Results and conjectures on the number of standard strong marked tableaux

Results and conjectures on the number of standard strong marked tableaux

Susanna Fishel

and Matjaˇz Konvalinka

1School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona, USA

2Department of Mathematics, University of Ljubljana, Slovenia

Abstract. Many results involving Schur functions have analogues involvingk-Schur functions. Standard strong marked tableaux play a role fork-Schur functions similar to the role standard Young tableaux play for Schur functions.

We discuss results and conjectures toward an analogue of the hook length formula.

Keywords:k-Schur functions, strong marked tableaux, enumeration

1 Introduction

In 1988, Macdonald Macdonald (1995) introduced a new class of polynomials and conjectured that they expand positively in terms of Schur functions. This conjecture, verified in Haglund et al. (2005), has led to an enormous amount of work, including the development of thek-Schur functions. Thek-Schur functions were defined in Lapointe et al. (2003). Lapointe, Lascoux, and Morse conjectured that they form a basis for a certain subspace of the space of symmetric functions and that the Macdonald polynomials indexed by partitions whose first part is not larger thankexpand positively in terms of the k-Schur functions, leading to a refinement of the Macdonald conjecture. Thek-Schur functions have since been found to arise in other contexts; for example, as the Schubert cells of the cohomology of affine Grassmannian permutations Lam (2006), and they are related to the quantum cohomology of the affine permutations Lapointe and Morse (2008).

One of the intriguing features of standard Young tableaux is the Frame-Thrall-Robinson hook length formula, which enumerates them. It has many different proofs and many generalizations, see e.g. (Stanley, 1999, Chapter 7), Greene et al. (1979), Ciocan-Fontanine et al. (2011) and the references therein.

In this extended abstract, we partially succeed in finding an analogue of the hook-length formula for standard strong marked tableaux (or starred tableaux for short), which are a natural generalization of standard Young tableaux in the context ofk-Schur functions. For a fixedn, the shape of a starred tableau (see Subsection 2.5 for a definition) is necessarily ann-core, a partition for which all hook-lengths are different fromn. In Lam et al. (2010), a formula is given for the number of starred tableaux forn= 3.

†Partially supported by NSF grant # 1200280 and Simons Foundation grant # 209806

‡was partially supported by Research Program L1-069 of the Slovenian Research Agency

subm. to DMTCS cby the authors Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

Proposition 1.1 (Lam et al. (2010), Proposition 9.17) For a3-coreλ, the number of starred tableaux of shapeλequals

m!

2b^m₂c,

wheremis the number of boxes ofλwith hook-length< n. 2

The number of2-hooks ism 2

. Therefore we can rewrite the result as m!

Y

i,j∈λ h_ij<3

h_ij .

Note that this is reminiscent of the classical hook-length formula.

The authors left the enumeration forn >3as an open problem. The main result (Theorem 3.1) of this extended abstract implies that for eachn, we can find(n−1)!rational numbers (which we callcorrection factors) so that the number of starred tableaux of shapeλfor anyn-coreλcan be easily computed. In fact, Theorem 3.1 gives the formula for the sum of certain powers oft, summed over all starred tableaux.

The theorem is “incomplete” in the sense that we were not able to find explicit formulas for the (weighted) correction factors. We have, however, been able to state some of their properties (some conjecturally), the most interesting of them being the unimodality conjecture (Conjecture 3.7).

Another result of interest is a new, alternative description of strong marked covers via simple triangular arrays of integers which we callresidue tablesandquotient tables(Theorem 4.2).

The extended abstract is structured as follows. In Section 2, we give the requisite background, notation, definitions, and results. In Section 3, we state the main results and conjectures. In Section 4, we give an alternative description of strong covers directly in terms of bounded partitions (instead of via cores, abacuses or affine permutations). We envision this description as the first steps toward an inductive proof of the main formula We finish with some remarks and open questions in Section 5.

2 Preliminaries

Here we introduce notation and review some constructions. Please see Macdonald (1995) for the defini- tions of integer partitions, ribbons, hook lengths, etc., which we omit in this extended abstract.

