MIXED -STIRLING NUMERS OF THE SEOND KIND DANIEL YAQUI, MADJID MIRZAVAZIRI AND YASIN SAEEDNEZHAD Abstact The Stilig umbe of the secod id { } couts the umbe of ways to patitio a set of labeled balls ito o-empty ulabeled cells We exted this poblem ad give a ew statemet of the -Stilig umbes of the secod id ad -ell umbes We also itoduce the -mixed Stilig umbe of the secod id ad -mixed ell umbes As a applicatio of ou esults we obtai a fomula fo the umbe of ways to wite a itege m > i the fom m 1 m 2 m, whee 1 ad m i s ae positive iteges geate tha 1 1 Itoductio Stilig umbes of the secod id, deoted by { }, is the umbe of patitios of a set with distict elemets ito disjoit o-empty sets The ecuece elatio { } { 1 1 } { 1 + is valid fo >, but we also equie the defiitios 1, ad, Fo > As a alteative defiitio, we ca say that the { } s ae the uique umbes satisfyig x }, x(x 1)(x 2) (x + 1) (11) A itoductio o Stilig umbes ca be foud i [5, 9] ell umbes, deoted by, is the umbe of all patitios of a set with distict elemets ito disjoit o-empty sets Thus 1 { } These umbes also satisfy the ecuece elatio +1 ( ) See [3, 1, 9] Values of ae give i Sloae s o-lie Ecyclopaedia of Itege Sequeces [12] as the sequece A11 The sequece A8277 also gives the tiagle of Stilig umbes of the secod id Thee ae a umbe of well-ow esults associated with them i [3, 8, 5] I this pape we coside the followig ew poblem 2 Mathematics Subject lassificatio 5A18, 1173 Key wods ad phases Multiplicative patitio fuctio; Stilig umbes of the secod id; mixed patitio of a set 1
2 DANIEL YAQUI, MADJID MIRZAVAZIRI AND YASIN SAEEDNEZHAD oside b 1 + b 2 + + b balls with b 1 balls labeled 1, b 2 balls labeled 2,, b balls labeled ad c 1 + c 2 + + c cells with c 1 cells labeled 1, c 2 cells labeled 2,, c cells labeled Evaluate the umbe of ways to patitio the set of these balls ito cells of these types I the peset pape, we just coside the followig two special cases of the metioed poblem: b 1 b 1, c 1,, c N ad b 1,, b N, c 1 c 1 As a applicatio of the above poblem, fo a positive itege m we evaluate the umbe of ways to wite m i the fom m 1 m 2 m, whee 1 ad m i s ae positive iteges 2 Mixed ell ad Stilig Numbes of secod ids Defiitio 21 A multiset is a pai (A, m) whee A is a set ad m : A N is a fuctio The set A is called the set of udelyig elemets of (A, m) Fo each a A, m(a) is called the multiplicity of a A fomal defiitio fo a multiset ca be foud i [1] Let A {1, 2,, } ad m(i) b i fo i 1, 2,, We deote the multiset (A, m) by A(b 1,, b ) Ude this otatio, the poblem of patitioig b 1 + b 2 + + b balls with b 1 balls labeled 1, b 2 balls labeled 2,, b balls labeled ito c 1 + c 2 + + c cells with c 1 cells labeled 1, c 2 cells labeled 2,, c cells labeled ca be stated as follows Defiitio 22 Let A(b 1,, b ) ad A(c 1,, c ) The the umbe of ways to patitio balls ito o-empty cells is deoted by { } These umbes ae called the mixed patitio umbes If cells ae allowed to be empty, the we deote the umbe of ways to patitio these balls ito these cells by { } The followig esult is staightfowad Popositio 23 Let A(b 1,, b ) ad A(c 1,, c ) The, J 1 i, j i c i whee J A(j 1,, j ) 3 Two Special ases I this sectio we coside two special cases The fist case is b 1 b 2 b 1 ad c 1,, c N ad the secod case is b 1,, b N ad c 1 c 1 Note that if b 1 b 2 b 1 ad c 1, c 2 c the { } { } ad { } i1 { i } We deote i1 { i } by { } Moeove, if b 1 b 2 b 1 ad c 1, c 2 c the { } Defiitio 31 Let, ad be positive iteges, b 1 b 2 b 1 ad c 1, c 2 c 1 The we deote { } by S(,, ) These umbes ae called the mixed Stilig umbes of the secod id I this case { } is also deoted by (,, ) ad is called mixed ell umbes
