Legendre-Stirling Permutations

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1 Legedre-Stirlig Permutatios Eric S. Egge Departmet of Mathematics Carleto College Northfield, MN 07 USA Abstract We first give a combiatorial iterpretatio of Everitt, Littlejoh, ad Wellma s Legedre-Stirlig umbers of the first id. We the give a combiatorial iterpretatio of the coefficiets of the polyomial (1 x) 3+1 { + } 0 x aalogous to that of the Euleria umbers, where { } are Everitt, Littlejoh, ad Wellma s Legedre-Stirlig umbers of the secod id. Fially we use a result of Beder to show that the limitig distributio of these coefficiets as approaches ifiity is the ormal distributio. Keywords: descet, Stirlig umber, Legedre-Stirlig umber 1 Itroductio Followig Kuth [6], let [ { ] ad } deote the (usiged) Stirlig umbers of the first ad secod ids, respectively, which may be defied by the iitial coditios [ ] [ ] 0 δ,0, δ,0 (1) 0 ad ad recurrece relatios ad [ ] { } { } δ,0, 0 [ ] 1 + ( 1) 1 { } [ 1 { 1 { } 0 δ,0 (2) ], (, Z), (3) }, (, Z). (4) It is well ow that [ { ] ad } have a variety of iterestig algebraic properties; for istace, [ ] { }, (, Z), () [ ]{ } i ( 1) j+ δ i,j, j (1 i, j ), (6) ad 1 { }[ ] i ( 1) j+ δ i,j, (1 i, j ). (7) j Mathematics Subject Classificatio: Primary 0A1; Secodary 0A18 1

2 [ The Stirlig umbers of each id also have combiatorial iterpretatios: for 1 ad 1 the quatity ] { is the umber of permutatios of [] with exactly cycles, while } is the umber of partitios of [] with exactly blocs. Recetly Everitt, Littlejoh, ad Wellma itroduced [4] the Legedre-Stirlig umbers of the secod id, which may be defied by the iitial coditios {{ }} {{ }} 0 δ,0, δ,0 (8) 0 ad recurrece relatio {{ }} {{ }} 1 + ( + 1) 1 It is ot difficult to show that whe 1 we have x j0 {{ 1 }}, (, Z). (9) {{ }} x j, (10) j where x j x(x 2)(x 6) (x (j 1)j). These umbers first arose i the study of a certai differetial operator related to Legedre polyomials, but Adrews ad Littlejoh [1] have give them the followig combiatorial iterpretatio. For each 1, let [] 2 deote the set {1 1, 1 2, 2 1, 2 2,..., 1, 2 }, which cosists of two distiguishable copies of each positive iteger from 1 to. By a Legedre-Stirlig set partitio of [] 2 ito blocs we mea a ordiary set partitio of [] 2 ito + 1 blocs for which the followig hold. 1. Oe bloc, called the zero bloc, is distiguished, but all other blocs are idistiguishable. 2. The zero bloc may be empty, but all other blocs are oempty. 3. The zero bloc may ot cotai both copies of ay umber. 4. Each ozero bloc cotais both copies of the smallest umber it cotais, but does ot cotai both copies of ay other umber. The Adrews ad Littlejoh have show [1] that the umber of Legedre-Stirlig set partitios of [] 2 ito blocs is { }, by showig that these two quatities satisfy the same iitial coditios ad recurrece relatio. I this paper we prove Legedre-Stirlig aalogues of a variety of results cocerig Stirlig umbers of the first ad secod ids. I sectio 2 we give a recursive defiitio of the Legedre-Stirlig umbers of the first id, which we deote by [[. We the prove aalogues of (), (6), ad (7) for the Legedre-Stirlig umbers, ad we give a combiatorial iterpretatio of [[ i terms }} of pairs of permutatios of [] with cycles. I sectios 3 ad 4 we tur our attetio to f () ad g (), which are the {{ + [[ 1 1 th orthwest to southeast diagoals of the secod ad first Legedre-Stirlig triagles, respectively. We show that f () is a polyomial of degree 3 i with f (0) f ( 1) f ( 1) 0; we show that similar results hold for g () by showig that g () ( 1) f ( ). These results, together with stadard facts cocerig ratioal geeratig