Asymptotics for the distributions of subtableaux in Young and

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Asymptotics for the distributions of subtableaux in Young and

the up-down tableau is actually a standard tableau, and approaches 0 if has fewer than k cells 1 Introduction Let be a partition of k,andT a standard Young tableau of shape WesaythatT is a subtableau of a larger Young tableau Y if the entries 1 through k of Y form the tableau T LetP(N;T) be the probability that a randomly chosen standard

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Young and up-down tableaux

David J. Grabiner

7318 Eden Brook Dr. #123

Columbia, MD 21046

grabiner@alumni.princeton.edu Submitted: Mar 21, 2005; Accepted: Apr 2, 2006; Published: Apr 11, 2006 Mathematics Subject Classications: primary 05E10, secondary 60G50

Abstract

Letbe a partition ofk,andTa standard Young tableau of shape. McKay, Morse, and Wilf show that the probability a randomly chosen Young tableau ofN cells containsTas a subtableau is asymptotic tof =k!asNgoes to innity, where f is the number of all tableaux of shape. We use a random-walk argument to show that the analogous asymptotic probability for randomly chosen Young tableaux with at mostnrows is proportional toQ 1i1 Introduction Letbe a partition ofk,andTa standard Young tableau of shape.WesaythatTis asubtableauof a larger Young tableauYif the entries 1 throughkofYform the tableau T.LetP(N;T) be the probability that a randomly chosen standard Young tableau ofN cells containsTas a subtableau.

McKay, Morse, and Wilf [12] show that

lim N!1

P(N;T)=f

k!;(1) wheref is the number of standard Young tableaux of shape. They apply this result to problems such as the asymptotic distribution of entries in random large Young tableaux. Stanley [14] gives an exact formula forP(N;T), and related asymptotics for the number of skew tableaux of a given shape. Jaggard [9] gives another proof. the electronic journal of combinatorics11(2)(2006), #R291 A Young tableau can be viewed as a walk in the regionx 1 x 2 x n ;the enumeration of walks to2Z n is the classicaln-candidate ballot problem. We use the random-walk view of Young tableaux, a local central limit theorem due to Kuperberg [10] which relates asymptotics for random walks and Brownian motion, and asymptotics for the Brownian motion from [6], to nd a formula analogous to (1) when the number of rows of theN-cell tableau is restricted ton. The asymptotic probability is proportional toY 1i2 Random walks in Weyl chambers We will give the necessary denitions and properties of Weyl groups from [8], and of random walks in Weyl chambers from [7]. ACoxeter groupWis a discrete group generated by reflections inR n . It is determined by theroot system, a set of vectors orthogonal to the hyperplanes of reflection, or by thepositive roots , choosing only one root for each hyperplane of reflection so that all the positive roots are on the same side of a given hyperplane. IfWstabilizes a lattice, it is called aWeyl group, and the hyperplanes of reflection partition space intoWeyl chambers.

For example, the rootsx

i -x j inR n give reflections in the hyperplanesx i =x j ,and the electronic journal of combinatorics11(2)(2006), #R292 these generate the symmetric groupS n (calledA n-1 as a Weyl group). The principal

Weyl chamber isx

1 >x 2 >>x n We consider the enumeration of walks which stay inside the Weyl chamber, or equiv- alently the probability distribution for a random walk which is killed when it hits a wall of the Weyl chamber. Such a random walk isreflectable[7] if the step setSis symmetric under the Weyl groupW, and the steps and starting point are such that a step which starts in the interior of the Weyl chamber cannot cross a wall. For example, on the lattice Z n , the walk with Weyl groupS n and stepse i in the positive coordinate directions is reflectable; a step starting at a point withx i >x j can go to a point withx i =x j but not apointwithx i x i+1

Denitions.Letb

;k be the number of walks of lengthkfromtowith a step set

SwhichstayintheWeylchamber.Letc

γ;k

be the number of walks of lengthkfrom the origin toγ(or equivalently with any start and end with dierenceγ) with the same step setS, but unconstrained by a chamber. Similarly, for Brownian motion, letb t (;)be the density function forn-dimensional Brownian motion started atto be atat time t, staying within the Weyl chamber through that time. Letc t (γ) be the density function forn-dimensional Brownian motion to go from the origin toγat timet, unconstrained by a chamber. For reflectable walks, Gessel and Zeilberger [5] and independently Biane [2] related the number of constrained walks to a signed sum of unconstrained walks of the same length.

The formula is

b ;k =X w2W sgn(w)c -w();k :(3) The proof is analogous to the reflection argument for the Catalan numbers [5]. Every walk from anyw()towhich does touch at least one wall has some last stepjat which it touches a wall; let the wall be the hyperplane perpendicular to i , choosing the largest iif there are several choices [13]. Reflect all steps of the walk up to stepjacross that hyperplane; the resulting walk is a walk fromw i w()towhich also touches walliat stepj. This clearly gives a pairing of walks, and sincew i has sign-1, these two walks cancel out in (3). The only walks which do not cancel in these pairs are the walks which stay within the Weyl chamber, and sincew() is inside the Weyl chamber only ifwis the identity, this is the desired number of walks. An analogous result with an analogous proof holds for Brownian motion, which is al- ways reflectable because it is continuous and symmetric under all reflections. The formula is [6] b t (;)=X w2W sgn(w)c t (-w()):(4) (This theorem is stated in [6] withc t (w()-) instead ofc t (-w()), reflecting the Brownian motion after the rst time it hits a wall rather than before the last time. The the electronic journal of combinatorics11(2)(2006), #R293 two forms are equivalent since Brownian motion is symmetric under the Weyl group, and we will need to use the theorem in the form (4) later.) The step sets which give reflectable walks are enumerated in [7]; they turn out to be precisely the Weyl group images of theminuscule weights[3], those weights with dot-product 0 or1 with every root. The reflectable walks include the weights which are minuscule for only one ofB n andC n , which are the same as Weyl groups but have dierent root systems. In the Bourbaki numbering [3], the allowed step sets are the Weyl group images of the following ! i , the duals of the fundamental roots. A n 1 n :All compatible. B n ;C n 1 n :Not compatible. D n 1 n-1 n :All compatible. E 6 1 6 :Compatible. E 7 7 E 8 ;F 4 ;G 2 :None. Any union of compatible step sets also gives a reflectable random walk.