MIXED -STIRLING NUMERS OF THE SEOND KIND 3 Let us illustate this defiitio by a example Example 32 We evaluate (2, 2, 2) Suppose that ou balls ae ad, ad ou cells ae, ad [ ] The patitios ae ( ) [ ], ( ) ( ) [ ], ( ) [ ], ( ) [ ], [ ] Thus (2, 2, 2) 5 Popositio 33 Let, ad be positive iteges ad { } 1 The l (,, ) ( 1) l l l Poof hoose l balls i ( l ) ways ad put them i cells i {l } ways We the have l diffeet balls ad 1 diffeet cells Each of these balls has 1 choices Thus the umbe of ways to put the emaiig balls ito the emaiig cells is ( 1) l Popositio 34 Let, ad be positive iteges The { l S(,, ) l +1 l }{ l 1 } ( 1)! Poof hoose l balls i ( l ) ways ad put them i o-empty cells i {l } ways We the have l diffeet balls ad 1 diffeet cells Thus the umbe of ways to put the emaiig balls ito the emaiig cells is ( 1)!{ l 1 } Note that we should have 1 l Popositio 35 Let, ad be positive iteges The S(,, ) ( 1) t+ε s 1 (, t, s), t s, t 1 whee Poof Let ε s { if s 1 othewise E i the set of all patitios i which i cells labeled 1 ae empty, i 1,, ; F j the set of all patitios i which the cell labeled j is empty, j 2,, The S(,, ) (,, ) ( i1 E i) ( j2 F j) + i1 E i + j2 F j We have E s F j1 F jt (, t, s), 1 s, 1 t 1, E s (,, s), 1 s, F j1 F jt (, t, ), 1 t 1 Olie Joual of Aalytic ombiatoics, Issue 11 (216), #5
4 DANIEL YAQUI, MADJID MIRZAVAZIRI AND YASIN SAEEDNEZHAD Now the Iclusio Exclusio Piciple implies the esult Popositio 36 Let, ad be positive iteges The S(,, ) S( 1,, 1) + ( 1)S( 1, 1, ) + ( 1 + )S( 1,, ) with the iitial values S(,, ) 1 ad S( 1,, ) ( 1)!{ 1 1 } Poof We have oe ball labeled 1 ad thee ae thee cases fo this ball: ase I This ball is the oly ball of a cell labeled 1 Thee ae S( 1,, 1) ways fo patitioig the emaiig balls ito the emaiig cells ase II This ball is the oly ball of a cell labeled j, 2 j Thee ae 1 ways fo choosig a cell ad S( 1, 1, ) ways fo patitioig the emaiig balls ito the emaiig cells ase III This ball is ot aloe i ay cell I this case we put the emaiig balls ito all cells i S( 1,, ) ways ad the ou ball has 1 + diffeet choices, sice all cells ae ow diffeet afte puttig diffeet balls ito them These esults ca be easily exteded to the followig geeal facts Theoem 37 Let b 1 b 1, c 1,, c N, A(b 1,, b ) ad A(c 1,, c ) The! l1 l2 l l l 1 ++l 1! l! c 1 c 2 c Theoem 38 Let b 1 b 1, c 1,, c N, A(b 1,, b ) ad A(c 1,, c ) The ( 1) (j 1,,j ), J 1 i, j i c i whee (j 1,, j ) is the umbe of i s such that j i ad J A(j 1,, j ) Theoem 39 Let b 1 b 1, c 1,, c N, A(b 1,, b ), A(c 1,, c ), A(b 2,, b ) ad j A(c 1,, c j 1, c j 1, c j+1,, c ) The { } { } (c 1 + + c ) + Theoem 31 Let b 1,, b N, c 1 c 1, A(b 1,, b ) ad A(c 1,, c ) The bj + 1 1 j1 Poof The umbe of ways to patitio b j balls labeled j ito labeled cells is ( b j+ 1 ) Now 1 the esult is obvious j1 j