fuctios, imply that there exist itegers B,j such that f ()x B,jx j (1 x) 3+1. We give two combiatorial iterpretatios of B,j, the secod of which ivolves descets i a certai family of permutatios, which we call Legedre-Stirlig permutatios. The results i these two sectios are aalogues of results of Gessel ad Staley [] cocerig the Stirlig umbers. I sectio we first show that for ay 1 the sequece {B,j } 2 1 is uimodal. We the tur our attetio to the radom variable X, which is the umber of descets i a uiformly chose Legedre-Stirlig permutatio. We show that E[X ] 6 1, ( 1), 2

3 ad ( 1)( ) V ar[x ], 2 17 { ( 1), ad we combie these results with a theorem of Beder to show that X E[X ] V ar[x ] } 1 coverges i distributio to the stadard ormal variable. These results are aalogues of results of Bóa [3] cocerig the Stirlig umbers. 2 Legedre-Stirlig Numbers of the First Kid Adrews ad Littlejoh [1] defie the Legedre-Stirlig umbers of the first id [[ via x [[ ( 1) +j j j0 x j, (11) where x j x(x 2)(x 6) (x (j 1)j) as above, but they say othig else about these quatities. I this sectio we give a recursive defiitio of [[, which we use to prove aalogues of (), (6), ad (7) ad to give a combiatorial iterpretatio of [[. Defiitio 2.1 For all, Z we write [[ to deote the (sigless) Legedre-Stirlig umbers of the first id, which are give by the iitial coditios [[ [[ 0 δ,0, δ,0, (12) 0 ad recurrece relatio [[ [[ 1 + ( 1) 1 [[ 1, (, Z). (13) It is ot difficult to show that (11) ad Defiitio 2.1 are equivalet for, 1, so we tur our attetio to a aalogue of (). Theorem 2.2 For all, Z, {{ }} [[ 1 ( 1) +. (14) 1 Proof. The Legedre-Stirlig umbers of the secod id [[ are uiquely determied by (8) ad (9), so it is sufficiet to show that the umbers L(, ) ( 1) also satisfy (8) ad (9). To prove L(, ) satisfies the left equatio i (8), first ote that L(1, 0) 0 by (12). Now if 1 the set 0 ad i (13) ad use (12) to fid that L(, 0) δ,0. The proof that L(, ) satisfies the right equatio i (8) is similar. To prove that L(, ) satisfies (9), ote that if 0 ad 0 the we have ad the result follows. [[ 1 L( 1, 1) ( 1) ( ( + 1) + [[ 1 ( 1) + 1 ( + 1) ( + 1)L( 1, ) + L(, ), [[ 1 1 ) + ( 1) + [[ 1 1 The followig aalogues of (6) ad (7) are clear from the relatioship betwee (10) ad (11), but for completeess we give a proof usig the recursive defiitios of [[ { ad }. 3

4 Theorem 2.3 If 1 the for all i, j with 1 i, j we have [[ {{ }} i ( 1) +j δ i,j (1) j 1 ad {{ }} [[ i ( 1) +j δ i,j. (16) j 1 Proof. To prove (1), first ote that if i < the [[ i 0, ad the result follows by iductio o. O the other had, if i the by (13), (9), ad iductio o we have [[ ( 1) +j {{ }} j 1 [[ 1 ( 1) (( +j 1) 1 1 ( 1) 1 [[ 1 ( 1) +j 1 δ j, 1 ( 1) [[ 1 1 {{ }} 1 + j [[ 1 ( 1) +j 1 δ j, 1 ( 1) + δ,j δ j, 1 j(j + 1) δ,j. The proof of (16) is similar to the proof of (1). 1 ) {{ }} j ({{ 1 j 1 [[ 1 ( 1) +j 1 }} + j(j + 1) {{ }} j {{ }}) 1 The Stirlig umbers of the first id cout permutatios of [] with cycles; we coclude this sectio with a aalogous iterpretatio of the Legedre-Stirlig umbers of the first id. Here the cycle maxima of a give permutatio are the umbers which are largest i their cycles. For example, if π (4, 6, 1)(9, 2, 3)(7, 8) is a permutatio i S 10, writte i cycle otatio, the its cycle maxima are, 6, 8, 9, ad 10. Defiitio 2.4 A Legedre-Stirlig permutatio pair of legth is a ordered pair (π 1, π 2 ) with π 1 S +1 ad π 2 S for which the followig hold. 1. π 1 has oe more cycle tha π The cycle maxima of π 1 which are less tha + 1 are exactly the cycle maxima of π 2. Theorem 2. For all 0 ad all with 0, the umber of Legedre-Stirlig permutatio pairs (π 1, π 2 ) of legth i which π 