Idia J Pure Appl Math 45): 75-89 February 204 c Idia Natioal Sciece Academy A GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS Mircea Merca Departmet of Mathematics Uiversity of Craiova A I Cuza 3 Craiova 200585 Romaia e-mail: mirceamerca@profifoeduro Received 3 September 202; after fial revisio 7 March 203; accepted 3 May 203) A geeralizatio for the symmetry betwee complete symmetric fuctios ad elemetary symmetric fuctios is give As corollaries we derive the iverse of a triagular Toeplitz matrix ad the expressio of the Toeplitz-Hesseberg determiat A very large variety of idetities ivolvig iteger partitios ad multiomial coefficiets ca be geerated usig this geeralizatio The partitioed biomial theorem ad a ew formula for the partitio fuctio p) are obtaied i this way Key words : Complete symmetric fuctios; elemetary symmetric fuctios; multiomial coefficiets; iteger partitios INTRODUCTION Let k ad be two positive itegers The k-th elemetary symmetric fuctio e k e k x x 2 x ) x i x i2 x ik i <i 2 <<i k
76 MIRCEA MERCA ad the k-th complete homogeeous symmetric fuctio h k h k x x 2 x ) x i x i2 x ik i i 2 i k are well-kow [5] I particular we have e 0 ad h 0 For k > or k < 0 we set e k 0 ad h k 0 The followig relatios e k det h i j+ ) ij k ad h k det e i j+ ) ij k are well-kow The first is a special case of Jacobi-Trudi idetity [4 eq 02] ad the secod is a special case of Nägelsbach-Kostka idetity [4 eq 03] O the other had the relatios h k ) k+t + +t t k + + t k e t t t e t 2 2 e t k k k e k t +2t 2 + +kt k k t +2t 2 + +kt k k ) k+t + +t t k + + t k h t t t h t 2 2 h t k k k where t + + t k t + + t k )! t t k t! t k! is the multiomial coefficiet ca be foud i [6 pp 3-4] Thus we ca write det f i j+ ) ij k ) k+t + +t t k + + t k f t t t f t 2 2 f t k k ) k t +2t 2 + +kt k k where f is ay of these complete or elemetary symmetric fuctios The startig poit of this paper is the followig geeralizatio for the symmetry betwee complete symmetric fuctios ad elemetary symmetric fuctios: Theorem Let {a } 0 ad {b } 0 be two sequeces such that ) k a k b k δ 0 2) t +2t 2 + +t k0 where δ ij is the Kroecker s delta The a ) +t + +t t + + t t b a 0 t t t b
A GENERALIZATION OF THE SYMMETRY BETWEEN SYMMETRIC FUNCTIONS 77 ad b t +2t 2 + +t ) +t + +t t + + t t a t t a 0 a a 0 ) t The iverse of a triagular Toeplitz matrix is give by Corollary Let A be a square matrix defied by a 0 a a 0 A with a 0 0 The A a a a 0 where /a 0 ad b b b b k a 0 for k { } t +2t 2 + +kt k k ) t t + +t k + + t t k a t t k a 0 ak a 0 ) tk This corollary is immediate from Theorem because i accordace with the relatio 2) we ca write a 0 a 0 a a 0 b b a a 0 a a a 0 b b b b a a a 0 I where a k )k a k for 0 k ad I is the idetity matrix O the other had the relatio 2) ca be rewritte as b b b a 0 a a 0 0
78 MIRCEA MERCA Applyig Cramer s rule to this matrix equatio we obtai b a b 0 b + 0 ) b 2 b + 0 b0 b b 0 b b 2 b Thus by Theorem we deduce the expressio of the Toeplitz-Hesseberg determiat: Corollary 2 For > 0 a a 0 a 2 a 0 a a 2 a a 0 ) t t t + + t a t t t a t 2 2 a t t +2t 2 + +t This corollary is kow as Trudi s formula ad the case a 0 of this formula is kow as Briochi s formula [7 Vol 3 Ch VII] We remark that the relatio ) is a special case of Corollary 2 Moreover the ext corollary follows immediately from Theorem ad Corollary 2 Corollary 3 Let {a } 0 be a sequeces such that a 0 0 If the sequece {b } 0 is defied by a 0 ad a a 0 b a 2 a + 0 a 0 a a 2 a b the a b 2 b + 0 b b 2 b Recall that the ivert trasform of a sequece {a } >0 is a sequece {b } >0 defied by + >0 b x a x >0