2.1 Cores and bounded partitions

Letnbe a positive integer. Ann-coreis a partitionλsuch thath^λ_ij 6=nfor all(i, j)∈λ. Core partitions were introduced by Nakayama (1941a,b) to describe when two ordinary irreducible representations of the symmetric group belong to the same block. There is a close connection between(k+ 1)-cores andk- bounded partitions, which are partitions whose first part (and hence every part) is≤k. Indeed, in Lapointe and Morse (2005), a simple bijection between(k+ 1)-cores andk-bounded partitions is presented. Given a(k+ 1)-core λ, letπi be the number of boxes in row iof λ with hook-length≤ k. The resulting π= (π1, π2, . . . , π`)is ak-bounded partition, we denote itb(λ). Conversely, given ak-bounded partition π, move from the last row ofπupwards, and in rowi, shift theπiboxes of the diagram ofπto the right until their hook-lengths are at mostk. The resulting(k+1)-core is denotedc(π). In this extended abstract, we will always usenas shorthand fork+ 1.

Example 2.1 On the left-hand side of Figure 1, the hook-lengths of the boxes of the5-coreλ= 953211 are shown, with the ones that are<5in bold. That means thatb(λ) = 432211.

14 11 9 7 6 4 3 2 1 9 6 4 2 1

6 3 1 4 1 2 1

Fig. 1:Bijectionsbandc.

The right-hand side shows the construction ofc(π) = 75221for the6-bounded partitionπ= 54221.

Of particular importance arek-bounded partitionsπthat satisfymi(π) ≤k−ifor alli = 1, . . . , k.

We call such partitionsk-irreducible partitions, see Lapointe et al. (2003). The number ofk-irreducible partitions isk!.

2.2 Young tableaux and the hook-length formula

Young’s latticeYtakes as its vertices all integer partitions, and the relation is containment. Ifλandµare partitions, thenµcoversλif and only ifλ⊆µand|µ|=|λ|+ 1. The rank of a partition is given by its size.

Asemistandard Young tableauT of shapeλis a Young diagram of shapeλwhose boxes have been filled with positive integers satisfying the following: the integers must be nondecreasing as we read a row from left to right, and increasing as we read a column from top to bottom. TheweightofT is the composition(α₁, α₂, . . .), whereα_iis the number ofi’s inT. The tableauT is astandard Young tableau if the entries are1, . . . ,|λ|in some order, i.e. if the weight is (1, . . . ,1). A standard Young tableau of shapeλrepresents a saturated chain in the interval[∅, λ]of the Young’s lattice. Let(λ⁽⁰⁾, λ⁽¹⁾, . . . , λ^(m)), λ⁽⁰⁾=∅,λ^(m)=λ, be such a chain. Then in the tableau corresponding to this chain,iis the entry in the box added in moving fromλ⁽ⁱ⁻¹⁾toλ⁽ⁱ⁾.

The Frame-Thrall-Robinson hook-length formula shows how to computefλ, the number of standard Young tableaux of shapeλ. We have:

fλ= |λ|!

Q

i,j∈λh^λ_ij. (2.1)

There exists a well-known weighted version of this formula. For a standard Young tableauT, define a descentto be an integerisuch thati+ 1appears in a lower row ofT thani, and define thedescent set D(T)to be the set of all descents ofT. Define themajor indexofT asmaj(T) =P

i∈D(T)i, and the polynomial

fλ(t) =X t^maj(T),

where the sum is over all standard Young tableaux of shapeλ. Then

fλ(t) = t^b(λ)(|λ|)!

Q

i,j∈λ(h^λ_ij) (2.2)

Hereb(λ) =P

i(i−1)λi =P

i λ⁰_i

2

,(i)= 1 +t+. . .+tⁱ⁻¹and(i)!=(1)·(2)· · ·(i). See e.g.

(Stanley, 1999, Corollary 7.21.5).

2.3 Strong marked and starred tableaux

Thestrongn-core posetCn is the subposet ofY induced by the set of alln-core partitions. That is, its vertices aren-core partitions andλ≤µinCnifλ⊆µ. The cover relations are trickier to describe inCn

than inY.

Proposition 2.2 (Lam et al. (2010), Proposition 9.5) Supposeλ≤µinCn, and letC1, . . . , Cmbe the connected components ofµ/λ. Thenµcoversλ(denotedλlµ) if and only if eachCiis a ribbon, and all the components are translates of each other with heads on consecutive diagonals with the same residue.

∅

2 2 2

2

2

2

2 2

2 2

2 2

Fig. 2:The4-core lattice up to rank6.