2 has exactly cycles is [[. Proof. Let a, deote the umber of Legedre-Stirlig permutatio pairs (π 1, π 2 ) of legth i which π 2 has exactly cycles. It is clear that a,0 δ,0 ad a 0, δ,0, so i view of (13) it is sufficiet to show that if > 0 ad > 0 the a, ( 1)a 1, + a 1, 1. To do this, first ote that by coditio 3 of Defiitio 2.4, if (π 1, π 2 ) is a Legedre-Stirlig permutatio pair of legth the 1 is a fixed poit i π 1 if ad oly if it is a fixed poit i π 2. Pairs (π 1, π 2 ) i which 1 is a fixed poit are i bijectio with pairs (σ 1, σ 2 ) of legth 1 i which σ 2 has 1 cycles by removig the 1 from each permutatio ad decreasig all other etries by 1. Each pair (π 1, π 2 ) i which 1 is ot a fixed poit may be costructed uiquely by choosig a pair (σ 1, σ 2 ) of legth 1 i which σ 2 has cycles, icreasig each etry of each permutatio by 1, ad isertig 1 after a etry of each permutatio. There are a 1, pairs (σ 1, σ 2 ), there are ways to isert a ew etry ito σ 1, ad there are 1 ways to isert a ew etry ito σ 2. Now the result follows. j 4

5 3 Legedre-Stirlig Polyomials It is atural to arrage the Legedre-Stirlig umbers of each id i a triagle; Figures 1 ad 2 show the first five rows of each of these triagles. Followig Gessel ad Staley s study [] of the Stirlig umbers of each id, i this sectio we give some elemetary properties of the sequeces parallel to the upper right sides of these triagles Figure 1: The First Legedre-Stirlig Triagle Figure 2: The Secod Legedre-Stirlig Triagle. Begiig with the secod Legedre-Stirlig triagle, it is ot difficult to show that {{ }} 1, ( 1), (17) ad {{ }} {{ }} ( ) ( ) + 72 These formulas suggest the followig result. Theorem 3.1 For all 0, the quatity ( + 2 {{ + ) + 36 ( ( ), ( 1), (18) ) + 4 ( ), ( 1). (19) }} is a polyomial of degree 3 i with leadig coefficiet 1 3!. We write f () to deote this polyomial; the for all 1 ad all Z we have f () ( + 1)f 1 () + f ( 1). (20) Proof. The result is immediate for 0, so suppose 1; we argue by iductio o. By (9) we have {{ }} {{ }} {{ }} ( + 1) 1 {{ }} for all Z. By iductio this implies that the first differece sequece for + is a polyomial of }} degree 3 1 i, so is a polyomial of degree 3 i. Let f () deote this polyomial; ow {{ + (20) is immediate from (21). Iteratig (21) ad usig the left equatio i (8) we fid that if 1 the f () j(j + 1)f 1 (j). Sice j0 j3 1 is a polyomial of degree 3 i with leadig coefficiet 1 3, by iductio the leadig 1 coefficiet of f () is 3!. Although oe ca use the same methods to prove a aalogue of Theorem 3.1 for the first Legedre- Stirlig triagle, we tae a differet approach. Theorem 3.2 For all 0, the quatity is a polyomial of degree 3 i with leadig coefficiet [[ !. We write g () to deote this polyomial; the for all 1 ad all Z we have (21) g () g ( 1) + ( 1)( 2)g 1 ( 1). (22)

6 Proof. By (14) we have [[ 1 ( 1) f ( ) (23) 1 for all 0; ow the result follows from Theorem 3.1. The relatioship betwee f ad g implied by (23) is worth otig, sice it will be useful later o. Corollary 3.3 For all 0 we have g () ( 1) f ( ). (24) Proof. This is immediate from (23). The forms of f 1 () ad f 2 () i (18) ad (19) also suggest the followig results cocerig the roots of f ad g. Theorem 3.4 If 1 the f (0) f ( 1) f ( ) f ( 1) 0 (2) ad g (0) g (1) g () g ( + 1) 0. (26) Proof. Whe 1 lie (2) is immediate from (18), so suppose > 1; we argue by iductio o. By the left equatio i (8) we have f (0) 0, ad by (20) we have f () f ( 1) ( + 1)f 1 (). By iductio the expressio o the right is zero for 0, ad the result follows. I view of (24), lie (26) is immediate from (2). 4 Legedre-Stirlig Permutatios We ow tur our attetio to the geeratig fuctios for f () ad g (), which are give by ad F (x) G (x) f ()x (27) 0 g ()x. (28) 0 By (26) ad stadard results cocerig ratioal geeratig fuctios (see [8, Cor. 4.6], for istace), there exist itegers B,j such that ad F (x) 2 1 B,jx j, ( 1), (29) (1 x) 3+1 G (x) x B,3 2 jx j (1 x) 3+1, ( 1). (30) I this sectio we give two combiatorial iterpretatios of B,j. We begi with a recurrece relatio for F (x), which we use to obtai a recurrece relatio for B,j. 6

