The Stirling triangles

Transcription

1 The Stilig tiagles Edyta Hetmaio, Babaa Smole, Roma Wituła Istitute of Mathematics Silesia Uivesity of Techology Kaszubsa, 44- Gliwice, Polad We will use hee oe moe, alteative, defiitio of the Stilig umbes of the fist id, it meas: hi sum of all possible poducts of ( ) diffeet iteges tae fom amog the - iitial oegative iteges, that is fom amog umbes,,, Abstact The pape is devoted to the elemetay discussio o the tiagles of Stilig umbes of the fist id ad the Stilig umbes as well Aim of ou ivestigatios was to extact the umeical sequeces coected with these tiagles, to veify thei pesece i OEIS ad to ty to geeate some popeties of these umbes Moeove, we have supplemeted with ew esults the pape witte by S Falco [4, the eseach object of which was the tiagle of umbes give by the biomial tasfomatios of -Fiboacci umbes Idex Tems Pascal tiagle, Stilig umbes of the fist id, -Stilig umbes of the fist id, Catala umbes, Fiboacci umbes So we have hi hi ( )!, X Y X ( )! i ( )!H, i i6 ( ) A7( ),,, X, I I NTRODUCTION Ceatio of this pape was diectly iflueced by aticle [7 which itigued us ad, i fact, imposed the subject of ou ivestigatios We became iteested i the state of owledge coceig the Pascal tiagles fo the Stilig umbes of the fist id ad the -Stilig umbes of the fist id Thus, as the goal of ou eseach we too the extactio of umeical sequeces coected with these tiagles (that is maily the sequeces of sums of elemets i the ows ad alog the atidiagoals of the give tiagle) Moeove, we veified the pesece of such sequeces i OEIS ad we udetoo the attempt to geeate some popeties of the ivestigated umbes It is woth to emphasize that we have also supplemeted with completely ew esults the pape witte by Segio Falco [4 coceig the tiagle of umbes give by the biomial tasfomatios of -Fiboacci umbes, whee H deotes the ( )th hamoic umbe, that is P H : Moeove we tae : fo evey N Let us otice that fom this defiitio, almost immediately afte multiplicatio of the moomials located o the left had side, the followig equality esults h i X x ( x)( x) ( x) x : x, () which was tae oigially by James Stilig i his moogaph Methodus Diffeetialis (7) as the defiitio of Stilig umbes of the fist id Poduct of the moomials located o the left had side of equality () defies today the so called Pochhamme symbol Fo compaiso we have the followig widely ow Newto biomial fomula X ( x) x, () II T HE S TIRLING TRIANGLE We begi by pesetig the defiitio of Stilig umbes of the fist id Defiitio : Stilig umbes of the fist id descibe the umbe of pemutatios o the elemet set possessig cycles (that is the pemutatios which may be decomposed ito exactly sepaated cycles) Thee is o oe stadad otatio fo these umbes ad fo deotig them oe ca use oe of the followig symbols: s(, ), () S, S (, ),, whee N, N, I this elaboatio we will use the last oe fom the above listed otatios descibig the geeatig fuctio fo the biomial coefficiets Let us otice that by usig defiitio oe ca easily deive the fomula (ecuece elatio fo the Stilig umbes of the fist id): h i ( ) () Copyight 7 held by the authos 5

