An Asymptotic Expansion for the Number of Permutations with a Certain Number of Inversions

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1 A Asymptotic Expasio for the Number of Permutatios with a Certai Number of Iversios Lae Clark Departmet of Mathematics Souther Illiois Uiversity Carbodale Carbodale, IL USA lclark@math.siu.edu Submitted: December 17, 1998; Accepted: August 8, Abstract Let b, k deote the umber of permutatios of {1,...,} with precisely k iversios. We represet b, k as a real trigoometric itegral ad the use the method of Laplace to give a complete asymptotic asio of the itegral. Amog the cosequeces, we have a complete asymptotic asio for b, k/! for a rage of k icludig the maximum of the b, k/!. AMS Subject Classificatio: 5A16, 5A15, 5A1 A permutatio σ =σ1,...,σ of [] ={1,...,} has a iversio at i, j, where 1 i<j, if ad oly if σi >σj. Let b, k deote the umber of permutatios of [] with precisely k iversios. The b, k =b, k for all itegers k, while, b, k ifadolyif k. Beder [; p. 11] showed that the b, k are log cocave i k. Hece, the maximum B oftheb, k occurs at k = /, aswellas / for odd. See [3; pps. 36 4] for further results. Radom permutatios show see [3; pps. 8 83], for example that the b, k satisfy a cetral limit theorem with µ = /adσ = 1 +5/7 see [; Theorem 1]. Beder [; p. 19] remarks that the theorems of Sectio 4 do ot apply to the b, k. He the shows [; p. 11] that the b, k are log cocave i k so that Lemma applies. This will give a first term asymptotic formula for b, k/! whe k = µ + xσ where x is a fixed real umber. I this paper, we represet b, k as a real trigoometric itegral. We the use the method of Laplace to give a complete asymptotic asio of this itegral i terms of the Beroulli umbers ad Hermite polyomials. Hece, we have the complete asymptotic 1

2 the electroic joural of combiatorics 7, #R5 asio b, k! { } m =6π 1/ 3/ e x / 1+ q S q H q 1/ x l m +1 m+3/ as, 1 whe k = ± x 3/ /3wherex = x l ad m is a fixed iteger at least. Here, H q are the Hermite polyomials defied before Theorem 1 ad the S q are defied i Theorem 3. I particular, we have a complete asymptotic asio for B/! whe is eve. See Corollaries, 4 for other asymptotic asios. I what follows, k, l ad are itegers with k ad l. We deote the oegative itegers by N. All asymptotic formulas are for. Muir [5] see also [3; p. 39] showed that b, k is the coefficiet of z k i l= 1 + z + + z l 1. The, b, k = 1 πi = 1 πi where C is the uit circle. Hece, b, k =! π π/ l= 1 + z + + zl 1 dz C z k 1 C l= l= z k+1 z l 1 z 1 dz, si lt cos l si t k t dt, upo parameterizig C z =e it ; t [, π] ad usig the symmetry of the itegrad. For a iteger ad real umbers a, b ad x, let I, x, a, b := b a si lt xt 3/ cos dt l si t 3 l= ad I, x :=I, x,, π where all discotiuities of the itegrad have bee removed. The gives b, k! = I, x, 3 π for all itegers k, where k, adk = ± x 3/ /3.

3 the electroic joural of combiatorics 7, #R5 3 For a oegative iteger q ad real umber x, let F q x := u /u q cosux du deote the Fourier cosie trasform of u /u q. The F q x = 1 q π 1/ q 1/ e x / H q 1/ x. Here H x are the Hermite polyomials give by H x = / k= 1k!x k/ k! k! see [4; pps. 6-64]. We use the followig Taylor series approximatios which are valid for all real umbers t. si t = t t3 6 + at; t4 at 4 for all real t ad at fort [,π]; 4 ad for a iteger m 1, cos t =1 t + bt; bt t3 for t [,π]; 5 e t =1+t + + tm 1 m 1! + c mt; c m t e t t m. 6 Of course, our error terms a, b ad c m are all ifiitely-differetiable fuctios over the reals. We also require the followig iequality itegratio by parts. For a real umber x>, We ow give our first result. x e t / dt 1 x e x /. 7 Theorem 1. For x = x l, we have the asymptotic asio π 1/ I, x =3 3/ e /{ x x4 19x x x x x } l 19 as. Proof. We use the method of Laplace. For <a 1 ad a iteger l, let M l a :=max{ si lt/ si t : t [a, π/]} ad b := cos a, 1. For all itegers l, M l a b l 1 +b l + +b+1 mi{l, 1 b 1 } by iductio o l, while a /3 1 b. Here, si lt l si t l= 1 b! 9/ 3e, a

