Explicit formulae for Kerov polynomials

DOI 10.1007/s10801-010-0239-3

Explicit formulae for Kerov polynomials

P. Petrullo·D. Senato

Received: 4 September 2009 / Accepted: 27 May 2010 / Published online: 16 June 2010

© Springer Science+Business Media, LLC 2010

Abstract We prove two formulae expressing the Kerov polynomialΣkas a weighted sum over the set of noncrossing partitions of the set{1, . . . , k+1}. We also give a combinatorial description of a family of symmetric functions specializing in the coefficients ofΣk.

Keywords Kerov polynomials·Noncrossing partitions·Symmetric group· Normalized characters·Symmetric functions

1 Introduction

Thenth free cumulantR_ncan be thought of as a functionR_n:λ∈Y→R_n(λ)∈Z, defined on the set of all Young diagramsY, which we identify with the corresponding integer partitions, and taking integer values. Indeed, after a suitable representation of a Young diagramλas a function in the planeR²[4], it is possible to determine the sequences of integersx0, . . . , xmandy1, . . . , ym, consisting of thex-coordinates of the local minima and maxima ofλ, respectively. In this way, if we set

Hλ(z)= _m

i=0(z−x_i) _m

i=1(z−yi),

thenR_n(λ)is the coefficient ofzⁿ⁻¹in the formal Laurent series expansion ofKλ(z) such thatKλ(Hλ(z))=Hλ(Kλ(z))=z. It can be shown thatR1(λ)=0 for allλ.

P. Petrullo·D. Senato (

Dipartimento di Matematica e Informatica, Università degli Studi della Basilicata, via dell’Ateneo Lucano 10, 85100 Potenza, Italy

e-mail:[email protected] P. Petrullo

e-mail:[email protected]

So, thekth Kerov polynomial is a polynomialΣ_k(R₂, . . . , R_k+1)which satisfies the following identity,

Σ_k

R2(λ), . . . , R_k+1(λ)

=(n)_kχ^λ(k,1^n−k) χ^λ(1ⁿ) ,

whereχ^λ(k,1^n−k)denotes the value of the irreducible character of the symmetric groupSnindexed by the partitionλonk-cycles. Two remarkable properties ofΣ_k have to be stressed. First, it is a “universal polynomial”, that is, it depends neither on λ nor onn. Second, its coefficients are nonnegative integers. A combinatorial proof of the positivity ofΣ_k is quite recent and is due to Féray [8]. An explicit combinatorial description of such coefficients is due to Doł˛ega, Féray and ´Sniady [6].

Until now, several results on Kerov polynomials have been proved and conjectured;

see, for instance, [5,10,12,17] and [9] for a more detailed treatment.

Originally, free cumulants arise in the noncommutative context of free probability theory [15]. To the best of our knowledge, their earliest applications in the asymptotic character theory of the symmetric group are due to Biane; see, for instance, [3]. In 1994, Speicher [16] showed that the formulae connecting moments and free cumu- lants of a noncommutative random variableXobey the Möbius inversion on the lat- tice of noncrossing partitions of a finite set. This result highlights the strong analogy between free cumulants and classical cumulants, which are related to the moments of a random variableX, defined on a classical probability space, via the Möbius inver- sion on the lattice of all partitions of a finite set. More recently, Di Nardo, Petrullo and Senato [7] have shown how the classical umbral calculus provides an alternative setting for the cumulant families which passes through a generalization of the Abel polynomials.

In 1997, it was again Biane [2] who showed that the lattice NCn of noncross- ing partitions of{1, . . . , n}can be embedded into the Cayley graph of the symmet- ric groupS_n. Thus it seems reasonable that a not too complicated expression of the Kerov polynomials involving noncrossing partitions, or the Cayley graph ofSn, should exist. In particular, such a formula, conjectured in [4], appeared with a rather implicit description in [6,8].

In this paper, we state two explicit formulae relating Σ_k and the set NCk+1 of noncrossing partitions of{1, . . . , k+1}. More precisely, if NC^irr_k+1denotes the subset of NCk+1of partitions having 1 andk+1 in the same block, then a new partial order

≤^irron NC^irr_k+1is considered, thanks to which we have

Σ_k=

τ∈NC^irr_k+1 τ≤^irrπ

(−1)^π−1W_τ(π )

R

τ.