2 I othe wods, the selectios of ( ) diffeet umbes fom amog iitial oegative iteges coespod with the sum of selectios of ( ) ( ) diffeet umbes fom amog ( ) iitial oegative iteges ad selectios of ( ) diffeet umbes fom amog ( ) iitial oegative iteges with the added umbe Let us costuct ow the Pascal tiagle fo the Stilig umbes of the fist id which will be called hecefowad as the Stilig tiagle of the fist id [ : [ [ [ [ [ [ [ [ [ that is the followig umeical tiagle zeo level fist level! secod level! thid level! fouth level!6 6 fifth level 4!4 5 5 Hece, as well as o the basis of fomula () ad i view of the alteative defiitio, the followig summatio fomula easily esults [ [ [! ( ) [ (4) Fo cotast, fom the Pascal tiagle fo the biomial coefficiets (also called by us the classic Pascal tiagle) we have the followig summatio fomula ( ) ( Obviously, the above fomula also aises easily fom the classic ecuece elatio (equivalet fomula fo () but fo the biomial coefficiet): ( ) ( ) ) ( ) Executig i the classic Pascal tiagle the summatio ove the atidiagoals we obtai the ext uexpected fomula (see [, pages 55-57): ( ) F, whee F deotes the th Fiboacci umbe Next, by executig i the Stilig tiagle of the fist id the summatio of elemets alog the atidiagoals, as it is peseted i the followig scheme (the fist colum cotaiig zeos ad oe digit oe at the zeo level is omitted hee): ,, 6 9, 4 6, , , , etc we obtai the sequece of atual umbes labeled by symbol A765 i the Sloae s OEIS ecyclopaedia I othe wods we have [ [ [ A765(), whe, ( )!, whe, fo evey,, Rema : I the classic Pascal tiagle the umbes at the give level (statig with the zeo level) epeset the coefficiets i the expasio of umbe (b ) i the give umeical base b (we assume that all the coefficiets at level ae b) which esults fom the biomial fomula () Wheeas i the Stilig tiagle of the fist id the umbes at the give level (statig with the zeo level) epeset the 6

3 coefficiets i the expasio of umbe (b ) i the give umeical base b (ude the assumptios that that all the coefficiets at level ae b) which esults fom fomula () (oe should substitute x b ad multiply by o both sides) What is iteestig, Sloe s OEIS epots the sequeces: A4477() ( ), A8548() (5 ), A7559() ( ) Rema : Segio Falco i pape [4 cosides the tiagle T of the polyomial coefficiets ( ) p () : F, j, N, j j which fom the biomial tasfoms of ows of the followig tiagle of -Fiboacci umbes whee F,, R, >, ad N : F, F, F, F, F, F, F, F, F, F, We have F, F, F,, F,, F,, fo evey N The followig esults give a supplemet fo the Falco s pape [4 We fid that the polyomials p () satisfy the double ecuece elatio { p () ( )p () q (), q () p () q (), fo evey N, whee p () q (), p (), q () Hece, afte simple algeba we get the ecusive elatio fo polyomials p () : p () p () p j (), N, j ad the ecuece elatio fo p () p () ( )p () p (), N The tiagle T has the fom We obseve also that the sum of elemets o atidiagoals of T fom thee ow sequeces defied i the OEIS, that is the sequece of all sums {,,, 4, 9, 4, 8, 47, 89, 55, 86, } A65 which satisfies the liea ecuece elatio of the thid ode a a a a, whee A65() : a,,, Ad ext the followig sequeces obtaied by bisectio of {a } : {a } {,, 9, 8, 89, 86, } A9479 {a } {, 4, 4, 47, 55, } A94789 which satisfy both the same ecuece elatio of the thid ode a 5a 6a a Let us set T [t N N If > the t We also veified that t A4( ), whee equality holds oly fo,,, 4 ad 5 t 5 A4(4) Rema : I the cotext of poblems discussed i this sectio it is also woth to metio the Naayaa umbes defied i the way give below (see [6): N(, ), N(, ), N(, ) ( )( ),, N, Ceatig the Naayaa tiagle we fid oe moe beautiful esult The sums of elemets i the ows of the Naayaa tiagle ae equal to the Catala umbes N(, ) C I esult of summatio ove the atidiagoals of Naayaa tiagle we get the geealized Catala umbes (C C CC, N, the sequece A448() i OEIS), ie N(, ) C 7

