Sémiaie Lothaigie de Combiatoie 63 (010), Aticle B63c IDENTITIES FOR THE NUMBER OF STANDARD YOUNG TABLEAUX IN SOME (k, l)-hooks A. REGEV Abstact. Closed fomulas ae kow fo S(k,0;), the umbe of stadad Youg tableaux of size ad with at most k pats, whee 1 k 5. Hee we study the aalogous poblem fo S(k,l;), the umbe of stadad Youg tableaux of size which ae cotaied i the (k,l)-hook. We deduce some fomulas fo the cases k + l 4. 1. Itoductio Give a patitio λ of, which we deote as usual by λ, let χ λ deote the coespodig ieducible S chaacte. Its degee is deoted by deg χ λ f λ ad is equal to the umbe of stadad Youg tableaux (SYT) of shape λ. (The eade is efeed to [8, 9, 13, 15] fo itoductios ito chaacte theoy of the symmetic goup ad symmetic fuctios.) The umbe f λ ca be calculated fo example by the hook fomula (see [8, Theoem.3.1], [13, Sectio 3.10], [15, Coollay 7.1.6]. We coside the umbe of SYT i the (k,l)-hook. Moe pecisely, give iteges k,l, 0, we wite H(k,l;) {λ (λ 1,λ,...) λ ad λ k+1 l} ad S(k,l;) f λ. λ H(k,l;) 1.1. The cases whee S(k,l;) ae kow. Fo the stip sums S(k, 0;) it is kow (see [11] ad [15, Ex. 7.16.b]) that ( ) S(, 0;) ad S(3, 0;) ( )( ) 1. + 1 0 ( Let C 1 ) +1 be the Catala umbes. Gouyou-Beauchamps [7] (see also [15, Ex. 7.16.b]) poved that S(4, 0;) C +1 C +1 ad S(5, 0;) 6 0 ( ) ( + )! C ( + )!( + 3)!. As fo the hook sums, util ecetly oly S(1, 1;) ad S(, 1;) S(1, ;) have bee calculated: 1. It easily follows that S(1, 1;) 1. 000 Mathematics Subect Classificatio. 05C30.
A. REGEV. The followig idetity was poved i [1, Theoem 8.1]: ( S(, 1;) 1 1 ( )( ) 4 0 + 1 k1! k! (k + 1)! ( k )! ( k 1) ( k) ) + 1. (1) 1.. The mai esults. I Sectio we pove Equatio (10), which gives (sot of) a closed fomula fo S(3, 1; ) i tems of the Motzki-sums fuctio. Fo the Motzkisums fuctio see [14, sequece A005043]. Equatio (10) is i fact a degee cosequece of a fomula fo S chaactes, of iteest i its ow ight, see Equatio (9). I Sectio 3 we fid some itiguig elatios betwee the sums S(4, 0;) ad the ectagula sub-sums S (,, ;) (see Sectio 3 fo thei defiitio), see idetities (1) ad (13) below. Fially, i Sectio 4 we eview some cases whee the hook sums S(k,l;) ae elated, i some athe mysteious ways, to hump eumeatios o Dyck ad o Motzki paths, see (14), (16), ad Theoem 4.1. As usual, i some of the above idetities it is of iteest to fid biective poofs, which might explai these idetities. Ackowledgemet. We thak D. Zeilbege fo veifyig some of the idetities hee by the WZ method. Defie the S chaacte χ(k,l;). The sums S(3, 1; ) ad the chaactes χ(3, 1; ) λ H(k,l;) χ λ, so that deg(χ(k,l;)) S(k,l;). ().1. The Motzki-sums fuctio. Defie the S chaacte Ψ() / k0 χ (k,k,1 k), ad deote deg Ψ() a(). (3) We call Ψ() the Motzki-sums chaacte. Note that deg χ (k,k,1 k) f (k,k,1 k)! (k 1)! k! ( k)! ( k) ( k + 1), hece a() / k1! (k 1)! k! ( k)! ( k) ( k + 1). (4) By [14, sequece A005043], it follows that a() is the Motzki-sums fuctio. The eade is efeed to [14] fo vaious popeties of a(). Fo example, a() + a( + 1)