Here,_π is the number of blocks ofπ,W_τ(π )is a suitable weight depending onτ andπ, and R

τ =

BR_|B|, withB ranging over the blocks of τ having at least 2 elements. The special structure of the weightWτ(π )allows us to give a combinato- rial description of the symmetric functionsg_μ(x₁, . . . , x_k)’s, that evaluated atx_i=i return the coefficient of

i≥2R_i^mi inΣ_k, for every integer partitionμof sizek+1 havingm_iparts equal toi.

A second formula expressing Σ_k as a weighted sum over the whole NCk+1 is proved by means of the notion of irreducible components of a noncrossing partition.

In particular, ifd_τis the number of irreducible components ofτ, then we have

Σk=

τ∈NCk+1

(−1)^dτ−1Vτ R

τ,

whereV_τ is a suitable weight depending onτ.

2 Kerov polynomials

Letnbe a positive integer and letλ=(λ₁, . . . , λ_l)be an integer partition of sizen, that is, 1≤λ₁≤ · · · ≤λ_l and

iλ_i=n. Denote byYnthe set of all Young diagrams of sizen, and setY=

nYn. From now on, an integer partition and its Young dia- gram will be denoted by the same symbolλ. Moreover, as is usual we writeλnif λis an integer partition of sizen.

After a suitable representation of a Young diagram λ as a function in the plane R² [4], it is possible to determine the sequences of integers x₀, . . . , x_m and y₁, . . . , y_m, consisting of thex-coordinates of its local minima and maxima, respec- tively. Then, by expanding the rational function

Hλ(z)= _m

i=0(z−x_i) _m

i=1(z−y_i) as a formal power series inz⁻¹one has Hλ(z)=z⁻¹+

n≥1Mn(λ) z⁻⁽ⁿ⁺¹⁾. The integer M_n(λ) is said to be the nth moment of λ. Now, define Kλ(z)=H⁻_λ¹(z), that isKλ(Hλ(z))=Hλ(Kλ(z))=z, and consider its expansion as a formal Laurent series,Kλ(z)=z⁻¹+

n≥1R_n(λ) zⁿ⁻¹.Then, the integerR_n(λ)is named thenth free cumulant ofλ. It is not difficult to see thatM₁(λ)=R₁(λ)=0 for allλ. By setting Mλ(z)=z⁻¹Hλ(z⁻¹)and Rλ(z)=zKλ(z), we obtain two formal power series inz,Mλ(z)=1+

n≥1M_n(λ) zⁿandRλ(z)=1+

n≥1R_n(λ) zⁿ, such that Mλ(z)=Rλ

zMλ(z)

. (2.1)

Letμnand denote byχ^λ(μ)the value of the irreducible character ofS_n in- dexed byλon the permutations of typeμ. So that, ifμ=(k,1^n−k), that is,μ₁=k andμ₂= · · · =μ_n−k+1=1, then the valueχ^λ(k,1^n−k)of the normalized character indexed byλon thek-cycles ofS_nis given by

χ^λ

k,1^n−k

=(n)k

χ^λ(k,1^n−k) χ^λ(1ⁿ) ,

where(n)_k=n(n−1)· · ·(n−k+1). Thekth Kerov polynomial is a polynomialΣ_k inkcommuting variables which satisfies the following interesting identity,

Σ_k

R₂(λ), . . . , R_k+1(λ)

=χ^λ k,1^n−k

.

If we think ofR_n(λ)as the image of a mapR_n:λ∈Y→R_n(λ)∈Z, then also Kerov polynomials become maps Σ_k =Σ_k(R₂, . . . , R_k+1), which are polynomials in the Rn’s, such thatΣk(λ)=χ^λ(k,1^n−k). Since the coefficients ofΣk depend neither on λnor on n, but only on k, such polynomials are said to be “universal”. A second remarkable property of Kerov polynomials is that all their coefficients are positive integers. This fact is known as the “Kerov conjecture” [11]. The first proof of the Kerov conjecture was given with combinatorial methods by Féray [8]. By using rather different techniques, the same author with Doł˛ega and ´Sniady [6] have then stated an explicit combinatorial interpretation of such coefficients. The following formula for Σ_kcan be found in Stanley [17]. It is also stated in Biane [4], where the author quotes it as a private communication with A. Okounkov.