4 III R-STIRLING TRIANGLE OF THE FIRST KIND By usig the aalogical popeties of -Stilig umbes we get a little bit moe geeal esults The -Stilig umbes of the fist id ae defied as follows Defiitio : We tae [ [ [, <, δ,,, [ [ ( ), > We also tae [ [ : [ The combiatoic desciptio of these umbes is peseted i pape [ So, the umbe [ deotes the umbe of pemutatios of set {,,, } possessig exactly mutually disjoit cycles ad such that the umbes,,, belog to diffeet cycles The above desciptio implies that [ umbe of all pemutatios of set {,,, } so that the umbes,, ae i diffeet, mutually disjoit cycles Moeove, by usig the defiitio of -Stilig umbes of the fist id we easily deive the ecuece elatio [ [ [ ( ) [ fo evey, which is the geealizatio of idetity (4) Oe ca chec that we have the [!!, fo Bode i [ gave the geeatig fuctio fo the -Stilig umbes of the fist id [ { x x (x ),,, othewise, whee, let us ecall, the Pochhamme symbol x is applied This fomula esults fom the alteative defiitio of the Stilig umbes of the fist id, amely [ sum of all possible poducts of exactly ( ) diffeet atual umbes fom amog the umbes,,,, (5) (6) (7) Let us otice that [ [ [ ( )( ), A96( ),, 4, ( )( ) A55998( ), 4, 5 A7( ), fo evey, ad, 4, [ A48( ), fo evey, 4, 5 ad,, O the basis of fomula (5) we ca costuct the -Stilig tiagle of the fist id [ [ [ [ [ [ [ [ [ [ Let us coside the case fo, that is Now, by summig the elemets alog the atidiagoals i the above tiagle,,,, 6 7, 4 5 9, 6 47, , etc 8

5 we obtai the sequece of atual umbes ot existig i OEIS We have, whe m, [ [ [ (m )(m ), whe m, A (), fo evey, Notatio A, N, 9, with the empty sets, is the otatio iveted by us to deote the sequeces {A (), N, }, N, 9 ot icluded i OEIS We have veified umeically (although basig o pemises esultig fom the algebaic estimatio) that fo 5 5 the above sequece satisfies the followig iequalities ( )! < A () < ( )! Next we costuct the aalogical tiagle fo the case Thus we have Summig the elemets alog the atidiagoals i the above tiagle we get,,,,, , , ad so o The obtaied sequece of atual umbes is ot icluded eithe i OEIS We have, whe m, [ [ [ (m )(m ), whe m A (), fo evey, 4 We have veified umeically (but agai, basig o pemises esultig fom the algebaic estimatio) that this sequece satisfies fo 7 5 the followig iequalities ( 5)! < A () < ( 4)! Rema : I case of the -Stilig umbes the umbes at oe level epeset the coefficiets i the expasio of umbe (b ) i the give umeical base b (ude the assumptio that all the coefficiets at level ae b) which esults fom fomula (7) (aalogically lie i the pevious case) IV TRIANGLES FOR THE POWERS OF STIRLING NUMBERS AND THE BINOMIAL COEFFICIENTS Popeties of the classic Pascal tiagle ae studied i [8 togethe with the pesetatio of Fiboacci sequece with the aid of biomial coefficiets Let us tu ou attetio ito the Pascal tiagle fo the powes of biomial coefficiets Fo example, fo the squaes of biomial coefficiets we eceive ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) o diectly Summig the elemets alog the atidiagoals we fid, 4 5, 9, 6 9 6, 5 6 6, 6 6 5, etc 9