IDENTITIES FOR THE NUMBER OF STANDARD TABLEAUX 3 M, whee M ae the Motzki umbes. Also a(1) 0, a() 1, ad a() satisfies the ecuece a() 1 ( a( 1) + 3 a( )), fo 3. (5) + 1 Note also that fo Equatio (1) ca be witte as ( S(, 1;) 1 1 ( )( ) ) 4 + a() 1 + 1. (6) 0 The asymptotic behavio of a() ca be deduced fom that of M. We deduce it hee, eve though it is ot eeded i the sequel. Remak.1. As teds to ifiity, 3 a() 8 π 1 3 ad a() 1 4 M. Poof. By stadad techiques it ca be show that a() has asymptotic behavio ( ) g 1 a() c α fo some costats c,g ad α which we ow detemie. By [11], we have ( ) 3/ 3 1 M π 3. Togethe with M a() + a( + 1) c (1 + α) this implies that α 3, that g 3/, ad that c 3 8 π. ( ) g 1 α,.. The oute poduct of S m ad S chaactes. Give a S m chaacte χ m ad a S chaacte χ, we ca fom thei oute poduct χ ˆ χ. The exact decompositio of χ m ˆ χ is give by the Littlewood Richadso ule, see [8, 9, 13, 15]. I the special case that χ χ (), this decompositio is give, below, by Youg s ule. Futhemoe, we have ( ) + m deg(χ ˆ χ () ) deg(χ ). (7) Youg s Rule (see [9, Ch. I, Sec. 7 ad (5.16)]): Let λ (λ 1,λ,...) m ad deote by λ + the followig set of patitios of m + : The λ + {µ + m µ 1 λ 1 µ λ }. χ λ ˆ χ () µ λ + χ µ.
4 A. REGEV Example.. Give, it follows that (see [11], [15, Ex. 7.16.b]) χ ( / ) ˆ χ ( / ) χ(, 0;), ad by takig degees, S(, 0;).3. A chaacte fomula fo χ(3, 1;). ( ). (8) / Popositio.3. With the otatios of () ad (3), [ ] χ(3, 1;) 1 χ(, 0,) + Ψ()ˆ χ ( ). (9) By takig degees, Example. togethe with (3) ad (7) imply that [ ( S(3, 1;) 1 ) ( ) ] + a(). (10) Poof. Deote Ω() 0 0 Ψ()ˆ χ ( ), ad aalyze this S chaacte. Youg s ule implies the followig: 0 If µ, the, by Youg s ule, χ µ has a positive coefficiet i Ω() if ad oly if µ H(3, 1;). Moeove, all these coefficiets ae eithe 1 o, ad such a coefficiet equals 1 if ad oly if µ is a patitio with at most two ows, say µ (µ 1,µ ). It follows that χ(, 0;) + Ω() χ λ. (11) λ H(3,1;) This implies (9) ad completes the poof of Popositio.3. 3. The sums S(4, 0;) ad S (, ;) Defiitio 3.1. (1) Let m, m, ad let H (, ; m) H(, ; m) deote the set of patitios H (, ; m) {(k+,k+, m k ) m k 0,...m } (the patitios i the (, )-hook with both am ad leg beig ectagula). Futhemoe, wite S (, ; m) f λ. λ H (,;m) () Let m + 1, m, ad let H (, ; m + 1) H(, ; m + 1) deote the set of patitios H (, ; m + 1) {(k + 3,k +, m k ) m + 1 k 0,...m } (the patitios i the (, )-hook with am ealy ectagula ad leg ectagula). Futhemoe, let S (, ; m + 1) f λ. λ H (,;m+1)
