π)

j=1

E[X|πj|]

,

the sum running over the set P(r)of set partitions π1tπ2t ··· t π`(π)of the integer interval

[[1, r]]. This combinatorial formula enables one to deﬁne the cumulants of Xassuming only

that Xhas moments of all order. Now, consider a ﬁnite set Tendowed with the uniform

distribution, and a statistics X:T→N. The generating function E[qX]is given by the ratio

P(q)

P(1) , where

P(q) = ∞

m=0 {T∈T|X(T) = m}qm;P(1) = ∞

m=0 {T∈T|X(T) = m}=|T|.

Lemma 2. Suppose that the polynomial P∈Z[q]is given by a ratio of q-integers:

P(q) = Ql

k=1[bk]q

k=1[ak]q

where {a1, . . . , al}and {b1, . . . , bl}are multisets of non-negative integers. Then, Xtakes its

values in [[0, n]] with n=Pl

k=1(bk−ak), and for any r≥1,

κ(r)(X) = Br

k=1

((bk)r−(ak)r),

where Bris the r-th Bernoulli number. Consequently, the odd cumulants of Xof order larger

than 3vanish.

Proof. Denote cm={T∈T|X(T) = m}, so that P(q) = Pn

m=0 cmqm. By hypothesis, Pis

a polynomial of degree nand is the ratio of two polynomials Ql

k=1(1 −qbk)and Ql

k=1(1 −qak)

with degrees Pl

k=1 bkand Pl

k=1 ak, so nis equal to the diﬀerence of the two sums. Now, the

Bernoulli numbers Brare deﬁned by their generating series

1−e−t=∞

r=0

r!= 1 + t∂

∂t ∞

r=1

r!!,

so the divided Bernoulli numbers Br

rhave for exponential generating series:

∞

r=1

r!= log et−1

t.

It suﬃces now to write, for q= ezwith zclose to 0:

log E[ezX ] =

k=1

log ezbk−1

ezak−1=

k=1

log ezbk−1

zbk−log ezak−1

zak

=∞

r=1

rzr l

k=1

((bk)r−(ak)r)!.

Since t

1−e−t−t

2=tcoth( t

2)is an even function, the odd Bernoulli numbers of order 2r+ 1 ≥3

all vanish. Therefore, κ(2r+1)(X) = 0 for any r≥1.

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ASYMPTOTICSOFTHEMAJORINDEXOFARANDOMSTANDARDTABLEAUPIERRE-LOÏCMÉLIOTANDASHKANNIKEGHBALIAbstract.Inthisarticle,weestablishthemod-convergenceofthemajorindexofauniformrandomstandardtableauwhoseshapeconvergesintheThomasimplex.Thisimpliesvariousprobabilisticestimates,inparticularspeedofconvergenceestimat...

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ASYMPTOTICS OF THE MAJOR INDEX OF A RANDOM STANDARD TABLEAU PIERRE-LOÏC MÉLIOT AND ASHKAN NIKEGHBALI

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