7 Theorem 4.1 We have ad Moreover, we also have B 1,j 2δ j,1 ad F 0 (x) 1 1 x d 2 (31) F (x) x 1 x dx 2 (xf 1(x)), ( 1). (32) B,j j(j + 1)B 1,j + 2j(3 1 j)b 1,j 1 + (3 j)(3 1 j)b 1,j 2. (33) Proof. Lie (31) is immediate from (17), ad by (20) we have F (x) ( + 1)f 1 ()x + f ( 1)x 0 0 x d2 dx 2 (xf 1(x)) + xf (x), from which (32) follows. Now set 1 i (32) ad use (31) to fid that F 1 (x) 2x (1 x) ; hece B 4 1,j 2δ j,1, as claimed. To obtai (33), first use (29) to elimiate F 1 (x) o the right side of (32) ad simplify the result to fid that F (x) 2 3 j 1 j(j + 1)B 1,jx j 2 3 2(3 2) j 1 (1 x) (j + 1)B 1,jx j+1 (1 x) 3 Now use (29) to elimiate F (x) ad clear deomiators to obtai B,j x j (1 x) 2 Fially, equate coefficiets of x j to complete the proof. + (3 2)(3 1) 2 3 j 1 B 1,jx j+2 (1 x) (j + 1)jB 1,j x j +2(1 x)(3 2) We have the followig aalogue of Theorem 4.1 for G (x). Theorem 4.2 We have G 1 (x) 1 1 x ad Proof. x3 d 2 (j + 1)B 1,j x j (3 2)(3 1) B 1,j x j+2. (34) G (x) 1 x dx 2 (G 1(x)), ( 1). (3) This is similar to the proof of (31) ad (32), usig (22). Sice B 1,j 2δ j,1, lie (33) implies that B,j is a oegative iteger for all. We give two combiatorial iterpretatios of B,j. The first is ispired by Riorda s iterpretatio [7, p. 9] of similar umbers arisig i the study of the usual Stirlig umbers, which he gives i terms of trapezoidal words. Defiitio 4.3 For ay positive iteger, a Legedre-Stirlig word o 2 letters is a word a 1 a 2 a 2 such that for all j with 1 j, the etries a 2j 1 ad a 2j are distict umbers from amog 1, 2,..., 3j 1. Theorem 4.4 The umber of Legedre-Stirlig words o 2 letters with exactly j + 1 differet etries is B,j. 7

8 Proof. Let b,j deote the umber of Legedre-Stirlig words o 2 letters with exactly j + 1 differet etries. The umbers B,j are determied by (33) ad the fact that B 1,j 2δ j,1, so it is sufficiet to show that b,j also satisfies these coditios. The oly two Legedre-Stirlig words o 2 letters are 12 ad 21, so b 1,j 2δ j,1. Now suppose > 1. Every Legedre-Stirlig word o 2 letters with exactly j + 1 differet etries may be uiquely costructed by choosig a Legedre-Stirlig word o 2 2 letters ad appedig two distict umbers a 2 1 ad a 2 from amog 1, 2,..., 3 1. To esure the resultig word has exactly j + 1 differet etries, we may start with a word with exactly j 1 differet etries ad apped two umbers which do ot already appear, we may start with a word with exactly j differet etries ad apped oe umber which already appears ad oe which does ot, or we may start with a word with exactly j + 1 differet etries ad apped two umbers which already appear. These costructios may be carried out i (3 j)(3 1 j)b 1,j 2, 2j(3 1 j)b 1,j 1, ad j(j + 1)b 1,j ways, respectively, ad the result follows. Our secod iterpretatio of B, is ispired by similar results cocerig the Euleria umbers ad the usual Stirlig umbers. I particular, if a () ad A (x) 0 a ()x the there are oegative itegers A,j such that A (x) A,jx j, ( 1). (1 x) +1 Moreover, these A,j are the Euleria umbers, so A,j is the umber of permutatios i S with exactly j descets. Similarly, Gessel ad Staley [] have show that if c () { } + ad C (x) 0 c ()x the there are oegative itegers C,j such that C (x) C,jx j, ( 1). (1 x) 2+1 Moreover, Gessel ad Staley have give a set of permutatios of a certai multiset such that C,j is the umber of these permutatios with exactly j descets. I view of these results, we would lie a iterpretatio of B,j ivolvig descets i a family of permutatios. Defiitio 4. For each 1, let M deote the multiset M {1, 1, 1, 2, 2, 2,...,,, }, i which we have two ubarred copies of each iteger j with 1 j ad oe ubarred copy of each such iteger. The a Legedre-Stirlig permutatio π is a permutatio of M such that if i < j < ad π(i) π() are both ubarred, the π(j) > π(i). A descet i a Legedre-Stirlig permutatio π is a umber i, 1 i 3, such that i 3 or π(i) > π(i + 1). Theorem 4.6 The umber of Legedre-Stirlig permutatios of M with exactly j descets is B,j. Proof. Let b,j deote the umber of Legedre-Stirlig permutatios of M with exactly j descets. As i the proof of Theorem 4.4, it is sufficiet to show that b,j satisfies the same recurrece ad iitial coditios as B,j. The oly two Legedre-Stirlig permutatios of M 1 are 111 ad 111; each of these has oe descet, so b 1,j 2δ j,1. Now suppose > 1. Every Legedre-Stirlig permutatio of M may be costructed by choosig a Legedre-Stirlig permutatio of M 1, isertig betwee two etries, ad the isertig the pair betwee two etries of this ew permutatio. We may esure the resultig permutatio has exactly j descets i four ways. The first way is to choose a permutatio of M 1 with j descets, isert immediately after a descet, ad isert immediately after a descet or immediately before. I this case there are b 1,j ways to choose the iitial permutatio, j ways to isert, ad j + 1 ways to isert. The secod way is to choose a permutatio of M 1 with j 1 descets, isert immediately after a descet, ad isert immediately after a odescet, but ot immediately to the left of. I this case there are b 1,j 1 ways to choose the iitial permutatio, j 1 ways to isert, ad 3 1 j ways to isert. 8

9 The third way is to choose a permutatio of M 1 with j 1 descets, isert immediately after a odescet, ad isert immediately after a descet or immediately to the left of. I this case there are b 1,j 1 ways to choose the iitial permutatio, 3 1 j ways to isert, ad j + 1 ways to isert. The fourth way is to choose a permutatio of M 1 with j 2 descets, isert immediately after a odescet, ad isert immediately after a odescet, but ot immediately to the left of. I this case there are b 1,j 2 ways to choose the iitial permutatio, 3 j ways to isert, ad 3 1 j ways to isert. Combiig all of these, we fid that as desired. b,j j(j + 1)b,j + 2j(3 1 j)b 1,j 1 + (3 j)(3 1 j)b 1,j 2, We coclude this sectio with a bijective proof of Theorem 4.6. I particular, we give a bijective proof that 2 1 f ()x b,jx j, (36) (1 x) where b,j is the umber of Legedre-Stirlig permutatios of M with exactly j descets. Recall from the Itroductio that we have a combiatorial iterpretatio of f () i terms of set partitios; we ow give a combiatorial iterpretatio of the coefficiet of x i the expressio o the right. For ay Legedre-Stirlig permutatio π, writte i oe-lie otatio, let the spaces of π be the spaces betwee cosecutive etries of π, alog with the space before the first etry ad the space after the last etry. The a slashed Legedre-Stirlig permutatio is a Legedre-Stirlig permutatio i which spaces may cotai oe or more slashes. For example, \\121\12\\\2 is a slashed Legedre-Stirlig permutatio of M 2. For ay, 0, let P, deote the set of slashed Legedre-Stirlig permutatios of M with slashes, i which every descet cotais at least oe slash. The we have the followig expressio for the geeratig fuctio for P,. Lemma 4.7 For all 1 we have 2 1 P, x b,jx