Eulerian numbers revisited : Slices of hypercube

Transcription

1 Eulera umbers revsted : Slces of hypercube Kgo Kobayash, Hajme Sato, Mamoru Hosh, ad Hroyosh Morta Abstract I ths talk, we provde a smple proof o a terestg equalty coectg the umber of permutatos of,..., wth k rus,.e., Eulera umbers to the volumes of slces betwee k ad k of the -dmesoal hypercube alog the dagoal axs. The proof s smple ad elegat, but the detal structures the problem are left to be uclear. I order to get more formato o ths problem, we gve the secod proof reled o the drect calculato of the related umbers ad the volumes. By computg codtoal probabltes wth respect to slces, we ca obta the kow recurrece relato o Eulera umbers. Itroducto It s qute terestg to otce relatos betwee combatoral cocepts ad probablstc oes. We provde a typcal example of such cases, that s, a strog relato betwee the umber of permutatos wth k rus ad the probablty that the sum of depedet uform radom varables takg values betwee ad wll be betwee k ad k. I the ext secto, we expla the ma theorem ad gve a smple proof for the theorem. Although the dea of the smple proof s deed elegat, but by the laborous computatos of related values the secto 3, we ca have much more formato aroud the problem, ad ecouter terestg dettes through a explct formula of the umber of permutatos of,..., wth k rus. I the secto 4, we compute the volume of slce betwee k ad k of the -dmesoal hypercube alog the dagoal axs. The combed results of secto 3 ad 4 gve aother proof of the ma theorem. The Uversty of Electro-Commucatos, Tokyo, JAPAN, E-mal: kgo@eee.org Seshu Uversty,Tokyo, JAPAN, E-mal: h-sato@sc.seshu-u.ac.jp The Uversty of Electro-Commucatos,Tokyo,JAPAN,E-mal:mmrhosh@gmal.com The Uversty of Electro-Commucatos, Tokyo, JAPAN, E-mal: morta@s.uec.ac.jp

2 2 Number of permutatos wth specfed umber of rus ad Slces of dmesoal Hypercube Let F,k be the umber of permutatos of,..., wth k rus, where the ru s a cosecutve ascet sequece followed by a descet or termated. For example, 2453 has three rus, ad 5432 has fve rus. Next, we cosder the sum of depedet radom varables U (,..., wth uform dstrbuto o the terval [,. Thus, we wll study the probablty: G,k Pr{k U + + U < k} (2. du... du, (2.2 k u + +u <k, u <,..., u < that s, the volume of slce whch s defed by a rego betwee k ad k of the -dmesoal hypercube alog the dagoal axs. I what follows, for smplcty we omt the codtos of u <,..., u < the tegral. The, we have a fascatg equalty coectg combatorcs ad probablty theory: Theorem 2. It holds that F,k! G,k. Remark It s qute terestg that ths s ot a approxmate, but exact equalty. I [2], p.659, t cotas a wrog expresso: [z u k ] F (z, u P { U + + U < k}, where F (z, u s the expoetal BGF of the umber of permutatos wth specfed umber of rus, that s, F (z, u k F,k z u k.! I the followg sectos, we wll compute precsely the cocered values F,k ad G,k, ad show the equalty. But, t s ot ecessary to evaluate the values oly for provg the theorem. It s suffcet to traslate related polyhedra to approprate polyhedra the hypercube. Here, we gve a smple proof of Theorem 2.: Proof: Let Q be the -dmesoal hypercube, that s, Q {u (u,..., u : u <,..., u < }. Defe the permutato σ (σ,..., σ σ(u for the pot u (u, u 2,..., u Q by (u, u 2,..., u u σ > u σ2 > > u σ.

