Advaces i Applied Mathematics 4 2008) 293 306 www.elsevie.com/locate/yaama O the maximum of -Stilig umbes Istvá Mező Depatmet of Algeba ad Numbe Theoy, Istitute of Mathematics, Uivesity of Debece, Hugay Received 3 Octobe 2007; accepted 29 Novembe 2007 Available olie 2 Apil 2008 Abstact Detemiig the locatio of the maximum of Stilig umbes is a well-developed aea. I this pape we give the same esults fo the so-called -Stilig umbes which ae atual geealizatios of Stilig umbes. 2008 Elsevie Ic. All ights eseved. MSC: B73 Keywods: Stilig umbes; -Stilig umbes; Uimodality; log-cocavity. Itoductio The Stilig umbe of the fist id [ m] gives the umbe of pemutatios of elemets fomed by exactly m disjoit cycles. They satisfy the ecuece elatio δ 0 0, ) +. ) m m m As a equivalet defiitio, the umbes ) ae the coefficiets of the ext polyomial: x xx + )x + 2) x + ). 2) E-mail addess: imezo@math.lte.hu. URL: http://www.math.lte.hu/algeba/mezo.htm. Peset addess: Uivesity of Debece, H-400, Debece, P.O. Box 2, Hugay. 096-8858/$ see fot matte 2008 Elsevie Ic. All ights eseved. doi:0.06/j.aam.2007..002
294 I. Mező / Advaces i Applied Mathematics 4 2008) 293 306 The Stilig umbe of the secod id, deoted by { m}, eumeates the umbe of patitios of a set with elemets cosistig of m disjoit, oempty sets. The followig ecuece elatio holds: δ 0 0, A alteative defiitio ca be give by the fomula x m +. 3) m m m xx )x 2) x + ). 4) A excellet itoductio to these umbes ca be foud i [9]. A sequece a,a 2,...,a is said to be uimodal [25] if its membes ise to a maximum ad the decease, that is, thee exists a idex such that ad a a 2 a, a a + a. A stoge popety, called log-cocavity, implies the uimodality. The sequece a,a 2,...,a is called log-cocave whe a 2 a +a 2,..., ), 5) ad it is called stogly log-cocave whe thee is stict iequality i the above expessio. Newto s iequality [8] gives a simple test to veify the stog log-cocavity. Theoem Newto s iequality). If the polyomial a x + a 2 x 2 + +a x has oly eal oots the a 2 a +a + 2,..., ). This immediately implies the stict vesio of 5). Cosideig 2), a immediate cosequece is that the sequece ) is stictly logcocave fo all. Accodig to the wo of Hammesley [] ad Edős [8], much moe is tue. Namely, the idex K of the maximal Stilig umbe of the fist id is uique fo all fixed >2: < < < < > > >. 2 K K K + Moeove, the maximizig idex is detemied by [ K log + ) + γ + ζ2) ζ3) log + ) + γ 3/2 + ] h log + ) + γ 3/2) 2,
I. Mező / Advaces i Applied Mathematics 4 2008) 293 306 295 whee [x] deotes the itege pat of x, ζ is the Riema zeta fuctio, γ 0.5772... is the Eule Mascheoicostat ad. <h<.5. As Edős emaed, this ca be simplified whe >88: [ log ] <K < [log ]. 6) 2 The situatio chages fo Stilig umbes of the secod id. Thee is o exact closed fom fo the maximizig idex K. What is moe, we do ot ow whethe it is uique o ot. Although K 2 is ot uique, sice { { 2 } 2 } 2, i 973 Wege [23] cojectued that fo all 3the idex K is uique. Accodig to the pape [4], thee is o couteexample fo 3 <<0 6.Oe thig is cetai, the Stilig umbes of the secod id fom a stogly log-cocave sequece [4,7,8,20]. The papes [0,2,4 6,23] cotai a umbe of estimatios fo K. The most exact without ay appoximative tem) was give by Wege [23]: K < log log log 3), log <K 8). 7) Asymptotic popeties of the maximizig idex wee poved i [9,2] ad eve by statistical tools i [3]: K log, i the sese that thei quotiet teds to as teds to ifiity. We ema that this appoximatio ca be give usig the esult i [4]. It is show that K { e ), e ) }, whee ) was defied implicitly by the equatio )e ). Sice ) is ow as the Lambet W fuctio [5] we ca use its appoximatio [5, p. 349]: This meas that Wz) log z log log z + e ) e W) log log z log z ) log log z 2 + O z > 3). log z log e log log log log.