j (1 x) Proof. Note that we ca uiquely costruct all slashed Legedre-Stirlig permutatios of M by choosig a Legedre-Stirlig permutatio of M, isertig a slash ito each descet, ad the isertig arbitrarily may slashes ito each of the spaces. Thus, 2 1 P, x b,j x j ( 1 + x + x 2 + )3+1 as desired b,jx j (1 x) 3+1, Bijective Proof of Theorem 4.6. I view of Lemma 4.7, it is sufficiet to give a bijectio betwee P, ad the set of Legedre-Stirlig set partitios of [ + ] 2 ito blocs. To begi, we first observe that every slashed Legedre-Stirlig permutatio i P, may be uiquely costructed as follows. Begi with a (possibly empty) row of slashes; these will be the slashes which do ot appear betwee ay two js i our fial slashed Legedre-Stirlig permutatio. Now for each j, 1 j, first isert j to the left of a slash, the isert jj to the left of j or to the left of a slash, ad the isert a (possibly empty) row of slashes betwee j ad j. To describe the image of a give slashed Legedre-Stirlig permutatio π uder our bijectio, we describe how to costruct this image as we costruct π. First umber the slashes i our iitial row of slashes 1, 2,..., m, from left to right, ad begi the Legedre-Stirlig partitio with blocs {i 1, i 2 }, where 1 i m. Whe we isert j immediately to the left of slash r, we put copy 1 of the smallest uused umber ito the 9

10 bloc whose smallest elemets are r 1 ad r 2. Whe we isert jj immediately to the left of slash s, we put copy 2 of the smallest uused umber ito the bloc whose smallest elemets are s 1 ad s 2. If that bloc also cotais copy 1 of same umber, the we move copy 1 of that umber to the zero bloc. Whe we isert jj immediately to the left of j, we put copy 2 of the smallest available umber ito the zero bloc. Fially, whe we isert slashes betwee j ad j, we umber them cosecutively from left to right, begiig with the smallest available umber. It is ot difficult to give a recursive descriptio of the iverse of this procedure, so this map is a bijectio. The Distributio of the Number of Descets Suppose 1, ad let X deote the radom variable whose value is the umber of descets i a Legedre- Stirlig permutatio of M, chose uiformly at radom. Figure 3 shows the distributio of X whe 8 i blue, alog with the ormal distributio with the same mea ad stadard deviatio i red. Ispired by examples lie this oe, ad by aalogous wor of Bóa [3] cocerig Gessel ad Staley s Stirlig permutatios, i this sectio we prove that for each 1 the sequece {B,j } 2 1 is uimodal, ad that X approaches a ormal variable as goes to ifiity Figure 3: The distributio of X 8 ad the ormal distributio. To prove {B,j } 2 1 is uimodal, we show that the polyomial B (x) 2 1 B,j x j has distict, real, opositive roots. To do this, let C (x) be give by C (x) (1 x) 3+2 d ( x(1 x) 1 3 B (x) ), ( 1). (37) dx The table i Figure 4 gives C (x) for 1 4. Sice B (x) is a polyomial of degree 2 1, we see that C (x) 1 4x(1 + x) 2 4x(2 + 23x + 36x 2 + 9x 3 ) 3 16x(1 + 49x + 31x x x x ) 4 16x(2 + 33x + 706x x x x x x 7 ) Figure 4: The polyomials C 1 (x), C 2 (x), C 3 (x), ad C 4 (x). 10

11 C (x) is a polyomial of degree 2. Moreover, sice every oempty Legedre-Stirlig permutatio has at least oe descet, we have B (0) 0 for all 1; ow it follows from (37) that C (0) 0 for all 1. We ca ow show that the ozero roots of B (x) ad C (x) are egative, by showig they are itertwied. Theorem.1 For all 1, the polyomials B (x) ad C (x) have distict, real, opositive roots. I particular, their sequeces of coefficiets are uimodal. Proof. The result is clear for 1, sice B 1 (x) 2x ad C 1 (x) 4x + 4x 2. Now suppose > 1 ad B 1 (x) ad C 1 (x) have distict, real, opositive roots; we