Theorem 2.1 LetR(z)=1+

n≥2R_nzⁿ. If F(z)= _R^z_(z) and G(z)=_F−1^z(z⁻¹), then we have

Σ_k= −1 k

z⁻¹ _∞

k−1 j=0

G(z−j ). (2.2)

More precisely, if[zⁿ]f (z) denotes the coefficient ofzⁿin the formal power se- ries f (z), then [z⁻¹]∞f (z)= [z]f (z⁻¹). This way, Identity (2.2) states that Σk

is obtained by expressing the right-hand side in terms of the free cumulants R_n’s.

Moreover, thanks to the invariance of the residue under translation of the variable, if M(z)=1+

n≥1Mnzⁿ, then by virtue of (2.1) we havezG(z)⁻¹=M(z⁻¹),and (2.2) can be rewritten in the following equivalent form,

Σk= −1 k

z^k+1 k

j=1

1−j z

M(₁₋^z_{j z}). (2.3)

For allj=1, . . . , k,we denote byλj the image of the diagramλunder the trans- lation of the plane given byx→x+j. Theith local minimum and maximum of λj arex_i+j andy_i+j,respectively, so that

Hλj(z)= _m

i=0z−(x_i+j ) _m

i=1z−(y_i+j ) and Mλj(z)= 1 1−j zMλ

z 1−j z

.

In this way, we may rewrite (2.3) as follows:

Σ_k= −1 k

z^k+1 k

j=1

1

Mλj(z). (2.4)

Denote byRλj(z)the formal power series such thatMλj(z)=Rλj(zMλj(z)).

It is immediate to verify that

Rλj(z)=j z+Rλ(z). (2.5)

3 Irreducible noncrossing partitions

A partition of a finite setS is an unordered sequenceτ = {A1, . . . , Al}of pairwise disjoint nonempty subsets ofS, called the blocks ofτ, such that

iA_i=S. We say thatτ refinesπ, writtenτ ≤π, if and only if each block ofπ is a union of blocks ofτ. IfT ⊂S, then the restriction ofτ toT is the partitionτ_|T obtained by removing fromτ all the elements inS\T.

Denote by [n]the set {1, . . . , n}. A partition τ = {A₁, . . . , A_l}of [n]is said to be noncrossing if and only ifa, c∈Ai andb, d∈Aj impliesi=j, whenever 1≤ a < b < c < d≤n. The set of all the noncrossing partitions of[n]is usually denoted by NCn. Its cardinality equals thenth Catalan number Cn= _n+¹_12n

n

. The reader interested in this subject may refer to [1] and references therein for further details.

Now, if we setR_τ=R_|A1|· · ·R_|Al|then we can state the following beautiful formula, due to Speicher [16], expressing the momentsM_n’s in terms of their respective free cumulantsR_n’s:

M_n=

τ∈NC_n

R_τ.

Following Lehner [13], if 1 andnlie in the same block of a partitionτof[n], then we say thatτ is irreducible. Moreover, we denote by NC^irr_n the set of all the irreducible noncrossing partitions of[n]. Note that a partition of NC^irr_n+1is obtained from a par- tition of NCnsimply by insertingn+1 in the block containing 1. By taking the sum of the monomialsR_τ’s,τ ranging in NC^irr_n instead of NCn, one defines a quantityB_n known as a boolean cumulant (see [13])

B_n=

τ∈NC^irr_n

R_τ. (3.1)

In particular, ifB(z)=

n≥1B_nzⁿ, then we have M(z)= 1

1−B(z). (3.2)