The classical Weyl groups areA

n-1 =S n ,B n =C n ,andD n . The Weyl groupA n-1 has rootsx i -x j and principal Weyl chamberx 1 >x 2 >>x n . The Weyl groupB n has rootsx i x j andx i . It contains all permutations with any number of sign changes, and has principal Weyl chamberx 1 >x 2 >>x n >0. The Weyl groupD n has rootsx i x j . It contains all permutations with an even number of sign changes, and has principal Weyl chamberx 1 >x 2 >>x n ;x n-1 >-x n

The groupA

n-1 =S n acts on the hyperplaneHgiven byPx i = 0, a subspace ofR n

Thus a random walk onR

n is reflectable if the steps project onto reflectable steps onH, and the step set is symmetric underS n . In particular, the stepse i give a reflectable random walk; this is the most important case because it is the walk we use for Young tableaux.

The stepe

1 projects to the fundamental weight ! 1 =((n-1)=n;-1=n;:::;-1=n), and the other steps project to itsS n -images. This step set is not reflectable forB n orD n because it is not symmetric under those Weyl groups. The walk we use for up-down tableaux has step sete i onZ n , which gives a reflectable random walk for all three groups. The stepe 1 is the fundamental weight ! 1 forB n and D n ,ande 1 and-e 1 project to the fundamental weights ! 1 and ! n-1 forA n-1

3 Asymptotics for random walks and Brownian mo-

tion We continue throughout the paper to letbe a partition ofk.Let=(-1;-2;:::;-n), so that the number of Young tableaux of shapeis the number of walks in the region x 1 >x 2 >>x n fromto+. A Young tableau withNcells and at mostnrows corresponds to a walk starting fromofNsteps, and the number of Young tableaux withNcells which contain a specic subtableau of shapeis thus the number of walks the electronic journal of combinatorics11(2)(2006), #R294 starting at+which stay in the Weyl chamberx 1 >x 2 >>x n forN-kmore steps. We now view this as a problem in random walks, taking each step randomly in one of the coordinate directions. The probability that a walk of lengthNwill reach a tableau of shapeinksteps isf =n k , since each of thef tableaux corresponds to one walk.

Applying Bayes' Theorem,

P(n;N;T)=P(at stepkjsurvive to stepN)=f

=P(survive to stepNjat stepk)P(at stepk)=f

P(survive to stepN)

P(survive to stepNjat stepk)

n k

P(survive to stepN):(5)

The denominator of the last fraction is independent of. Thus the probabilityP(n;N;T) for tableaux of shapeis proportional to the probability that a random walk which reaches at stepkwill survive to stepN. It is more natural to compute the asymptotic for the probability that a random walk started atwill survive forNmore steps, since this is a more natural problem and the notation will be simpler; we will replaceNin the asymptotic byN-kwhen we apply (5) to our specic problem for Young tableaux. We can compute asymptotics for these probabilities up to a constant factor; we can then eliminate the constant becauseP T P(n;N;T) = 1 by denition. We will compute the asymptotics for this case, and by an analogous process for the other classical Weyl groups. Theorem 1For any reflectable random walk in any of the classical Weyl chambers, where the Weyl group hasmpositive roots, the asymptotic probability that a walk starting at will stay in the chamber for some large numberNof steps is CN -m=2 Y 2 ();(6) for a constantCdepending on the walk and the chamber. This is the same as the asymp- totic for Brownian motion (appropriately normalized so that the random walk and the Brownian motion have the same variance) started atin the same chamber to remain in the chamber up to timeN. We conjecture that this theorem also holds for reflectable walks for the exceptional groupsE 6 andE 7 We will start with the asymptotics for the corresponding problem in Brownian motion, the probability that a Brownian motion started atwill stay within the Weyl chamber up to a large timet. These asymptotics are given by the following lemmas from [6]. Lemma 1For Brownian motion in any of the classical Weyl chambers inR n , where the Weyl group hasmpositive roots, we have the following asymptotic for the densityb t the electronic journal of combinatorics11(2)(2006), #R295 for the motion started atto stay inside the chamber up to timet, with dierent constants

Cfor the dierent chambers:

b t (;)Ct -m-n=2 Y 2 ()()exp-jj 2 -jj 2 2t :(7)

Forjjof orderO(t

+1=2 ), this asymptotic is valid to a factor of1+O(t -1=2 We omit the details of the computation because it is a separate, very long computationquotesdbs_dbs5.pdfusesText_10

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Chapter 6Linear Programming: The Simplex Method