The rank of ann-core is the number of boxes of its diagram with hook-length< n. Ifλlµandµ/λ consists ofmribbons, we say thatµcoversλin the strong order with multiplicitym. Figure 2 shows the strong marked covers for4-cores with rank at most6. Only multiplicities6= 1are marked.

Astrong marked coveris a triple(λ, µ, c)such thatλlµand thatcis the content of the head of one of the ribbons. We callcthemarkingof the strong marked cover. Astrong marked horizontal strip of

sizer and shapeµ/λis a sequence(ν⁽ⁱ⁾, ν⁽ⁱ⁺¹⁾, c_i)^r−1_i=0 of strong marked covers such thatc_i < c_i+1, ν⁽⁰⁾ = λ,ν^(r) = µ. Ifλis ann-core, astrong marked tableauT of shapeλis a sequence of strong marked horizontal strips of shapesµ⁽ⁱ⁺¹⁾/µ⁽ⁱ⁾,i = 0, . . . , m−1, such thatµ⁽⁰⁾ = ∅andµ^(m) = λ.

Theweightof T is the composition(r1, . . . , rm), whereri is the size of the strong marked horizontal stripµ⁽ⁱ⁾/µ⁽ⁱ⁻¹⁾. If all strong marked horizontal strips are of size1, we callTastandard strong marked tableauor astarred tableaufor short. For ak-bounded partitionπ(recall thatn = k+ 1), denote the number of starred tableaux of shapec(π)byFπ^(k).

1^∗ 2^∗ 3^∗ 1^∗ 2^∗ 4^∗ 1^∗ 2^∗ 4 1^∗ 3^∗ 4^∗ 1^∗ 3^∗ 4 1^∗ 4 4^∗

4^∗ 3^∗ 3^∗ 2^∗ 2^∗ 2^∗

4 4 4^∗ 4 4^∗ 3^∗

Fig. 3:All starred tableaux of shape311.

Figure 3 illustratesF₂₁₁⁽³⁾ = 6.

Ifλis ak-bounded partition that is also ann-core (i.e., ifλ1+`(λ) ≤ k+ 1), then strong marked covers on the interval[∅, λ]are equivalent to the covers in the Young lattice, strong marked tableaux of shapeλare equivalent to semistandard Young tableaux of shapeλ, and starred tableaux of shapeλare equivalent to standard Young tableaux of shapeλ.

2.4 Schur functions

For the definition ofΛ, the ring of symmetric functions, see Macdonald (1995) or Stanley (1999). For a partitionλ, define themonomial symmetric function

mλ=mλ(x1, x2, . . .) =X

α

x^α,

where the sum is over all weak compositionsαthat are a permutation ofλ, andx^α =x^α₁¹x^α₂²· · ·. For partitionsλandµof the same size, define theKostka numberKλµas the number of semistandard Young tableaux of shapeλand weightµ. Define theSchur function

s_λ=X K_λµm_µ

with the sum over all partitionsµ. The Schur functions form the most important basis ofΛand have nu- merous beautiful properties. See for example (Stanley, 1999, Chapter 7) and (Macdonald, 1995, Chapter 1).

2.5 k-Schur functions

There are at least three conjecturally equivalent definitions ofk-Schur functions. Here, we give the def- inition from Lam et al. (2010) via strong marked tableaux. Fork-bounded partitionsπandτ, define the k-Kostka numberKπτ^(k)as the number of strong marked tableaux of shapec(π)and weightτ. Then we define thek-Schur function

s^(k)_π =X

τ

K_πτ^(k)m_τ, (2.3)

where the sum is over allk-bounded partitionsτ.

Ifπis also a(k+1)-core, then strong marked tableaux of shapeπare equivalent to semistandard Young tableaux of shapeπ, and therefore in this cases^(k)π =sπ.

The original definition ofk-Schur functions was viaatomsLapointe et al. (2003), which we will not use here (but see5.2). Note that in full generality, thek-Schur functions (in any definition) have a parameter t. In this extended abstract,t= 1.

3 Main results and conjectures

For a starred tableauT, define thedescent set ofT,D(T), as the set of allifor which the marked box at iis strictly above the marked box ati+ 1. Define themajor index ofT,maj(T), byP

i∈D(T)i. For a k-bounded partitionπ, define the polynomial

F_π^(k)(t) =X

T

t^maj(T), (3.1)

where the sum is over all starred tableaux of shapec(π). Recall that Fπ^(k) denotes the number of such starred tableaux, i.e.Fπ^(k)=Fπ^(k)(1).