argue by iductio o. To see that B (x) has distict, real, opositive roots, first use (32) ad the fact that F (x) B (x) (1 x) 3+1 to show that B (x) x(1 x) 3 d ( (1 x) 1 3 C 1 (x) ). (38) dx By Rolle s Theorem, B (x) has a root strictly betwee each pair of cosecutive roots of C 1 (x); icludig 0, this accouts for 2 2 of the 2 1 roots of B (x). To fid the last root, let α < 0 deote the leftmost root of C 1 (x); by (38) we have B (α) α(1 α)c 1 (α). Sice the degree of C 1(x) is 2 2 we have lim x C 1 (x). Now sice the roots of C 1 (x) are distict we fid C 1 (α) < 0; hece B (α) > 0. But the degree of B (x) is 2 1, so lim x B (x), ad therefore B (x) has a root which is less tha α. Now it follows that B (x) has distict, real, opositive roots. The proof that C (x) has distict, real, opositive roots is similar, usig (37). It is well ow that if a polyomial has oly real, egative roots the its sequece of coefficiets is uimodal; see Wilf s boo [9, Prop ad Thm. 4.27] for a proof of this fact. We ow tur our attetio to the distributio of the umber of descets i a radomly chose Legedre- Stirlig permutatio. To state our result precisely, we itroduce some otatio. For all 1, let p (x) be the probability geeratig fuctio for X, so that p (x) 2 1 P (X j)x j, where P (X j) is the probability that X j. I additio, for all 1 let Z be the radom variable give by Z X E[X ]. Here V ar[x ] is the usual expected value of X ad E[X ] 2 1 jp (X j) V ar[x ] is the usual variace of X. We recall that (E(X ) j) 2 P (X j) 2 1 V ar[x ] E[X 2 ] E[X ] 2, ( 1). (39) I our mai result we prove that {Z } 1 coverges i distributio to the stadard ormal variable; to prove this, we use the followig result of Beder. Theorem.2 [2] Suppose X ad p (x) are as above. If all of the roots of p (x) are real ad lim V ar[x ] (40) the {Z } 1 coverges i distributio to the stadard ormal variable. ( 2 1 ) Sice p (x) B,j B (x), Theorem.1 implies all of the roots of p (x) are real. To prove (40), we first set some additioal otatio. For all positive itegers ad j, let Y B,j be the idicator variable for the evet that j is ot the bottom of a descet i a uiformly chose Legedre-Stirlig permutatio of 11

12 M. Similarly, let Y L,j (resp. Y R,j ) be the idicator variable for the evet that the left (resp. right) j is ot the bottom of a descet i a uiformly chose Legedre-Stirlig permutatio of M. Observe that X (Y B,j + Y L,j + Y R,j ). (41) We prove (40) by first obtaiig a explicit formula for V ar[x ]; as a first step, we obtai recurreces for the expected values of Y B,j, Y L,j, ad Y R,j. Lemma.3 Fix 2 ad let Y be oe of Y B, Y L, ad Y R. The we have E[Y, ] 1 ad E[Y,j ] E[Y 1,j], (1 j < ). (42) Proof. The fact that E[Y, ] 1 is immediate. For ease of expositio, suppose that Y Y B; the proof is idetical i the other two cases. To obtai (42), first ote that E[Y B,j ] is the probability that j is ot the bottom of a descet i a radomly chose Legedre-Stirlig permutatio of M. We ca obtai such a permutatio by choosig a Legedre-Stirlig permutatio of M 1 i which j is ot a descet, isertig aywhere except immediately to the left of j, ad the isertig aywhere except immediately to the left of j. Thus E[Y B,j ] E[Y B 1,j], ad (42) follows. Lemma.3 allows us to compute E[X ], which will be useful i our computatio of V ar[x ]. Propositio.4 For all 1 we have E[X ] 6 1. (43) Proof. The result is immediate for 1, so suppose > 1; we argue by iductio o. Sice expectatio is liear, by (41), Lemma.3, ad iductio we have as desired. E[X ] (E[Y B,j ] + E[Y L,j ] + E[Y R,j ]) (E[Y B 1,j ] + E[Y L 1,j ] + E[Y R 1,j ]) (3 2 E[X 1]) 6 1, The variace V ar[x ] also ivolves expected values of products of our idicator variables, so we ow fid recurrece relatios for these quatities. Lemma. Fix 2, let Y be oe of Y B, Y L, ad Y R, ad let Z be oe of Y B, Y L, ad Y R. The we have (3 4)(3 3) E[Y,i Z,j ] (3 2)(3 1) E[Y 1,iZ 1,j ], (1 i < j < ). (44) Proof. This is similar to the proof of Lemma.3. We ow have eough iformatio to compute V ar[x ]. Propositio.6 For all 1 we have V ar[x ] ( 1)( ). (4)