A GENERALIZATION OF THE SYMMETRY BETWEEN SYMMETRIC FUNCTIONS 79 ad its combiatorial iterpretatios ca be see i [2] We remark that the ivert trasform of the sequece { ) + a } >0 is the case a 0 i Theorem Theorem geerates a very large variety of idetities ivolvig iteger partitios ad multiomial coefficiets Some examples are preseted i the last sectio of this paper 2 PROOF OF THEOREM We ote that a 0 ad the we cosider Ax) a x ad Bx) b x a 0 0 0 Clearly A x)bx) We have A x) b )x + j b )x 2) j For each j the coefficiet of x i b )x j is t + +t j t +2t 2 + +t t + + t t t ) b ) t b ) t So the coefficiet of x i the right side of 2) is j t + +t j t +2t 2 + +t t + + t t t ) b ) t b ) t
80 MIRCEA MERCA or t +2t 2 + +t t + + t t t ) b ) t b ) t The first idetity is proved The secod idetity follows easily from the relatio ) k b k ) a k δ 0 ad from the first idetity k0 3 SOME APPLICATIONS I this sectio we use Theorem to prove some idetities ivolvig iteger partitios ad multiomial coefficiets We start with a iterestig versio of the well-kow biomial theorem 3 Partitioed biomial theorem Let {a } 0 ad {b } 0 be two sequeces defied by y if 0 a x otherwise ad y if 0 b y )+ xx + y) otherwise with y 0 It is a easy exercise to show that ) k a k b k δ 0 k0 Accordig to Theorem we obtai Corollary 4 [The partitioed biomial theorem] For > 0 x + y) t + + t x t + +t y t t t t t +2t 2 + +t
A GENERALIZATION OF THE SYMMETRY BETWEEN SYMMETRIC FUNCTIONS 8 I the case x ad y by Corollary we have 2 0 2 2 0 2 2 2 2 0 2 3 2 2 2 2 0 By Corollary 3 we obtai x y x x y x x x y x x x x xx + y) By Corollary 4 we easily deduce Corollary 5 For > 0 x + y) y + t +2t 2 + +t 32 Sequece of atural umbers t + + t t + + t Let {a } 0 be a positive iteger sequece defied by if 0 a otherwise Because t t ) x t + +t y t t a a a 2 + a 4 + a 5 a 7 a 8 + δ 0 we cosider the sequece {b } 0 defied by if 0 b 3 + otherwise 3
82 MIRCEA MERCA where x deotes the largest iteger ot greater tha x Applyig Theorem we obtai the idetity that ca be writte as t +2t 2 + +t t 3k 0 Corollary 6 For > 0 t q +t 2 q 2 + +t q ) +t +t 4 +t 7 + ) +t +t 3 +t 5 + t + + t t t t + + t t t where 3k q k 2 By Corollary we get 2 3 2 4 3 2 5 4 3 2 6 5 4 3 2 0 0 0 0 0 where o the diagoals of the right side the matrix elemet is ) k + ) k k 3 k > 0 3 the label of the first row is zero)
A GENERALIZATION OF THE SYMMETRY BETWEEN SYMMETRIC FUNCTIONS 83 For > 0 by Corollary 3 we have 2 3 2 4 3 2 5 4 3 2 6 5 4 3 2 + 3 3 ad 0 0 0 0 0 33 Square umbers We cosider the sequece {a } 0 defied by a + ) 2 The recurrece relatio a 3a + 3a 2 a 3 0 is well-kow [8 A000290] It is immediate from this recurrece that a 4a + 7a 2 + 8 ) k a k δ 0 3) k3 The followig result is a cosequece of Theorem
84 MIRCEA MERCA Corollary 7 Let be a positive iteger The ) +t + +t t + + t 4 t 7 t 2 8 t 3+ +t + ) 2 t t t +2t 2 + +t By Corollary we get 2 2 3 2 2 2 4 2 3 2 2 2 5 2 4 2 3 2 2 2 6 2 5 2 4 2 3 2 2 2 4 7 4 8 7 4 8 8 7 4 8 8 8 7 4 where o the diagoals of the right side the matrix elemet is ) k 8 for k > 2 the label of the first row is zero) By Corollary 3 we have 4 7 4 8 7 4 8 8 7 4 8 8 8 7 4 + ) 2 ad 2 2 3 2 2 2 4 2 3 2 2 2 5 2 4 2 3 2 2 2 6 2 5 2 4 2 3 2 2 2 8 The last idetity is true oly if > 2 34 Catala umbers I terms of biomial coefficiets the th Catala umber is give directly by C 2 + The Catala umbers form a sequece of atural umbers that occur i various coutig problems ofte ivolvig recursively defied objects [3 Sectio 8] For istace Seger