Our main result is the following theorem.

Theorem 3.1 Letπbe ak-bounded partition, and write

π=hk^a1+1·w1,(k−1)^a2+2·w2, . . . ,1^ak+k·wki,

for0≤a_i < i. Then

F_π^(k)(t) = t^Pki=1wi(₂ⁱ)^(k−i+1)(|π|)!Fσ^(k)(t) (|σ|)!Qk

j=1(j)

Pk

i=1w_imin{i,j,k+1−i,k+1−j},

whereσ=hk^a1,(k−1)^a2, . . . ,1^aki.

By plugging int= 1, we get the following.

Corollary 3.2 Letπbe ak-bounded partition, and write

π=hk^a1+1·w1,(k−1)^a2+2·w2, . . . ,1^ak+k·wki,

for0≤ai < i. Then

F_π^(k)= |π|!Fσ^(k)

|σ|! Qk

j=1j^Pki=1wimin{i,j,k+1−i,k+1−j},

whereσ=hk^a1,(k−1)^a2, . . . ,1^aki. 2

The theorem (respectively, corollary) implies that in order to computeFπ^(k)(t)(resp.,Fπ^(k)) for allk- bounded partitionsπ, it suffices to computeFσ^(k)(t)(resp.,Fσ^(k)) only fork-irreducible partitionsσ; recall that there arek!such partitions.

The proof is omitted in the extended abstract. Let us just mention that it uses the expansion ofk-Schur functions in terms of fundamental quasisymmetric functions, and the stable principal specialization (i.e., evaluation at1, t, t², . . .) of fundamental quasisymetric functions.

Example 3.3 The following gives the formulas fork≤3.

1. Fork= 1, we haveF₁⁽¹⁾0 (t) = 1and therefore F₁⁽¹⁾w1(t) = (w1)!·1

(0)!·(1)^w1 =(w1)!.

This is consistent with (Lam et al., 2010,§9.4.1), which states thatF₁⁽¹⁾w1 =w1!.

2. Fork= 2, we haveF₂⁽²⁾01⁰(t) = 1andF₂⁽²⁾01¹(t) = 1. Therefore,

F₂⁽²⁾_w112w2(t) =t^w2(2w₁+ 2w₂)!·1

(0)!·(2)^w1+w2 = t^w2(2w₁+ 2w₂)!

(2)^w1+w2 .

F₂⁽²⁾_w111+2w2(t) =t^w2(2w1+ 2w2+ 1)!·1

(0)!·(2)^w1+w2 = t^w2(2w1+ 2w2+ 1)!

(2)^w1+w2 .

This is consistent with (Lam et al., 2010, Proposition 9.17), reprinted here as Proposition 1.1.

3. Fork = 3, we have F₃⁽³⁾₀₂₀₁₀ =F₃⁽³⁾₀₂₀₁₁ =F₃⁽³⁾₀₂₁₁₀ = 1,F₃⁽³⁾₀₂₀₁₂ = t,F₃⁽³⁾₀₂₁₁₁ =t(1 +t)and, F₃⁽³⁾02¹1² =t t²+ 1

t²+t+ 1

. So, among other formulas, we have

F₃⁽³⁾_{w121+2w211+3w3}(t) = t^2w2+3w3+1·(3w3+ 4w2+ 3w1+ 3)!

(2)^w1+2w2+w3·(3)^w1+w2+w3+1 .

Using a computer, it is easy to obtain formulas for largerk.

For ak-bounded partitionπ, letHπ^(k)(t) =Q

(hij), where the product is over all boxes(i, j)of the (k+ 1)-corec(π)with hook-lengths at mostk, and letHπ^(k)=Hπ^(k)(1)be the product of all hook-lengths

≤kofc(π). Furthermore, ifb_jis the number of boxes in thej-column ofc(π)with hook-length at most k, writeb^(k)π =P

j b_j

2

.

Example 3.4 For the6-bounded partitionπ= 54211from Example 2.1, we have Hπ⁽⁶⁾(t) =(1)⁴(2)³(3)²(4)²(5)(6)²,Hπ⁽⁶⁾= 207360andb⁽⁶⁾π = 2 ³₂

+ 3 ²₂ + 2 ¹₂

= 9.