296 I. Mező / Advaces i Applied Mathematics 4 2008) 293 306 2. Notio of -Stilig umbes I the pevious sectio we have itoduced the poblems o the maximum of Stilig umbes ad peseted the solutios. Now we exted the poblem to the atual geealizatio of Stilig umbes as follows. Fo ay positive itege the symbol [ ] m deotes the umbe of those pemutatios of the set {, 2,...,} that have m cycles such that the fist elemet ae i distict cycles. The ecuece elatio is the same that of odiay Stiligs 0, <, m δ m,, m ] ) [ m +, >. 8) m m A double geeatig fuctio is give i [3]: ) + x z + 0!. 9) z) +x Let us itoduce the -Stilig umbes of the secod id. { } m deotes the umbe of those patitios of the set {, 2,...,} that have m oempty, disjoit subsets, such that the fist elemets ae i distict subsets. The usual ecuece is agai the same. 0, <, m δ m m,, m +, >. 0) m m m The idetity 4) tus to be x + ) + xx ) x + ). ) + Oe ca idetify the odiay Stiligs to -Stiligs via m m m 0 m m m A ice, itoductoy pape was witte by Bode [3]. 0,.
I. Mező / Advaces i Applied Mathematics 4 2008) 293 306 297 The questio aises immediately: what is tue fom the esults of the fist sectio with espect to -Stilig umbes? I the followig sectios we give the aswe. 3. Results fo -Stilig umbes of the fist id Theoem 2. The sequece [ +] + ) is stogly log-cocave ad thus uimodal). Poof. Let us defie the followig polyomial: P, x) : + x. 2) + It is woth to shift the idices by to avoid the edudat zeos, sice 0if<.The expoetial geeatig fuctio of P, x) is give i 9), whece Compaig the coefficiets, z) +x 0 + x + ) z 0 P, x) z.! ) + x + P, x)! x + )x + + ) x + + ). 3) Theefoe the oots of P, x) ae eal. Applyig Newto s iequality, the poof is complete. I what follows let K, deote the maximizig idex of the sequece ) the uppe idex efes to the id). To fid the estimatio of K, we have to ema that the umbes fo [ a fixed, ae the elemetay symmetic fuctios of the umbes,...,, while the umbes ] ae the elemetay symmetic fuctios of the umbes,..., see [3,8]). That is, fo a fixed, the 0-)Stilig umbes ae the sums of the poducts of the fist atual umbes tae at a time ad -Stilig umbes ae the sums of the poducts of the,..., atual umbes tae at a time. This was detailed i [3]: i i 2 i, 0). 4) i <i 2 < <i < Now we cite a theoem of Edős ad Stoe [8]. Theoem 3 P. Edős ad A.H. Stoe). Let u <u 2 < be a ifiite sequece of positive eal umbes such that i u i ad <. u 2 i i
298 I. Mező / Advaces i Applied Mathematics 4 2008) 293 306 Deote by Σ, the sum of the poduct of the fist of them tae at a time ad deote by K the lagest value of fo which Σ, assumes its maximum value. The K [ i u i u 2 i i + ) ] + o). u i It is obvious fom 4) that Σ, 5) with the sequece u, u 2 +,... As a cosequece, we get the paallel esult of 6): Theoem 4. The lagest idex fo which the sequece ) assumes its maximum is give by the appoximatio ) K, [log + ] + o). Poof. If we choose u, u 2 +,...the, by 5), the maximizig idex K, equals to + [ + + + + ] + i ) + i) + o) i + [ log ) + o) ], sice it is well ow that + 2 + + log + γ + o). The additive tem comes fom the fact that the fist ozeo symmetic fuctio belogs to the idex i the sequece ). Example 5. We give a elemetay applicatio: the maximal elemet of the sequece ) 30 30 3 belogs to the idex ) 30 K30,3 [log 3 + 3 ] 3 + o) 5. Ideed, 30.259 0 3 5 3 is maximal, as oe ca see with ay compute algeba system usig the ecuece elatios 8).