IDENTITIES FOR THE NUMBER OF STANDARD TABLEAUX 5 Recall fom Sectio 1.1 that S(4, 0; m 1) C m ad S(4, 0; m) C m C m+1. We have the followig itiguig idetities. Popositio 3.. (1) Let m. The S(4, 0; m ) C m 1 C m S (, ; m). Explicitly, we have the followig idetity: ( ) ( ) 1 m m C m 1 C m m (m + 1) m 1 m m k0 () Let m + 1. The m + 1 m + C m m k0 (m)! k! (k + 1)! (m k )! (m k 1)! (m 1) m (m + 1). m + 1 m + Explicitly, we have the followig idetity: ( )( ) 1 m m + 1 (m + 1) (m + ) m m S(4, 0; m 1) m + 1 m + C m S (, ; m + 1). (1) (m + 1)! k! (k + )! (m k )! (m k 1)! (m 1) m (m + 1) (m + ). (13) Poof. Equatio (1) is the specializatio of Gauß s F 1 (a,b;c; 1) with a m,b 1 m,c (cf. [1]), ad (13) is simila. Alteatively, the idetities (1) ad (13) ca be veified by the WZ method (cf. [10, 16]). 4. Hook sums ad humps fo paths A Dyck path of legth is a lattice path, i Z Z, fom (0, 0) to (, 0), usig up-steps (1, 1) ad dow-steps (1, 1) ad eve goig below the x-axis. A hump i a Dyck path is a up-step followed by a dow-step. 1 A Motzki path of legth is a lattice path fom (0, 0) to (, 0), usig flat-steps (1, 0), up-steps (1, 1) ad dow-steps (1, 1), ad eve goig below the x-axis. A hump i a Motzki path is a up-step followed by zeo o moe flat-steps followed by a dow-step. We ow cout humps fo Dyck ad fo Motzki paths ad obseve the followig itiguig pheomea: The hump eumeatio i the Dyck case associates the ectagula shape λ (,) to the (1, 1)-hook shape µ (, 1 ). Moeove, i the Motzki case we show below that it associates the (3, 0) stip shape patitios H(3, 0;) to the (, 1)-hook shape patitios H(, 1;). 1 I the Dyck path cotext, humps ae usually called peaks. Howeve, we pefe the tem hump because, i the cotext of Motzki paths, this tem will ideed diffe fom peak.
6 A. REGEV 4.1. The Dyck case. The Catala umbe C ()!!( + 1)! is the cadiality of a vaiety of sets (see [15, Ex. 6.19]); hee we ae iteested i two such sets. Fist, C f (,), the umbe of SYT of shape (,). Secod, C is the umbe of Dyck paths of legth. Let HD deote the total umbe of humps i all Dyck paths of legth. The ( ) 1 HD, see [3, 4, 6]. Sice ( ) 1 f (,1 ), we have We deote this associatio by C f (,) ad HD f (,1). H : (,) (, 1 ). (14) 4.. The Motzki case. Like the Catala umbes, also the Motzki umbes M ae the cadiality of a vaiety of sets (cf. [15, Ex. 6.38], [14, sequece A001006]). The esult fom [11] that M S(3, 0;) gives the Motzki umbes a SYT itepetatio. Moeove, M is the umbe of Motzki paths of legth. Let HM deote the total umbe of humps i all Motzki paths of legth. The, accodig to [14, sequece A097861], HM 1 1 ( )( ). (15) We show below that this implies the itiguig idetity HM S(, 1;) 1, which gives a SYT-itepetatio of the umbes HM. Thus, the hump eumeatio i the Motzki case associates the (3, 0) stip shape patitios H(3, 0;) to the (, 1)-hook shape patitios H(, 1;). We deote this by H : H(3, 0;) H(, 1;). (16) Theoem 4.1. The umbe of humps of all Motzki paths of legth satisfies HM S(, 1;) 1. Combiig Equatios (1) ad (15), the poof of Theoem 4.1 will follow oce the followig biomial idetity of iteest i its ow ight is poved. Lemma 4.. Fo, we have / 1 0 ( )( ) 1 1 ( )( ) + 0 ( 1 k1 )( ) + a() 1! k! (k + 1)! ( k )! ( k 1) ( k). (17)