4 the electroic joural of combiatorics 7, #R5 4 ad, hece, for all 9adallrealumbersx, e I, x, 3.5,π/. 8 3 For all itegers l ad all real umbers t with si t, 4 gives si lt/l si t = 1 l 1t /6+dl, t where dl, t l 3 t 3 /1 for t, 1] ad l. Hece, si lt < l si t 1 l t 4 l t for lt [, 1] ad l. 9 4 Naturally, we defie si lt/l si t =1whet = to remove that discotiuity. For all 144 ad all real umbers x, 9 gives I, x,.7, 3.5 I, x, 1, /3 l=.7 l= si lt dt l si t si lt dt l si t , 1, 11 ad I, x, 3/ l, 1 1 3/ l l= si lt dt l si t l 7. 1 Recall that cot t = t 1 + k=1 4k B k t k 1 /k!, for real t with < t <π.here B are the Beroulli umbers defied by z/e z 1 = = B } z /! for complex z d with z < π see [3; pps. 48, 88]. The, dt{ lsi lt/l si t = l cot lt cot t = k=1 4k B k l k 1t k 1 /k! for < lt <π, hece, l si lt = l si t 4 k B k l k t k 1 kk! k=1 for lt <π. 13 For a oegative iteger m, lt 1adl 1 see [1; p. 85], 4 k B k l k t k 1 kk! lm+ t m+. 14 k=m+1 For adθ k := l= lk 1 see [3; p. 155], 13, 14; m =3ad6;m =1

5 the electroic joural of combiatorics 7, #R5 5 give I, x,, 3/ l 3/ l { } l 1 = t + l t4 + l6 1 xt 835 t6 8 t 8 3/ cos dt 3 l= 3/ l = { θ t θ 4t 4 θ } 6t 6 xt t 8 3/ cos dt 3 3/ l = { θ t θ 4t 4 θ } 6t 6 xt 3/ l 9 cos dt / l /3 u { R u + R 4 u 4 + R 6 u 6} cosux du = 3 3/ l 9 9/, 15 upo settig u = 3/ t/3, where R = 3/4 +5/4, R 4 = 9/1 9/4 3/ 3 +93/ 5 ad R 6 = 9/45 9/7 3 9/7 4 +3/ /49 8.It is readily see that the error term i 15 is at most e 9/ l 9 for all ad all real umbers x. For u l /3, 6; m =3gives { R u + R 4 u 4 + R 6 u 6} l 18 =1+S u + S 4 u 4 + S 6 u 6 + S 8 u 8, 16 where S = 3/4 +5/4, S 4 = 9/1 +9/16, S 6 = 63/196 ad S 8 =81/. Hece, 15 ad 16 give I, x,, 3/ l = 3 l /3 { u 1+S 3/ u + S 4 u 4 + S 6 u 6 + S 8 u 8 l 18 } l 9 cosux du 3 9/ = 3 l /3 u {1+S u + S 3/ 4 u 4 + S 6 u 6 + S 8 u 8} cosux du l 19 9/ = 3 u {1+S u + S 3/ 4 u 4 + S 6 u 6 + S 8 u 8} cosux du u l 19 du 4 9/ l /3 3

6 the electroic joural of combiatorics 7, #R5 6 = 3 3/ u {1+S u + S 4 u 4 + S 6 u 6 + S 8 u 8} cosux du l 19, 17 9/ where the last equatio follows from 7. The error term i the first equatio holds uiformly for all real umbers x by the commets after 15 ad, sice cosux 1, the error term i the secod equatio holds uiformly for all real umbers x by 16 as does the error term i the third equatio ivolvig the itegral. The 8, 1 1 ad 17 give I, x = 3 { F x+s 3/ F x+s 4 F 4 x+s 6 F 6 x+s 8 F 8 x } l 19, 18 9/ where our error term holds uiformly for all real umbers x. Hece, after simplifyig 18 we obtai π 1/ I, x =3 3/ e /{ x 1 1 9x 4 19x x x x x } l 19, 19 where our error term holds uiformly for all real umbers x. Our result follows sice, apart from the error term, the smallest term i 19 has order of magitude at least 4 for x = x l. We ote several cosequeces of Theorem 1. Corollary. For x = x l, we have the asymptotic asio { b, k =6π 1/ 3/ e x / 1 1! 1 9x4 19x x x x x } l 19 as, whe k = ± x 3/ /3. We also have the asymptotic asio b, k =6π 1/ 3/ 1 51! o as, 98 7/ provided k = + o 1/ l 3/. I particular, B/! has the same asymptotic asio. 9/ 9/