Ifμ=(μ1, . . . , μl) is an integer partition of size n, set Rμ=Rμ₁· · ·Rμ_l and define NC^irr_μ to be the subset of NC^irr_n consisting of all the irreducible noncrossing partitions of typeμ. From (3.1) we have

B_n=

μn

NC^irr_μR_μ. (3.3)

Moreover, thanks to the Lagrange inversion formula, we recover B_n= 1

n−1

zⁿ R(z)ⁿ⁻¹=

μn

(n−2)_μ−1

m(μ)! R_μ, (3.4)

wherem(μ)! =m₁(μ)! · · ·m_n(μ)!, andm_i(μ)is the number of occurrences ofias a part ofμ. By comparing (3.3) and (3.4), we deduce

NC^irr_μ=(n−2)_μ−1

m(μ)! . (3.5)

The notion of noncrossing partition can be given for any totally ordered setS. In particular, NC^irr_S will denote the set of all the noncrossing partitions ofS, such that the minimum and the maximum ofSlie in the same block. Let us introduce a partial order on NC^irr_S.

Definition 3.1 (Irreducible refinement) Letτ, π ∈NC^irr_S. We say thatτ refinesπ in an irreducible way, and writeτ ≤^irrπ, if and only ifτ≤πand the restrictionπ_|A, of πto each blockAofτ, is in NC^irr_A. In particular, we say thatπcoversτ if and only if τ≤^irrπandπ is obtained by joining two blocks ofτ.

For instance, letτ = {{1,5},{2,3},{4}}, π= {{1,2,3,5},{4}}and σ = {{1,5}, {2,3,4}}. Thenτ, π, σ ∈NC^irr₅ andτ refines bothπandσ. However,τ≤^irrπ and in particularπ coversτ, while it is not true thatτ≤^irrσ, sinceτ_|{2,3,4}= {{2,3},{4}}is not irreducible.

The singletons of the noncrossing partitions will play a special role. For allτ ∈ NCn, we denote byU (τ )the subset of[n]consisting of all the integersisuch that {i}is a block ofτ, while τ will be the partition obtained from τ by removing its singletons. Whenτ, π ∈NC^irr_n andτ ≤^irrπ, thenπ_τ is the restriction ofπ toU (τ ).

Note thatπ_τ∈NCU (τ ).

We define a tree-representation for the partitions of NC^irr_n in the following way.

Assumeτ = {A₁, . . . , A_l} ∈NC^irr_n and minA_i<minA_i+1. Construct a labeled rooted treet_τ by the following steps:

• ChooseA₁as the root oft_τ;

• If 2≤i < j≤lthen draw an edge betweenA_iandA_jif and only ifiis the biggest integer such that minAi<minAj≤maxAj<maxAi;

• Label each edge{A_i, A_j}with minA_j.

For example, ifτ = {{1,2,7,12},{3,5,6},{4},{8,9},{10,11}}thentτ is the fol- lowing tree,

{1,2,7,12} 3

8

10

{3,5,6}

4

{8,9} {10,11}

{4}

Now, letE(τ )be the set of labels oft_τ, and choosej∈E(τ ). We denote byt_τ,j the tree obtained fromt_τ by deleting the edge labeled byj and joining its nodes (i.e.,

joining the blocks). In the following, we will say thatt_τ,j is the tree obtained fromt_τ by removingj. Hence,t_τ,3is given by

{1,2,3,5,6,7,12} 4

8

10

{4} {8,9} {10,11}

Now,tτ,j is the tree-representation of an irreducible noncrossing partition, here de- noted byτ_{j}, whose blocks are the nodes oftτ,j. By construction, we haveτ≤^irrτ_{j}

andE(τ_{j})=E(τ )\ {j}. More generally, given a subsetS⊆E(τ ), we denote by τ_S the only partition whose treet_τS is obtained fromt_τ by successively removing all labels inS. We note thatτ_∅=τ and thatτ_S depends only on the setSand not on the order in which labels are chosen. The following proposition is easy to prove.