MIXED -STIRLING NUMERS OF THE SEOND KIND 5 Theoem 311 Let b 1,, b N, c 1 c 1, A(b 1,, b ) ad A(c 1,, c ) The bj + i 1 1 i( 1) i( i) j1 i 1 4 Applicatio to - Stilig Numbes of the secod id ad Multiplicative Patitioig The -Stilig umbes of the secod id wee itoduced by ode [4] fo positive iteges,, The -Stilig umbes of the secod ids { } ae defied as the umbe of patitios of the set {1, 2,, } that have o-empty disjoit subsets such that the elemets 1, 2,, ae i distict subsets They satisfy the ecuece elatios i { } <, ii { } { 1 } 1 + { 1 1 } >, iii { } { } ( 1){ 1 1 } 1 1 The idetity 11 exteds ito (x + ) + x(x 1)(x 2) (x + 1) + The odiay Stilig umbes of the secod id ae idetical to both -Stilig ad 1- Stilig umbes We give a theoem i which we expess the -Stilig umbes i tems of the Stilig umbes Theoem 41 Fo positive iteges, ad S(, + 1, ) Poof Fistly, we put the umbes of {1, 2,, } ito cells as sigletos Now, we patitios elemets to o-empty cells such that cells ae labeled The umbe of patitios of these elemets is equal to S(, + 1, ) oollay 42 Let, ad be positive iteges The { l + l l 2 l oollay 43 Fo positive itege, ad 1 1 + 1 Poof It is clea by Popositio 36 1 }! 1 + Olie Joual of Aalytic ombiatoics, Issue 11 (216), #5
6 DANIEL YAQUI, MADJID MIRZAVAZIRI AND YASIN SAEEDNEZHAD Defiitio 44 Let, ad be positive iteges The -mixed Stilig umbe of secod id is the umbe of o-empty patitios of the set {1, 2,, } to A(c 1,, c ) such that the elemets 1, 2,, ae i distict cells We deote the -mixed Stilig umbe of secod id by { } Example 45 We evaluate { 4 2 }A(2,1) 2 Suppose that set of ou balls is {1, 2, 3, 4} ad ou cells ae, ad [ ] The patitios ae (1, 4) (2) [3], (1) (2, 4) [3], (1) (2) [3, 4], (1, 4) (3) [2], (1) (3, 4) [2], (1) (3) [2, 4], (3, 4) (2) [1], (3) (2, 4) [1], (3) (2) [1, 4], (1, 3) (2) [4], (2, 3) (4) [1], (1) (2, 4) [3], (1) (2, 3) [4], (2) (4) [1, 3], (1) (4) [2, 3], so that { 4 2 }A(2,1) 2 15 Theoem 46 Let, ad be positive iteges If c 1 t, c 2 c 1, the ( 1 l i i mi{t,} i l 1+t ) ( i)!( l) S(l, 1 + t, t i) Poof We have balls with labels 1 to ad cells such that t cells ae ulabeled We patitios this balls i two steps i such a way that the umbes of {1, 2,, } ae put ito cells as sigletos Step I Let i be the umbe of the t cells of the fist id which cotais some of the balls 1, 2,, The i mi{t, } We choose i balls of the balls 1, 2,, ad put them ito i cells of the t cells of the fist id i ( i ) ways The we choose i cells of the 1 diffeet cells ad put the emaiig i balls ito them i )( i)! ways Thus if N is the umbe of patitios of these balls ito cells i ( 1 i this way, the N i1 ( i )( 1 i ) ( i)! Step II Now, we have diffeet balls ad t + 1 cells amog which thee ae t i cells ae of the fist id ad the emaiig cells ae diffeet Note that we ow have 1 + t empty cells Pio to aythig, we fill these empty cells by l balls of the balls Thus 1 + t l hoose l balls i ( l ) ways ad put them ito cells i such a way that thee ae o empty set The umbe of ways is S(l, 1 + t, t i), by Popositio 34 The put the emaiig l diffeet balls ito cells which cotais the balls 1, 2,, oollay 47 Let, ad be positive iteges If c 1,, c N ad A(c 1,, c ), the! i i 1 ++i 1! i!