7 the electroic joural of combiatorics 7, #R5 7 Proof. The asymptotic asio for b, k/! whek = ± x 3/ /3wherex = x l follows immediately from 3 ad Theorem 1. For all e 141 ad all real umbers x, 8 ad 1 1 give π/ { } si lt xt 3/ 1 cos dt 1 l. l si t 3 7 3/ l l= For a iteger l ad all t [,π/l], si lt/l si t [, 1] by iductio o l. The, for all ad all x [, l 1 ], 5 gives 3/ l 3/ l { } si lt xt 3/ 1 cos dt l si t 3 l= x t 3 18 dt = x l / Hece, for all e 141 ad all x [, l 1 ], ad 1 give I, I, x x l l. 3/ 7 Assume is eve odd is similar ad e 141.Letl:= /+ 3/ /6l so that l = + x 3/ /3withx [, l 1 ]. For k l, log cocavity of the b, k implies b, b, k b, l, so that 3 ad give b, k I, π! Hece, Theorem 1 gives b, k! π I, x l 3 7π3/ π l. 7 =6π 1/ 3/ o, 7/ for k = + o 1/ l 3/. Remark. We ca replace the o 7/ error term i the asymptotic asio of B/! witho 9/ l 19. The followig extesio of Theorem 1 the case m = 3 givig a complete asymptotic asio of I, x ca be immediately read out of its proof.

8 the electroic joural of combiatorics 7, #R5 8 Theorem 3. Fix a iteger m. For x = x l, we have the asymptotic asio { } π m 1/ I, x =3 3/ e x / 1+ q S q H q 1/ x l m +1 as. m+3/ The S q are defied i the proof. Proof. For l ad t [, 1 ], 13 ad 14 give l si lt = l si t m c k l k 1t k m+ t m+, 3 k=1 where c k := 4 k B k /kk! <, while, θ k = l k 1 = 1 k +1 l= Hece, 3 ad 6; m =1give k j= k +1 B j +1 k+1 j. j I, x,, 3/ l = 3 { l /3 m } l m+3 9 k c 3/ k θ k u k 3k cosux du m+3/ = 3 3/ l /3 k=1 u m+3/ { R u + + R m u m} cosux du l m+3, 4 where R = 3/4 +5/4 ad, for k m, R k := 36k B k kk +1! k j= k +1 B j 3k +1 k+1 j 36k B k j kk! 3k+1. The error term i 4 holds uiformly for all real umbers x. For k m 1, crude estimates see [1; p. 85] give R k 6k +1! k+1, 5

9 the electroic joural of combiatorics 7, #R5 9 i fact, R k ivolves k+1 ad smaller iteger powers of. For all m + 1 ad all u l /3, 5 gives l m R u + + R m u m mm +1!. 6 Hece, 6 ad 6 give { R u + + R m u m} =1+ where S q isthatpartof m 1 r=1 e,...,e m N m e + +e m =r e + +me m =q m S q u q R e Re m m e! e m! l m, 7 m ivolvig oly 1,..., m+1 upo asio. Here R e Re m m ivolves e + +m 1e m = q r+e ad smaller iteger powers of while q r + e m if q m 1. The, 4 ad 7 give I, x,, 3/ l = 3 3/ = 3 3/ l /3 u { 1+ m S q u q } cosux du { } u m 1+ S q u q cosux du l m +1 m+3/ l m +1 m+3/, 8 where our error term holds uiformly for all real umbers x. Hece, after simplifyig, 8, 1 1 ad 8 give { } π m 1/ I, x =3 3/ e x / 1+ q S q H q 1/ x l m +1, 9 m+3/ where our error term holds uiformly for all real umbers x. Our result follows sice, apart from the error term, the smallest term i 9 has order of magitude at least m 1 for x = x l. As a cosequece of Theorem 3, we have a complete asymptotic asio for b, k/! whek = ± x 3/ /3wherex = x l, aswellasforb/! whe is eve.

10 the electroic joural of combiatorics 7, #R5 1 Corollary 4. Fix a iteger m. For x = x l, we have the asymptotic asio { } m b, k =6π 1/ 3/ e x / 1+ q S q H q 1/ x! l m +1 as, m+3/ whe k = ± x 3/ /3. I particular, we have the asymptotic asio } m B =6π 1/ {1+ 3/ q q! l m +1 S q as,! q! m+3/ whe is eve. I the followig table we compare the exact value of B/! foud by adig the geeratig fuctio for the b, k with the approximatios give by Corollary 4 for m =, 3 for = 4 ad 8. B4/4! B8/8! Exact Value Approximatio m = Relative Error %.3539% Error as a fuctio of Approximatio m = Relative Error.9176%.1166% Error as a fuctio of Ackowledgemet. I wish to thak the referee for umerous commets ad suggestios which have led to a substatially improved paper.

11 the electroic joural of combiatorics 7, #R5 11 Refereces [1] M. Abramowitz ad I.A. Stegu, Eds., Hadbook of Mathematical Fuctios with Formulas, Graphs ad Mathematical Tables, Dover Publicatios, New York, [] E.A. Beder, Cetral ad Local Limit Theorems Applied to Asymptotic Eumeratio, J. Combiatorial Theory A , [3] L. Comtet, Advaced Combiatorics, D. Reidel, Bosto, [4] N.N. Lebedev, Special Fuctios ad Their Applicatios, Dover Publicatios, New York, 197. [5] T. Muir, O a Simple Term of a Determiat, Proc. Royal Society Ediburg ,