I. Mező / Advaces i Applied Mathematics 4 2008) 293 306 299 4. Results fo -Stilig umbes of the secod id To fomulate ou esults, we pove the followig theoem. Theoem 6. The sequece { +} + ) is stogly log-cocave. Poof. As befoe, we defie the polyomial Usig the ecuece elatio 0), B, x) : + B, x) + ) x + + x + ) + x. 6) + + + ) x + + x + + xb +, x) x x B, x) ) + xb, x). x Fom this we get a ecuece elatio to the polyomials B, x): ) B, x) x x B,x) + B, x) + B, x). 7) This equatio implies the idetity e x x B, x) x e x x B, x) ). 8) x Moeove, by the defiitio 6), B, x) > 0ifx 0. We pove the emaiig pat by iductio. Sice B, x) x +, its oot is eal ad egative). Now assume that all of the oots of B, x) ae eal ad egative. So Rolle s theoem gives that o the ight-had side of 8) thee ae egative oots beside the oot x 0 with multiplicity. Because the fuctio o the left-had side must have exactly + fiite oots, the missig oe caot be complex. Sice B, x) > 0ifx 0, it must be egative, too. Newto s theoem completes the poof. Rema 7. The Bell polyomials ae defied as B x) x. The Bell umbes ae B B 0). Theefoe the defiitio 6) ca be cosideed as a geealizatio of these umbes ad polyomials i the special case B x) B,0 x).
300 I. Mező / Advaces i Applied Mathematics 4 2008) 293 306 We cotiue with the followig lemma which is a patial geealizatio of the so-called Bofeoi iequality see [23,24]). Lemma 8. We have m + ) m! m + ) m )! + < < m + m + ), m! fo all m>0. Poof. Eq. ) yields that m + ) m + m! + m )!. Hece m m + ) m! + + + m )! +, m + theefoe the iequality o the ight-had side is valid. Applyig ) agai, we get + m + ) > m + m! m + ) m! m + + m )! m + ). m )! Sice the sequece ) is stogly log-cocave, thee exists a idex K2, fo which < K, 2 K, 2 > K, 2 + >. Now we give estimatios of the maximizig idex K, 2 fo -Stilig umbes of the secod id. Theoem 9. Let K, 2 be the geatest maximizig idex show above. The K, 2 < + 3), log ) log log ) ) log ) <K2, + max 8, log 2/ log + /).
I. Mező / Advaces i Applied Mathematics 4 2008) 293 306 30 Poof. It is coveiet agai to shift the idices. To pove the uppe estimatio, we apply [3, Eq. 32)]: theefoe + + + + + + j0 j0 ){ j j } ) [ { j j j, } ] j j. The tems { j } { j } ae suely egative if >K2, because of the stog log-cocavity of Stilig umbes of the secod id ad the fact that K, 2 K2, fo all see [7,20]). Thus { + + } < { + } + fo all >K2,, whece K2 +, <K2, follows. Wege s estimatio i 7) validates the uppe estimatio. To pove the lowe estimatio, we use the geealized Bofeoi s iequality stated i Lemma 8 above. Fo the sae of simplicity, let us defie M by K+, 2. The Let us itoduce the fuctio ad its logaithm + + 0 > M + M M + ) M + )! M M )! M M )! M + ) ) M )! M + 2M. 9) f, x) : x /x 2x ), g, x) : log f, x) ) log x logx ) log 2 log ). x The it is obvious that the last pat of 9) ca be witte i the fom M + ) ) M )! M + 2M M )! f,m + ). 20) Fist, we detemie the umbe of oots of f, x). Iff, x) 0 the + ) 2x ). x
302 I. Mező / Advaces i Applied Mathematics 4 2008) 293 306 The left-had side is a stictly deceasig ad the ight-had side is a stictly iceasig fuctio of x, so thee is at most oe solutio. But f, + )>0 if> log 2 log + /) 2) accodig to 9), all of the iteestig values of x ae ot less tha + ) ad lim x f,x), theefoe f, x) must have at least oe oot. Cosequetly, f, has exactly oe oot Z,, say, ad f, x) > 0ifx<Z, ad f, x) < 0ifx>Z,. Cosideig 20) we get that M + >Z,. O ca easily see that the sig of g, x) is the same as of f, x) fo all x ad thus g, Z, ) 0, too. We collect these esults i the ext fomula: > 0, <x<z,, f, x), g, x) 0, x Z,, < 0, x >Z,, if the coditio ude 2) holds fo. The fuctio g, x) ca be sepaated ito the tems g, x) h x) log ). x The fuctio h x) was examied i the pape of Wege [23] ude the otatio g,2 x) ad he poved that ) h log + > 0 8) 22) ad thus the zeo of h is geate tha log +. Sice h has the same mootoicity as g, see [23] agai), the oot of g, x) is geate tha the oot of h x) because the secod tem log /x) is positive fo x>. Thus / log + <Z, <M+. Collectig the ecessay coditios o see 2), 22)) ad cosideig that M K+, 2, the poof is complete. Example 0. We give a applicatio of this case, too. The theoem states that 2.78 50 log 50 <K2 58,8 < 50 log 50 log log 50 9.62. I fact, ad this is eally the maximal. 58 9.687 0 55, 9 8
I. Mező / Advaces i Applied Mathematics 4 2008) 293 306 303 Rema. We metioed i the poof of Theoem 9 that K+, 2 K2, o K2, +. Thee ae two poofs i [7] ad [20]. The poof i [7] ca be used without ay modificatio to pove that K+, 2 K2, o K, 2 + > ). 5. A asymptotic fomula fo -Stilig umbes of the secod id Lemma 8 eables us to give aothe esult. It is ow [7] that m m m!. Bofeoi s geealized esult yields that this asymptotic fomula has the fom The poof is staightfowad, sice + m + m + ). m! ) m + m! + m< m + m + ) <, m + ad the left-had side teds to as teds to ifiity m 0,,...). 6. Some otes o Daoch s theoem The followig useful theoem was poved by Daoch [2,6]. Theoem 2 J.N. Daoch). Let Ax) a x be a polyomial that has eal oots oly that satisfies A)>0. I othe wods, Ax) has the fom Ax) a x + j ), whee j > 0. Let K be the leftmost maximizig idex fo the sequece a 0,a,...,a ad let The we have j μ A ) A) j +. j K μ <.