3 Next, we troduce a trasformato T from the pot u Q to the pot v Q as follows: v u u 2 v 2 u 2 u 3 T (u.. + c u, v u u v u where ad c u (c : c [u + > u ] for,...,, ad c. The, f the umber of rus the permutato σ(u s k, the t holds that < v <,. < v <, k < v + v v < k. Here, we ote that the umber of c u s the umber of descets plus oe, or the umber of rus the permutato σ(u. Thus, the u, σ(u of whch has k rus, s trasformed to the slce k v + + v < k as v T (u. Now, let U be the radom vector dstrbuted uformly Q, ad V (V,..., V T (U. The, t s clear that V s also dstrbuted uformly Q, ad t holds that Pr { σ(u has k rus } Pr {k V + + V < k}. Remark 2 Fgure depcts the stuato of 3 showg how the trasformato of polyhedra s performed. 3 Evaluato of Number of permutatos wth specfed umber of rus ad Eulera umbers We start by the explct expresso of BGF F (z, u:

4 <u < <u 2 < <u 3 < u >u 2 >u 3 u >u 3 >u 2 u 3 >u >u 2 u 2 >u >u 3 u 2 >u 3 >u u 3 >u 2 >u u 3 u u v v 2 v 3 u u 2 u 2 u 3 u A A v 3 v 2 v <v + v 2 + v 3 u 3 < <v + v 2 + v 3 2 u 3 < 2 <v < <v 2 < <v 3 < Fgure : Trasform of polyhedra the cube 2 <v + v 2 + v 3 3 u 3 < 3 Proposto 3. F (z, u u( u e (u z u. (3. Proof: Omtted. From ths BGF F (z, u, we ca get the formula o the umber of permutatos of,..., wth k rus. Thus, we ca show the followg theorem. Theorem 3. The coeffcet of z u k ( of the geeratg fucto F (z, u s gve by [z u k ]F (z, u k ( + ( (k. (3.2! Proof: By the argumet by Davd ad Barto([], we wll proceed as follows: A(z, u u e (u z u ( ue( uz e ( uz ( u u k e (k+( uz ( u k k u k! (k + ( u z

5 Thus, we have [z ] A(z, u ( u u k! (k + ( u k ( u+! s (k + u k k + ( + ( j u j (k + u k! j j k s ( + u s ( j (s j +! j! u s s j s ( + ( j (s j +. j j where we eed at the last equalty the fact that for s, t holds s ( + ( j (s j +, j j that s deduced from the proposto 3.2 at the ed of ths secto. Therefore, we ca coclude [z u k ]F (z, u [z u k ]ua(z, u k ( + ( (k.! Note that Davd ad Barto([], there s ot the explct statemet equvalet to the proposto 3.2, ad do t meto about the term of Eulera umbers. Remark 3 Note that F,k! [z u k ]F (z, u s commoly expressed as follows: where k s called as Eulera umbers. F,k k k ( ( +, (3.3 (k + Remark 4 Moreover, F,k of (3.3 s the umber of permutatos wth k rus amog! permutatos of,...,. Thus, the value of (3.2 gves the probablty

6 of occurece of permutato wth k rus whe permutatos of,..., are equally produced. From the expoetal BGF F (z, u, we ca derve the mea ad the varace of the umber of rus permutatos of,..., as µ + 2, σ Ad ts dstrbuto asymptotcally obeys the Gaussa law(refer to Proposto IX.9 of [2]. Compare ths result wth the bomal dstrbuto case that has the BGF 2 2 z(+u ( k 2 k z u k, from whch the mea ad the varace are µ 2, σ2 4, respectvely. Proposto 3.2 For all, 2,..., ad ay real α, we have for teger c satsfyg c, ( ( ( + α c. (3.4 Moreover, we have ( ( ( + α (!. (3.5 Proof: We ca prove by the ducto o, but wll omt the proof due to the restrcto of space. But we would lke to meto that t mght be dffcult to solve oly the equato (3.4, or the equato (3.5. It s effectve to solve jotly both of equatos by ducto. 4 The Volumes of Slces of Hypercube ad ther Recurso Now, we cosder the volume G,k of slce whch s defed by a rego betwee k ad k of the -dmesoal hypercube alog the dagoal axs. We start by establshg the covolutoal formula for uform dstrbuto. It s coveet to troduce a fucto called the rght-half power fucto: x m + { x m, x,, x <. (4. Proposto 4. The -th covoluto g (x of the uform dstrbuto (x [ x < ] o the terval [, s gve by the fucto: g (x (! ( ( (x +, (4.2 where [ ] s the dcator fucto takg the value f the codto brackets s true, otherwse.