MIXED -STIRLING NUMERS OF THE SEOND KIND 7 The -ell umbes, with paametes is the umbe of the patitios of a set {1, 2,, } such that the elemets 1, 2,, ae distict cells i each patitio Hece +, + It obvious that, The ame of -Stilig umbes of secod id suggests the ame fo -ell umbes with polyomials +, (x) x, + which is called the -ell polyomials [11] Theoem 48 Let, ad be positive iteges The l, l l l Poof Put the umbes 1, 2,, i cells as sigletos Now, we patitio elemets ito cells such that cells ae labeled Thus, (, + 1, ) ( } l l ){ l l Defiitio 49 Let, ad be positive iteges The -mixed ell umbe is the umbe of patitios of the set {1, 2,, } to A(c 1,, c ) such that the elemets 1, 2,, ae i distict cells We deote the -mixed ell umbe by, Theoem 41 Let, ad be positive iteges If c 1 t, c 2 c 1, the, i ( i)! i (, + i 1, t i) i i1 Poof We have balls with labels 1 to ad cells such that t cells ae ulabeled We patitio these balls ito cells such that the umbes 1, 2,, ito cells ae as sigletos Thee ae ( i ) ways to choose elemet i 1, 2,, The umbe of patitios of these balls is equal to i ( i)! i i i1 y Popositio 33, the umbe of patitios of labeled balls ito + i 1 cells such that t i cells ae ulabeled is equal to (, + i 1, t i) Olie Joual of Aalytic ombiatoics, Issue 11 (216), #5
8 DANIEL YAQUI, MADJID MIRZAVAZIRI AND YASIN SAEEDNEZHAD oollay 411 Let, ad be positive iteges, the {, Popositio 412 Let ad be positive iteges If A(c 1,, c ), the,! i 1! i! i 1 ++i We ow give a applicatio of Theoem 31 ad Theoem 311 i multiplicative patitioig Let be a positive itege A multiplicative patitioig of is a epesetatio of as a poduct of positive iteges Sice the ode of pats i a patitio does ot cout, they ae egisteed i deceasig ode of magitude Theoem 413 Let m p α 1 1 pα be a positive itege, whee p i s ae diffeet pime umbes The i the umbe of ways to wite m as the fom m 1 m, whee 1 ad m i s ae positive iteges is αj + 1 ; j1 1 ii the umbe of ways to wite m as the fom m 1 m, whee 1 ad m i s ae positive iteges geate tha 1 is αj + i 1 1 i( 1) i( i) j1 [1] M Aige, ombiatoial Theoy, Spige, 1979 Refeeces } i 1 [2] H elbachi, M Mihoubi, A geealized ecuece fo ell polyomials: A alteate appoach to Spivey ad Gould Quaitace fomulas Euopea J ombi 3 (29), 1254 1256 [3] D aso, Stilig umbes ad ell umbes, thei ole i combiaoics ad pobability, Mathematical Scietist, 25 (2), 1-31 [4] A Z ode, The -Stilig umbes Discete Math, 49(1984), 241 259 [5] A haalambides, Eumeative ombiatoics, hapma & Hall, 22 [6] J N Daoch, O the distibutio of the umbe of successes i idepedet tials, A Math Stat (1964), 1317-1321 [7] R L Gaham, D E Kuth ad O Patashi, ocete Mathematics, Secod Editio, Addiso-Wesley, 1998 [8] T Masou, ombiatoics of Set Patitios, hapma & Hall/R, Taylo & Facis Goup, oca Rato, Lodo, New Yo, 212 [9] T Masou ad M Scho, ommutatio Relatios, Nomal Odeig ad Stilig Numbes, hapma & Hall/R a impit of Taylo & Facis LL, 215 [1] I Mezo, The -ell umbes, J Itege Seq 14 (211), Aticle 1111
MIXED -STIRLING NUMERS OF THE SEOND KIND 9 [11] M Mihoubi, H elbachi, Liea ecueces fo -ell polyomials, To appea i Austalas, J ombi [12] N J A Sloae, edito, The O-Lie Ecyclopedia of Itege Sequeces, published electoically at https://oeisog, [216] Depatmet of Pue Mathematics, Fedowsi Uivesity of Mashhad, P O ox 1159, Mashhad 91775, Ia E-mail addess: mizavazii@umaci, mizavazii@gmailcom E-mail addess: daiel_yaqubi@yahooes Olie Joual of Aalytic ombiatoics, Issue 11 (216), #5