IDENTITIES FOR THE NUMBER OF STANDARD TABLEAUX 7 Equatio (17) was fist veified by the WZ method. About this method, see [10, 16]. Hee is a elemetay poof which is due to Ia Gessel [5]. Poof. Note fist that a() is the th Rioda umbe, [14, sequece A005043], defied (fo example) by a()x 1 + x + 1 x 3x. 0 Addig to both sides of (17) gives the equivalet idetity / ( )( ) ( )( ) + a(). (18) 0 0 0 Now let us eplace by i the sum o the ight-had side of (18), theeby gettig ( )( ), ad the sepaate the eve ad odd values of so that this sum is equal to u()+v() whee u() ( )( ) ad ( )( ) + 1. + 1 v() Notig that the left-had side of (18) is u(), we see that the idetity to be poved is equivalet to u() v() + a(). (19) It is staightfowad to show that u() is the coefficiet of x i (1 + x + x ) [14, sequece A0046, cetal tiomial coefficiets] ad that v() is the coefficiet of x 1 (o of x +1 ) i (1 + x + x ) [14, sequece A005717]. With these itepetatios fo u() ad v(), a combiatoial poof of the idetity u() v() a() has bee give by David Calla [Rioda umbes ae diffeeces of tiomial coefficiets, 006, http://www.stat.wisc.edu/~calla/otes/ioda/ioda.pdf]. Alteatively, Equatio (19) follows easily fom the kow geeatig fuctios fo u(), v(), ad a(), which ca all be foud i [14] (o deived diectly). This completes the poof of Theoem 4.1. Refeeces [1] G. E. Adews, R. Askey ad R. Roy, Special Fuctios, Ecyclopedia of Mathematics ad its Applicatios, Cambidge Uivesity Pess (1999). [] C. Daasathy as A. Yag, A tasfomatio o odeed tees, Compute J. 3 (1980), 161 164. [3] N. Deshowitz ad S. Zaks, Eumeatio of odeed tees, Discete Math. 31 (1980), 9 8. [4] N. Deshowitz ad S. Zaks, Applied tee eumeatio, Lectue Notes i Compute Sciece, vol. 11, Spige, Beli, 1981, pp. 180 193.
8 A. REGEV [5] I. Gessel, pivate lette. [6] E. Deutsch, Dyck path eumeatio, Discete Math 04 (1999), 167 0. [7] D. Gouyou-Beauchamps, Stadad Youg tableaux of height 4 ad 5, Euop. J. Combi. 10 (1989), 69 8. [8] G.D. James ad A. Kebe, The Repesetatio Theoy of the Symmetic Goup, Ecyclopedia of Mathematics ad its Applicatios, vol. 16, Addiso Wesley, Readig, MA (1981). [9] I.G. Macdoald, Symmetic Fuctios ad Hall Polyomials, d editio, Oxfod Uivesity Pess (1995). [10] M. Petkovšek, H.S. Wilf ad D Zeilbege, AB, A.K. Petes Ltd. (1996). [11] A. Regev, Asymptotic values of degees associated with stips of Youg diagams, Adv. Math. 41 (1981), 115 136. [1] A. Regev, Pobabilities i the (k,l) hook, Isael J. Math. 169 (009), 61 88. [13] B. E. Saga, The Symmetic Goup: Repesetatios, Combiatoial Algoithms, ad Symmetic Fuctios, d editio, Gaduate Texts i Mathematics 03, Spige-Velag (000). [14] N.J.A. Sloae, The O-Lie Ecyclopedia of Itege Sequeces, http://www.eseach.att.com/~as/sequeces. [15] R. P. Staley, Eumeative Combiatoics, vol., Cambidge Uivesity Pess, Cambidge (1999). [16] D. Zeilbege, The method of ceative telescopig, J. Symbolic Comput. 11 (1991), 195 04. Mathematics Depatmet, The Weizma Istitute, Rehovot 76100, Isael E-mail addess: amitai.egev at weizma.ac.il