Proposition 3.1 Letτ, π∈NC^irr_n. Then, we haveτ ≤^irrπ if and only ifπ=τS for someS⊆E(τ ). In particular, if(τ )is the number of blocks ofτ, then we have

σ|τ≤^irrσ=2^{E(τ )}=2^{(τ )−1}, where 2^{E(τ )}is the powerset ofE(τ ), and

{σ|σ coversτ}=E(τ )=(τ )−1.

4 Kerov polynomial formulae

By means of the results of previous sections, we are able to give new formulae for the Kerov polynomialsΣ_k. In particular, the first formula is related to the partial order

≤^irr on the irreducible noncrossing partitions of the set [k+1], instead the second formula expressesΣ_kas a weighted sum over NCk+1.

4.1 Kerov polynomials and irreducible refinement

Letτ, π∈NC^irr_k+1withτ≤^irrπ, and letπ_τ = {A1, . . . , A_l}. Moreover, setA_i= ∅for i > land defineW_τ(π )to be such that

W_τ(π )= 1 k!

w∈Sk

w(1)^|A1|· · ·w(k)^|Ak|.

Theorem 4.1 We have

Σk=

τ∈NC^irr_k+1 τ≤^irrπ

(−1)^π−1Wτ(π )

R

τ. (4.1)

Proof Let B_n(j )= −[zⁿ](Mλj(z))⁻¹ for all n≥1, and set B₀(j )=1. Since R₁(λ)=0, then from (2.5), (3.1) and (3.2) we deduce

B_n(j )=

π∈NC^irr_n

j^{u(π )}R

π(λ), (4.2)

whereu(π )= |U (π )|.The right-hand side in (2.4) is equal to

−1 k

a1,...,ak≥0 a1+···+ak=k+1

k

j=1

z^aj 1

M_λj(z)=1 k

μk+1 μ≤k

(−1)^μ−1 m(μ)!(k−μ)!

w∈S_k

k

i=1

B_μi w(i)

,

whereμ₁, . . . , μ_μ are the parts ofμandμ_i=0 wheni > _μ. However, by taking into account (3.5), we may rewrite it in the following form:

1 k!

π∈NC^irr_k+1

(−1)^π−1

w∈S_k

k

i=1

B_|Ai| w(i)

,

whereA₁, . . . , A_μ are the blocks ofπandA_i= ∅ifi > _π. Moreover, by means of identity (4.2), we obtain

w∈S_k

k

i=1

B_|Ai| w(i)

=

τ1,...,τk

w∈S_k

w(1)^u(τ1)· · ·w(k)^u(τk)R

τ1

(λ)· · ·R

τ_k(λ),

whereτiranges over all NC^irr_A

i, with NC^irr_∅ = {∅}.

Now, if we set τ =τ1∪ · · · ∪τk, then τ ∈ NC^irr_k+1, τ ≤^irr π and R

τ(λ) = R

τ1

(λ)· · ·R

τk

(λ).Finally,u(τ_i)is the number of singletons inτ_i=τ_|Ai, that is, the cardinality of the setA_i∩U (τ ), which if nonempty is a block ofπ_τ. This completes

the proof.

Remark 4.1 Consider the polynomialΩk(x1, . . . , xk)defined by

Ω_k(x₁, . . . , x_k)= −1 k

z^k+1 k

j=1

1−x_jz M(₁₋^z_x

jz).

Of course,Ωkis symmetric with respect to thexi’s. Moreover, by virtue of (2.3), we obtain Ωk(1,2, . . . , k)=Σk. A formula for Ωk(x1, . . . , xk)is obtained from (4.1) simply by replacingw(j )withxw(j ) inWτ(π ). More precisely, via Proposition 3.1, ifμis an integer partition of sizek+1 then the coefficient ofR

μinΩ_k(x₁. . . , x_k)is g_μ(x₁, . . . , x_k)=

τ∈NCirrμ

S⊆E(τ )

(−1)^{|E(τ )|−|S|}W_τ(S;x₁, . . . , x_k), (4.3)

whereR

μ=

i≥2R_i^mi(μ), and where W_τ(S;x1, . . . , x_k)is obtained by replacing w(j )withx_{w(j )}inW_τ(τ_S). Now, letλ_τ(S)denote the integer partition corresponding

to the type ofπ_τ, withπ=τ_S. Then, it is not difficult to see that k!W_τ(S;x₁, . . . , x_k)=m