A GENERALIZATION OF THE SYMMETRY BETWEEN SYMMETRIC FUNCTIONS 85 developed i 76 a recursive formula for C usig the triagulatio problem: C 0 ad C + C k C k 0 for 0 Accordig to this recurrece relatio ad Theorem for if k 0 a k ad b k C k ) k C k otherwise k0 we obtai the followig covolutio idetity for Catala umbers Corollary 8 Let be a positive iteger The C ) +t + +t t + + t C t t t Ct 2 2 Ct t +2t 2 + +t By Corollary we get C 0 C C 0 C 2 C C 0 C 3 C 2 C C 0 C 4 C 3 C 2 C C 0 C 0 C C 0 C 2 C C 0 C 3 C 2 C C 0 ad by Corollary 3 we ca write C C 0 C 2 C C 0 C ) C 3 C 2 C C 0 C 4 C 3 C 2 C C 0 C 5 C 4 C 3 C 2 C 35 Fiboacci umbers Recall that the sequece {F } 0 of Fiboacci umbers [3 Sectio 26] is defied by the recurrece relatio F F F 2 0
86 MIRCEA MERCA with seed values F 0 0 ad F We cosider two sequeces {a } 0 ad {b } 0 defied by if 0 if 0 a ad b mod 2 otherwise F otherwise It is a easy exercise to show that b b b 3 b 5 δ 0 Accordig to Theorem the Fiboacci umber F ca be expressed as a sum over iteger partitios with odd parts Corollary 9 Let be a positive iteger The t + + t F 2 t t 2 t q +t 2 q 2 + +t 2 q 2 where q k 2k ad x deotes the smallest iteger ot less tha x By Corollary we have F F 2 F F 3 F 2 F F 4 F 3 F 2 F 0 0 0 0 I accordace with Corollary 3 we ca write F F 2 F F 3 F 2 F mod 2 ad F 4 F 3 F 2 F F 5 F 4 F 3 F 2 F 0 0 0 0 0 0 F
A GENERALIZATION OF THE SYMMETRY BETWEEN SYMMETRIC FUNCTIONS 87 36 Euler s partitio fuctio The petagoal umber theorem relates the product ad the series represetatios of the Euler fuctio I other words x ) ) k x k3k )/2 k x) x 2 ) x 3 ) ) k 2 x q k where the expoets q k are called geeralized petagoal umbers By the petagoal umber theorem we have k0 p)x 0 x ) ) 2 x q 0 We cosider the sequece {a } 0 defied by a p) ad the sequece {b } 0 where b is the coefficiet of x i the Euler fuctio x) x ) ie ) m if 2 b 3m2 ± m) m N 0 otherwise It is clear that Takig ito accout that ) k a k b k δ 0 k0 b qk ) k 2 a ew formula for p) follows easily from Theorem
88 MIRCEA MERCA Corollary 0 Let be a positive iteger The p) t q +t 2 q 2 + +t q ) t 3+t 4 +t 7 +t 8 + t + + t t t where q k k 3k + 2 2 2 are geeralized petagoal umbers ad x deotes the smallest iteger ot less tha x Accordig to Corollary we ca write p) p2) p) p3) p2) p) p4) p3) p2) p) p5) p4) p3) p2) p) 0 0 0 0 0 where o these diagoals the matrix elemet is ) k 2 or 0 The oly o-zero diagoals of this matrix are those which start o a row labeled by a geeralized petagoal umber q k The label of the first row is 0 Fially by Corollary 3 we have p) p2) p) p3) p2) p) p4) p3) p2) p) p5) p4) p3) p2) p) ) b
A GENERALIZATION OF THE SYMMETRY BETWEEN SYMMETRIC FUNCTIONS 89 ad p) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 where o these diagoals the matrix elemet is ) k b k The label of the first row is ACKNOWLEDGEMENT The author expresses his gratitude to Oaa Merca for the careful readig of the mauscript ad helpful remarks Special thaks go to the aoymous referee for may suggestios to ad commets o the origial versio of this paper REFERENCES G E Adrews The Theory of Partitios Addiso-Wesley Publishig 976 2 P J Camero Some sequeces of itegers Discrete Math 75 989) 89-02 3 T Koshy Elemetary Number Theory with Applicatios Secod editio Academic Press 994 4 I G Macdoald Schur fuctios: theme ad variatios i Sémiaire Lotharigie de Combiatoire Publ IRMA Strasbourg 498 992) 5-39 5 I G Macdoald Symmetric Fuctios ad Hall Polyomials 2d ed Claredo Press Oxford 995 6 P A MacMaho Combiatory aalysis Two volumes boud as oe) Chelsea Publishig Co New York 960 7 T Muir The theory of determiats i the historical order of developmet Four volumes Dover Publicatios New York 960 8 N J A Sloae The O-Lie Ecyclopedia of Iteger Sequeces Published electroically at http://oeisorg 203