7 Proof: We wll prove ths theorem by the ducto o wth the help of the followg useful relato: for ay x ad, For, we have (x t + dt ( x + (x +. (4.3 g (x x + (x + (x. (4.4 Thus, the formula (4.2 s obvously true ths case. Now we assume that the formula (4.2 s true for up to m. The (m + -th covoluto g m+ (x of the uform dstrbuto (x s the covoluto of g m wth that s computed as follows: g m+ (x g m (x g m (x ydy (m! m ( m ( (x y m + dy m ( m ( (m! ( m ( m+ m ( (x m + + m! m+ ( m + ( (x m +, m! ( (x m m + (x m + ( ( m (x m + (4.5 where we use (4.3 at the equalty (4.5. Therefore, for m +, we have show that the formula (4.2 s vald. Remark 5 Ths fucto g (x s actually a pecewse-polyomal fucto that s expressed as follows: for k x < k (k,..., g (x f,k (x def k (! ( ( (x. (4.6 Moreover, the fucto g (x s actually detcal to the scalg fucto of B-sple wavelet. By usg f,k (x we ca provde the explct formula of the volume G,k of slce of -dmesoal hypercube, ad reestablsh the ma theorem 2., F,k! G,k.

8 The secod proof of Theorem 2.: G,k k k k k f,k (xdx k (! k ( ( (! ( ( k (x dx (x dx k k ( ( {(k (k }! k ( + ( (k (4.7!,! k that s detcal to F,k /!. Moreover, wth respect to the codtoal probablty, we have a smple formula as follows: Proposto 4.2 It holds the detty, Pr{k U + + U < k k U + + U < k} Pr{k U + + U < k, k U + + U < k} Pr{k U + + U < k} k. Proof: Frst of all, we ote that t holds Pr{k U + + U < k, k U + + U < k} Pr{k U + + U < k} For the deomator, we have by the prevous equato (4.7, k k f,k (xdx k (! ( ( k α k k k f,k (xdxdα. f,k (xdx (k. (4.8

9 Next, the umerator ca be deduced as follows: k α k f,k (xdxdα k α k k ( 2! ( ( (x 2 dxdα k ( ( { (k α (k } dα (! k ( {(k ( (k (k } (! k (! (! { k k (! ( k ( } ( (k ( (k { k ( k ( k ( (k ( (k k ( } ( (k ( ( (k. (4.9 Summarzg (4.8 ad (4.9, we have establshed the proposto. From Proposto 4.2, we ca deduce the recurso of the slces of hypercube: Corollary 4. G,k k + I other words by Eulera umbers, ( k k k G,k + k G,k. + (k + k Remark 6 The meag of ths Corollary s depcted the Fgure 2. Z G,k du...du kx + ( (k A + B k appleu+ +u<k!. u A k + G,k k + k k k k k u + u u B k G,k Fgure 2: The meag of recurrece

10 Here, we show Euler s tragle: Fgure 3: Euler s tragle Ackowledgmet Oe of the authors would lke to thak Natoal Isttute of Iformato ad Commucatos Techology (NICT of Japa for the facal support to partcpate ITA24. Refereces [] F.N. Davd ad D.E.Barto, Combatoral Chace, Charles Grff & Compay Lmted, Lodo, 962. [2] P. Flajolet ad R. Sedgewck, Aalytc Combatorcs, Cambrdge Uversty Press, 29.