13 Proof. The result is immediate for 1, so suppose > 1; we argue by iductio o. I view of (39) ad (43), it is sufficiet to fid E[X 2 ]. To do this, first use (41) ad liearity of expectatio to obtai 2 E[X] 2 E (3 + 1) 2 2(3 + 1) (Y B,j + Y L,j + Y R,j ) + (Y B,j + Y L,j + Y R,j ) 2 (3 + 1) 2 2(3 + 1)E (Y B,j + Y L,j + Y R,j ) + E (Y B,j + Y L,j + Y R,j ). Now use (41) ad (43) to elimiate the expected value i the middle term o the right side, obtaiig 2 E[X] E (Y B,j + Y L,j + Y R,j ). (46) To evaluate the last term o the right, first observe that 2 (Y B,j + Y L,j + Y R,j ) Q 1 () + 2Q 2 () + Q 3 (), (47) where ad Q 2 () Q 3 () Q 1 () ( Y B 2,j + Y L 2,j + Y R,j 2 ), (Y B,i Y L,j + Y L,i Y R,j + Y R,i Y B,j ), i, (Y B,i Y B,j + Y L,i Y L,j + Y R,i Y R,j ). i, i j Sice Y B,j, Y L,j, ad Y R,j are always equal to 0 or 1, by (41) ad (43) we have Now observe that Q 2 () 2 (Y B,i + Y L,i + Y R,i ) 3 + i1 so by (41), (43), ad Lemma. we have Similarly, we fid that E[Q 1 ()] (48) 1 i, (Y B,i Y L,j + Y L,i Y R,j + Y R,i Y B,j ), E[Q 2 ()] 3 (3 4)(3 3) (6 1) + (3 2)(3 1) E[Q 2( 1)]. (49) E[Q 3 ()] 18 (3 4)(3 3) ( 1) + (3 2)(3 1) E[Q 3( 1)]. (0) Now combie (46), (47), (48), (49), ad (0) to fid that E[X 2 ] (3 4)(3 3) (3 2)(3 1) E[2Q 2( 1) + Q 3 ( 1)]. (1) 13

14 To obtai a expressio for E[2Q 2 ( 1) + Q 3 ( 1)], first replace with 1 i (46) ad (47) to obtai V ar[x 1 ] E[X 2 1] E[X 1 ] 2 1 ( ) + E[Q 1 ( 1)] + E[2Q 2 ( 1) + Q 3 ( 1)] E[X 1 ] 2. Now replace with 1 i (48) ad (43) ad use the results to elimiate E[Q 1 ( 1)] ad E[X 1 ] 2, respectively. Usig iductio to elimiate V ar[x 1 ] we fid that E[2Q 2 ( 1) + Q 3 ( 1)] 3(3 2)( ) Use this to elimiate E[2Q 2 ( 1) + Q 3 ( 1)] i (1), obtaiig Now the result follows from (43) ad (39). { } X Corollary.7 The sequece E[X ] Proof. E[X] V ar[x ] 1 coverges i distributio to the stadard ormal variable. This is immediate from Theorem.2, Theorem.1, ad Propositio.6. Acowledgmet This wor was completed durig the author s sabbatical, which was made possible by a Carleto College Hewlett Mello ad Elledge Fellowship; the author thas Carleto College for this support. The author also thas the Uiversity of Pesylvaia Mathematics Departmet, ad especially Jim Haglud, for their support ad hospitality. Refereces [1] G. E. Adrews ad L. L. Littlejoh, A combiatorial iterpretatio of the Legedre-Stirlig umbers, Proc. Amer. Math. Soc. 137(2009), [2] E. A. Beder, Cetral ad local limit theorems applied to asymptotic eumeratio, J. Combiatorial Theory, Ser. A, 1(1973), [3] M. Bóa, Real zeros ad ormal distributio for statistics o Stirlig permutatios defied by Gessel ad Staley, SIAM J. Discrete Math. 23(2008/09), [4] W. N. Everitt, L. L. Littlejoh, ad R. Wellma, Legedre polyomials, Legedre-Stirlig umbers, ad the left-defiite spectral aalysis of the Legedre differetial expressio, J. Comput. Appl. Math. 148(2002), [] I. M. Gessel ad R. P. Staley, Stirlig polyomials, J. Combiatorial Theory, Ser. A 24(1978), [6] D. E. Kuth, Two otes o otatio, Amer. Math. Mothly 99(1992), [7] J. Riorda, The blossomig of Schröder s fourth problem, Acta Math. 137(1976), [8] R. P. Staley. Geeratig fuctios. I Studies i Combiatorics. Mathematical Associatio of America, [9] H. S. Wilf. geeratigfuctioology. A. K. Peters, 3rd editio,