304 I. Mező / Advaces i Applied Mathematics 4 2008) 293 306 I the poof of Theoem 2 we ca fid that P, x) : theefoe we immediately get the ext coollay. + x x + )x + + ) x + + ), + Coollay 3. Daoch s theoem yields that K +, + + + 2 + + ) <, + which is the same as the cosequece of Edős theoem Theoem 4). The case of Stilig umbes of the secod id is a bit moe difficult. We poved ealie see 7)) that Thus B, x) B +,x) x B,x) x B, x). μ B, ) B, ) B +, B, + ). Coollay 4. We have ) K2 +, B+, + ) <, B, which is a staight geealizatio of Hape s esult [3]. 7. Nomality of -Stilig umbes As a othe applicatio of the eal zeo popety of the polyomials 2) ad 6) we pove that the coefficiets of these polyomials the -Stilig umbes ae omally distibuted. Let a ) be a tiagula aay of oegative eal umbes,, 2,...; 0,,...,mm depeds o ). Let X be a adom vaiable such that PX ) p ) a ) mj0 a j), ad let g x) p )x.
I. Mező / Advaces i Applied Mathematics 4 2008) 293 306 305 We use the otatio X X EX))/ VaX). Fially, X N 0, ) meas that X coveges i distibutio to the stadad omal vaiable. Oe ca ead moe o these otios i [22]. A applicatio of the followig theoem will be give. Theoem 5. See E.A. Bede [].) Usig the otatios as above, if g x) has eal oots oly, ad σ VaX ) m i ) i the X N 0, ).Hee ) i ) s ae the oots of g x). ) i + ), 2 The Stilig umbes of the fist ad secod id ae omal i this sese. These facts wee poved by Gochaov ad Hape [22], espectively. We pove that these statemets stad fo -Stilig umbes, too. Fist, let a ) [ +] +. The, because of 3), + σ + + ) 2. So the coditios of Bede s theoem ae fulfilled. A esult of Rucisi ad Voigt [22, p. 223] says that if x a )x c 0 ) x c ) holds fo some oegative aithmetic pogessio c 0,c,..., the the aay a ) is omal. Eq. ) with the substitutio x x immediately yields that a ) { +} + is omal. Acowledgmet I would lie to tha Pofesso Milós Bóa fo his valuable advice ad impovemets. Refeeces [] E.A. Bede, Cetal ad local limit theoems applied to asymptotic eumeatio, J. Combi. Theoy Se. A 5 973) 9. [2] M. Bóa, Real zeos ad patitios without sigleto blocs, mauscipt. [3] A.Z. Bode, The -Stilig umbes, Discete Math. 49 984) 24 259. [4] E.R. Cafield, C. Pomeace, O the poblem of uiqueess fo the maximum Stilig umbes) of the secod id, Iteges 2 2002), pape A0. [5] R.M. Coless, G.H. Goet, D.E.G. Hae, D.J. Jeffey, D.E. Kuth, O the Lambet W fuctio, Adv. Comput. Math. 5 996) 329 359. [6] J.N. Daoch, O the distibutio of the umbe of successes i idepedet tials, A. Math. Stat. 35 964) 37 32. [7] A.J. Dobso, A ote o Stilig umbes of the secod id, J. Combi. Theoy 5 968) 22 24. [8] P. Edős, O a cojectue of Hammesley, J. Lodo Math. Soc. 28 953) 232 236.
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