By introducingweighted correction factorsCσ^(k)(t)for ak-irreducible partitionσ, we can, by Theorem 3.1, expressFπ^(k)(t)(for allk-bounded partitionsπ) in another way which is reminiscent of the classical hook-length formula. More precisely, define a rational functionCσ^(k)(t)so that

F_σ^(k)(t) =t^b(k)σ (|σ|)!Cσ^(k)(t) Hσ^(k)(t)

. (3.2)

Note that this implies, in the notation of Theorem 3.1, that

F_π^(k)(t) = t^{b(k)σ +Pki=1wi}(ⁱ₂)(k+1−i)(|π|)!Cσ^(k)(t) Hσ^(k)(t)·Qk

j=1(j)^Pki=1wimin{i,j,k+1−i,k+1−j}.

Thecorrection factorCσ^(k)is defined asCσ^(k)(1).

Fork≤3, all weighted correction factors are1. Fork= 4, all but four of the24weighted correction factors – for4-bounded partitions2211,321,3211and32211– are1, and the ones different from1are

1 + 2t+t²+t³

(2)(3) , 1 +t+ 2t²+t³

(2)(3) , 1 + 2t+ 2t²+ 2t³+t⁴

(3)² , 1 +t+ 3t²+t³+t⁴

(3)² ,

respectively.

We state some results and conjectures about the weighted correction factors. For ak-bounded partition π, denote by∂k(π)the boxes ofc(π)with hook-length≤ k. If∂k(π)is not connected, we say thatπ splits. Each of the connected components of∂_k(π)is a horizontal translate of∂_k(πⁱ)for somek-bounded partitionπⁱ. Callπ¹, π², . . .thecomponentsofπ.

Proposition 3.5 The weighted correction factors are multiplicative in the following sense. If ak-irreducible partitionσsplits intoσ¹, σ², . . . , σ^m, thenCσ^(k)(t) =Qm

i=1C_σ^(k)i (t).

Conjecture 3.6 For ak-irreducible partitionσ, the weighted correction factor is1if and only ifσsplits intoσ¹, σ², . . . , σ^l, where eachσⁱis ak-bounded partition that is also a(k+ 1)-core.

The “if” direction is easy: if ak-bounded partitionσis also a(k+ 1)-core, then strong covers on the interval[0, σ]are precisely the regular covers in the Young lattice, the starred tableaux of shapeσare standard Young tableaux of shapeσ, and the major index of a starred tableau of shapeσis the classical major index for standard Young tableaux; the fact that the weighted correction factor is1then follows from the classical weighted version of the hook-length formula (2.2). Ifσsplits into cores, we can use (Denton, 2012, Theorem 1.1).

The most interesting conjecture about the weighted correction factors is the following. Recall that a sequence(α_i)_iisunimodalif there existsIso thatα_i ≤α_i+1fori < Iandα_i≥α_i+1fori≥I, and a unimodal polynomialis a polynomial whose sequence of coefficients is unimodal.

Conjecture 3.7 For ak-irreducible partitionσ, we can write 1−C_σ^(k)(t) = P1(t)

P₂(t),

whereP₁(t)is a unimodal polynomial with non-negative integer coefficients andP₂(t)is a polynomial of the formQk−1

i=1 (j)^wj for some non-negative integersw_j. In particular, we have0< Cσ^(k)≤1for allσ.

4 Strong covers and k-bounded partitions

Our proof of Theorem 3.1, omitted in the extended abstract, closely follows one of the possible proofs of the classical (non-weighted and weighted) hook-length formula, see e.g. (Stanley, 1999,§7.21). Note, however, that the truly elegant proofs (for example, the celebrated probabilistic proof due to Greene, Nijenhuis and Wilf Greene et al. (1979)) are via induction. In this section, we show the first steps toward such a proof.

In the process, we present a new description of strong marked covers in terms of bounded partitions (previous descriptions included cores – at least implicitly, viak-conjugation – affine permutations and abacuses). See the definition of residue and quotient tables below, and Theorem 4.2.

We identify a bounded partitionπ = hk^p1,(k−1)^p2, . . . ,1^pkiwith the sequencep= (p1, . . . , pk).