6 This is the sequece labeled by symbol A586 i the Sloae s OEIS ecyclopaedia I othe wods we have ( ) ( ) ( ), whe m, (m ), A586() whe m, Summig the elemets alog the atidiagoals i the aalogical tiagle fo the cubes of biomial coefficiets we obtai,, 9, 9, 9, 4, 8, 47,, that is the sequece deoted by A8545 i Sloae s OEIS ecyclopaedia I othe wods ( ) ( ) ( ), whe m, (m ), A8545() whe m, Let us coside ow the aalogical tiagles fo the Stilig umbes of the fist id Fo the stat we tae the squaes of these umbes Executig the summatio of elemets ove the atidiagoals, similaly as i the cases of pevious umeical tiagles, we get the sequece,, 4 5, , , , ad so o, that is the sequece ot icluded i OEIS [ [ [, whe, (( )!), whe, A (), fo evey,, Moeove, we have checed umeically that fo 4 5 the above sequece satisfies the iequalities! 5 < A () < (! 5 ) Next, by summig the elemets alog the atidiagoals i the Pascal tiagle fo [ we eceive the sequece,, 8 9, 6 7 4, , ad so o, that is [ [ [ A 4(),, whe, (( )!), whe, agai ot peset i OEIS As befoe we have checed umeically that fo 4 5 the above sequece satisfies the iequalities )! < A 4() < (! V CONCLUSION I the pape the umbe of elemetay popeties of the umeical tiagles coected with the Stilig umbes ad the -Fiboacci umbes is peseted We lied the discussed sequeces with the sequeces icluded i OEIS The obtaied esults summaize some state of eseach I the futue we ited to efe to these umeical tiagles fom the algebaic side, similaly lie, amog othes, the authos of papes [, [, [9, [ ad [4 did it i case of the classic Pascal tiagle I the futue we also pla to aalyze pape [ ad the daw coclusios will be icluded i ou ext publicatio Acowledgmets The Authos wat to expess thei sicee thas to the Refeees fo thei costuctive epots with may essetial emas ad advices coceig the efeeces REFERENCES [ A Z Bode, The -Stilig umbes, Discete Math 49 (984), 4 59 [ GS Call, DJ Vellema, Pascal s matices, Ame Math Mothly No 4 (99), 7 76 [ K S Davis, W A Webb, Pascal s tiagle modulo 4, Fiboacci Quat 9 (99), 79-8 [4 S Falco, O the complex -Fiboacci umbes, Coget Mathematics (6), :944 [5 RL Gaham, DE Kuth, O Patashi, Cocete Mathematics, Addiso Wesley, Readig, MA, 989 [6 RP Gimaldi, Fiboacci ad Catala Numbes, Wiley, [7 R Gzymowsi, R Wituła, Complex fuctios ad Laplace tasfomatios - examples ad execises, Pacowia Komputeowa Jaca Salmiesiego (PKJS), Gliwice (i Polish) [8 R Hoshio, Rivetig Popeties of Pascal a Tiagle, Caadia Mathematical Society, Cux Mathematicoum, 4 (Mach), 998 [9 J G Huad, B K Speama, K S Williams, Pascal s tiagle (mod 8), Euop J Combiatoics 9 (998), 45 6 [ J G Huad, B K Speama, K S Williams, Pascal s tiagle (mod 9), Acta Aith 78 No 4 (997), -48 [ C Napoli, G Pappalado, E Tamotaa, A mathematical model fo file fagmet diffusio ad a eual pedicto to maage pioity queues ove bittoet, Iteatioal Joual of Applied Mathematics ad Compute Sciece 6 No (6),

7 [ T Koshy, Fiboacci ad Lucas Numbes with Applicatios, Wiley, New Yo [ W Lag, O geealizatios of the Stilig umbe tiagles, J Itege Seq (), Aticle 4 [4 B Lewis, Revisitig the Pascal Matix, Ame Math Mothly 7 (), 5 66 [5 G Capizzi, G Lo Sciuto, C Napoli, E Tamotaa, M Wozia, Automatic classificatio of fuit defects based o co-occuece matix ad eual etwos, IEEE Fedeated Cofeece o Compute Sciece ad Ifomatio Systems (FedCSIS), [6 ASofo, Fiite sums i Pascal s tiagle, The FIboacci Quately 5 No 4 (), 7 45 [7 T Wight, Pascal s tiagle gets its gees fom Stilig umbes of the fist id, College Math J 6 No 5 (995),