λ_τ(S)

! k−

λ_τ(S)

!m_λτ(S)(x₁, . . . , x_k), withmλ_τ(S)(x1, . . . , xk)being the monomial symmetric function indexed by the par- titionλ_τ(S)[14]. This way, the coefficient ofR

μinΩ_k(x1, . . . , x_k)is a symmetric functiong_μ(x1, . . . , x_k)of degreem1(μ). Assume that

g_μ(x₁, . . . , x_k)=

λ

g_μ,λm_λ(x₁, . . . , x_k).

The left-hand side of (4.3) says that, for everyλof sizem1(μ),we have gμ,λ= 1

k!

τ∈NC^irr_μ

S⊆E(τ ) λτ (S)=λ

(−1)^{|E(τ )|−|S|}m(λ)!(k−λ)!,

thus we have provided a combinatorial formula for theg_μ,λ’s.

Finally, we stress that the coefficients in the expressions ofg_μ in terms of the classical basis, and Schur basis, are not positive integers. Indeed, we have

g_(3,1,1,1)=4

5m_(1,1,1)−3

5m_(1,2)+4 5m₍₃₎.

4.2 Kerov polynomials via irreducible components of noncrossing partitions We will state a second formula expressingΣ_k as a weighted sum over the whole NCk+1. To this end, we introduce the notion of an irreducible component of a non- crossing partition.

Givenτ ∈NCn, letj₁be the greatest integer lying in the same block as 1. Set τ₁=τ_|[j1] so that τ₁ is an irreducible noncrossing partition of [j₁]. Now, let j₂ be the greatest integer lying in the same block of j1+1 and set τ2=τ_|[j1+1,j2]. By iterating this process, we determine the sequence of irreducible noncrossing partitions τ1, . . . , τ_d, which we name the irreducible components of τ, such that τ =τ1∪ · · · ∪τ_d. For all τ ∈NCn, we denote byd_τ the number of its irreducible components. Note thatd_τ=1 if and only ifτ is an irreducible noncrossing partition.

Theorem 4.2 We have

Σk=

τ∈NC_k+1

(−1)^dτ−1Vτ R

τ, (4.4)

where

Vτ=1 k

1≤i1<···<id≤k

i1u(τ1)· · ·idu(τd), ifd=d_τ.

Proof LetBλj(z)=1(Mλj(z))⁻¹. From (2.4) we obtain

Σ_k= −1 k

z^k+1 k

j=1

1−B_λj(z)

= k

d=1

(−1)^d−1 k

1≤i1<···<i_d≤k

z^k+1 Bλi₁(z)· · ·Bλi_d(z).

Of course, the complexB_n(j )= [zⁿ]Bλj(z)is then-boolean cumulant ofλjand satisfies (4.2). This way we deduce

z^k+1 Bλi1(z)· · ·Bλid(z)=

a1,...,ad≥1 a1+···+ad=k+1

π₁,...,π_d

i1u(π1)· · ·idu(πd)R

π₁· · ·R

π_d,

whereπi ranges over NC^irr_a

i. Leta0=0 and consider the intervalsAi = [a0+ · · · + a_i−1+1, a0+ · · · +a_i]fori=1, . . . , k+1. Each translationh∈ [1, ai] →h+a0+

· · · +a_i−1∈A_i induces a bijectionπ∈NC^irr_ai→τ ∈NC^irr_Ai. Hence, in the identity above we may replace eachπ_i with the correspondingτ_i obtaining

z^k+1 Bλi1(z)· · ·Bλid(z)=

a1,...,ad≥1 a1+···+ad=k+1

τ1,...,τd

i₁^u(τ1)· · ·i_d^u(τd)R

τ1· · ·R

τd

.

Now, setτ =τ1∪· · ·∪τdso thatτ∈NCk+1,τiis theith irreducible component ofτ,

and (4.4) follows.

Acknowledgement The authors thank the referees for their useful remarks and suggestions improving the technical quality of the paper.

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