Giveni, j, m,0 ≤ m < i ≤ j ≤ k, definep^i,j,m as follows: ifi 6= j, define p^i,j,m_i−1 = pi−1 +m, p^i,j,m_i =pi−m,p^i,j,m_j =pj−m−1,p^i,j,m_j+1 =pj+1+m+ 1,p^i,j,m_h =phforh6=i−1, i, j, j+ 1.

Definep^i,i,m_i−1 =p_i−1+m,p^i,i,m_i =p_i−2m−1,p^i,i,m_i+1 =p_i+1+m+ 1,p^i,i,m_h =p_hforh6=i−1, i, i+ 1.

In other words, to getp^i,j,mfromp, addmcopies ofk+ 2−i, removemcopies ofk+ 1−i, remove m+ 1copies ofk+ 1−j, and addm+ 1copies ofk−j. (Ifj =k, then we are addingm+ 1copies ofk−j = 0, which does not change the partition. Ifi = 1, we havem = 0, so addingmcopies of k+ 2−i=k+ 1also does not change the partition.) To put it another way: to getp^i,j,mfromp, increase the firstmcopies ofk+ 1−iby1, and decrease the lastm+ 1copies ofk+ 1−jby1. See Example 4.3.

Define upper-triangular arraysR= (r_ij)_{1≤i≤j≤k},Q= (q_ij)_{1≤i≤j≤k}by

• rjj =pjmodj,rij = (pi+ri+1,j) modifori < j,

• qjj =pjdivj,qij = (pi+ri+1,j) diviforj < i.

We callRtheresidue tableandQthequotient table.

Example 4.1 Takek= 4andp= (1,3,2,5). Then the residue and quotient tables are given by 0 0 0 0

1 1 1 2 0 1

1 2 2 2 1 2 1 0 1 1

It is easy to reconstructpfrom the diagonals ofRandQ:p1= 0 + 1·1,p2= 1 + 1·2,p3= 2 + 0·3,

p4= 1 + 1·4.

It turns out that the residue and quotient tables determine strong marked covers (and probably other important relations as well, see5.5).

Theorem 4.2 Takep= (p₁, . . . , p_k)and1≤i≤j≤k. Ifr_ij < r_i+1,j, . . . , r_jj, thenpcoversp^i,j,rij in the strong order with multiplicityq_ij+. . .+q_jj. Furthermore, these are precisely all strong covers.

In particular, an element of the(k+ 1)-core lattice covers at most ^k+1₂

elements.

Example 4.3 Takek = 4andp= (1,3,2,5)as before. Let us underline the entriesrij in the residue tableRfor whichrij < ri+1,j, . . . , rjj.

0 0 0 0 1 1 1 2 0 1

By Theorem 4.2,pcovers (exactly) the following elements in the strong order:

p^1,1,0= (0,4,2,5)with multiplicity1,p^1,2,0= (1,2,3,5)with multiplicity2+1 = 3,p^2,2,1= (2,0,4,5) with multiplicity1,p^1,3,0= (1,3,1,6)with multiplicity2+2+0 = 4,p^2,3,1= (2,2,0,7)with multiplicity 2 + 0 = 2,p^3,3,2= (1,5,−3,8)with multiplicity0,p^3,4,0 = (1,3,2,4)with multiplicity1 + 1 = 2, and p^4,4,1= (1,3,3,2)with multiplicity1.

Note that while(1,5,−3,8)does not represent a valid partition, the multiplicity of the cover is0, so

we can ignore this cover relation.

For ak-bounded partitionπ, we clearly have

F_π^(k)=X

τ

mτ πF_τ^(k),

where the sum is over allk-boundedτ that are covered byπ, andm_{τ π} is the multiplicity of the cover.

Therefore the theorem can be used to prove Corollary 3.2 for small values ofkby induction. First, we need the following corollary.

Corollary 4.4 Letp = (p1, . . . , pk),pi < i, with corresponding residue and quotient tablesRandQ.

Assume that for1≤i≤j≤k, we haverij < ri+1,j, . . . , rjj. Forsi ∈N, writes= (s1,2s2, . . . , ksk).

Thenp+scoversp^i,j,rij+swith multiplicityqij+. . .+qjj+si+. . .+sj.

The corollary implies that in order to prove Corollary 3.2, all we have to do is checkk!equalities. The authors did all such calculations with a computer for smallk(k≤8).

5 Final remarks

5.1

There are also notions ofweak horizontal stripsandweak tableaux. Forn-coresλandµ,λ⊆µ, we say thatµ/λis a weak horizontal strip ifb(µ)/b(λ)is a horizontal strip andb(µ⁰)/b(λ⁰)is a vertical strip. If in addition|b(µ)|=|b(λ)|+ 1, we say thatµcoversλin the weak order. A weak tableau of shapeλis a sequence of weak horizontal stripsµ⁽ⁱ⁺¹⁾/µ⁽ⁱ⁾,i = 0, . . . , m−1, such thatµ⁽⁰⁾ =∅andµ^(m) =λ.

Definefπ^(k)to be the number of weak tableaux of shapec(π). In Lam et al. (2010), it was proved that f₂⁽²⁾w11^2w2 =f₂⁽²⁾w11^1+2w2 = ^(w_w^1+w2)!

1!w2! . It is not hard to prove by induction that

f₃⁽³⁾w12^2w21^3w3 =2^2w2(w1+w2)!(w2+w3)!(w1+ 2w2+w3−1)!(2w1+ 2w2+ 2w3)!

w₁!w₂!w₃!(w₁+w₂+w₃−1)!(2w₁+ 2w₂)!(2w₂+ 2w₃)! ;

similar formulas exist for

f₃⁽³⁾w12^2w21^1+3w3, f₃⁽³⁾w12^2w21^2+3w3, f₃⁽³⁾w12^1+2w21^3w3, f₃⁽³⁾w12^1+2w21^1+3w3, f₃⁽³⁾w12^1+2w21^2+3w3.

We were unable to find formulas fork≥4, and it seems unlikely that simple formulas exist. For example, the simplest recurrence relation thatg(i, j) =f₂⁽⁴⁾_3i14j seem to satisfy is

a(i, j)g(i, j) +b(i, j)g(i, j+ 1)−c(i, j)g(i+ 1, j) = 0,

where aandbare fourth degree polynomials iniandjwith rational coefficients andc, also fourth degree, is a polynomial with integer coefficients.

5.2

Our work has led us to consider (weighted) correction factors. They seem to be mysterious objects that deserve further study. The unimodality conjecture (Conjecture 3.7) is certainly intriguing and could hint that the factors have some geometric meaning.

Let us give another perspective on these factors. Sincek-Schur functions are symmetric, they can be expanded in terms of Schur functions; in fact, the original definition ofk-Schur functions via atoms gives precisely such an expansion. For example,s⁽⁴⁾₂₂₁₁=s2211+s321. Take the stable principal specialization and multiply by (6)!(1−t)⁶. By calculations done in our proof of Theorem3.1 and (Stanley, 1999, Proposition 7.19.11), we have

F₂₂₁₁⁽⁴⁾ (t) =f2211(t) +f321(t).

Then, by (3.2) and (Stanley, 1999, Corollary 7.21.5),

C₂₂₁₁^(k) (t) =(2)(3)(4) t³

(2)²(4)(5)+ 1 (3)²(5)

!

=1 + 2t+t²+t³ (2)(3) .

5.3

There is also a formula for the principal specialization ofs_λ of orderi(i.e. evaluation at1, t, . . . , tⁱ⁻¹, see e.g. (Stanley, 1999, Theorem 7.21.2)), in which both hook-lengths and contents of boxes appear. By imitating5.2, we can get rational functions (which depend oni) which converge to the weighted correction factors asi→ ∞. These rational functions also seem interesting and worthy of further study.

5.4

As we already mentioned, it would be preferable to prove Corollary 3.2 by induction, using the cover relations in Section 4 for a generalkand in a way that would make apparent the meaning of hook-lengths and correction factors (the ideal being a variant of the probabilistic proof from Greene et al. (1979)). It seems likely that one would need to know a formula for the correction factors before such a proof would be feasible.

5.5

We showed (in Theorem 4.2) how to interpret the residue and quotient table to find strong covers. We feel that residue (and quotient) tables could prove important in other aspects of thek-Schur function theory.

These tables can also be used to describe weak covers, weak horizontal and vertical strips and at least one of the possible cases of LLMS insertion for standard strong marked tableaux (see Lam et al. (2010)).

Acknowledgments

We thank Jennifer Morse for pointing us to the right version of the problem, Sara Billey for encouraging our collaboration, and Mike Zabrocki for sharing (Lam et al., 2012, Chapter 2) with us.

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