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The degree of SO(n) MadelineBrandt,DJBruce,TaylorBrysiewicz,RobertKrone,ElinaRobeva 7 1 0 2 n a J AbstractWeprovideaclosedformulaforthedegreeofSO(n)overanalgebraically 3 closedfieldofcharacteristiczero.Inaddition,wedescribesymbolicandnumerical 1 techniqueswhichcanalsobeusedtocomputethedegreeofSO(n)forsmallvalues ] ofn.Asanapplicationofourresults,wegiveaformulaforthenumberofcritical G points of a low-rank semidefinite programming optimization problem. Finally, we A providesomeevidenceforaconjectureregardingthereallocusofSO(n). . h t a m 1 Introduction [ 2 ThespecialorthogonalgroupSO(n,R)isthegroupofautomorphismsofRnwhich v preservethestandardinnerproductandhavedeterminantequaltoone.Thecomplex 0 specialorthogonalgroupisthecomplexificationofthespecialorthogonalgroupand 0 canbethoughtofmoreexplicitlyasthegroupofmatrices 2 3 0 MadelineBrandt . 1 DepartmentofMathematics,UniversityofCalifornia,Berkeley,970EvansHall,Berkeley,CA, 0 94720e-mail:[email protected] 7 DJBruce 1 DepartmentofMathematics,UniversityofWisconsin,480LincolnDrive,Madison,WI,53706 : v e-mail:[email protected] i X TaylorBrysiewicz DepartmentofMathematics,TexasA&MUniversity,155IrelandSt,CollegeStation,TX77840 r e-mail:[email protected] a RobertKrone DepartmentofMathematicsandStatistics,Queen’sUniversity,48UniversityAvenue,Kingston, ON,K7L3N6e-mail:[email protected] ElinaRobeva Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge,MA02139e-mail:[email protected] 1 2 MadelineBrandt,DJBruce,TaylorBrysiewicz,RobertKrone,ElinaRobeva SO(n):=SO(n,C)=(cid:8)M∈Mat (C)| detM=1, MtM=Id(cid:9). n,n Astheseconditionsarepolynomialsintheentriesofsuchamatrix,weviewSO(n) asacomplexvariety. RecallthatthedegreeofacomplexvarietyX isthegenericnumberofintersec- tion points of X with a linear space of complementary dimension. Problem 4 on Grassmannians in [Stu16] asks for a formula for the degree of the of SO(n). Our primaryresultisthefollowingtheorem,whichanswersthisquestioncompletely. Theorem1.1.ThedegreeofSO(n)isgivenby (cid:18)(cid:18) (cid:19)(cid:19) 2n−2i−2j degSO(n)=2n−1det . n−2i 1≤i,j≤(cid:98)n(cid:99) 2 OurproofofTheorem1.1usesaformulaofKazarnovskij[Kaz87](seealsoThe- orem2.4)forthedegreeoftheimageofarepresentationofaconnected,reductive, algebraicgroupoveranalgebraicallyclosedfield.Byapplyingthisformulatothe case of the standard representation of SO(n) we are able to express the degree of SO(n)intermsofitsrootdataandotherinvariants. Inadditiontothisresult,Theorem4.2providesacombinatorialinterpretationof thisdegreeintermsofnon-intersectinglatticepaths.IncontrasttoTheorem1.1,the combinatorialstatementhastheimmediatebenefitofbeingobviouslynon-negative. Remark1.2.Let k be a field of characteristic zero. We can define SO(n,k) using the same system of equations since they are defined over the prime field Q. For k that is not algebraically closed, the degree of a variety can be defined in terms of theHilbertseriesofitscoordinatering.SincetheHilbertseriesdoesnotdependon thechoiceofk,thedegreedoesnoteither.WechoosetoworkoverCnotonlyfor simplicity,butalsosothatwemayusetheabovedefinitionofdegree. Remark1.3.Our methods are not specific to SO(n). The same approach can be usedtocomputethedegreeofotheralgebraicgroups.Forexample,towardtheend of Section 3 we provide a similar closed formula for the degree of the symplectic group.ThisformulaisalsointerpretedcombinatoriallyinSection4. InordertoverifyTheorem1.1,aswellasexplorethestructureofSO(n)infurther depth,itisusefultocomputethisdegreeexplicitly.Wewereabletodothisforsmall nusingsymbolicandnumericalcomputations.Acomparisonofthesuccessofthese twoapproaches,togetherwithourformulafromTheorem1.1,isillustratedbythe followingtable. ThedegreeofSO(n) 3 n Symbolic Numerical Formula 2 2 2 2 3 8 8 8 4 40 40 40 5 384 384 384 6 - 4768 4768 7 - 111616 111616 8 - - 3433600 9 - - 196968448 Table1 DegreeofSO(n)computedinvariousways Thisprojectstartedinthespringof2014,whenBenjaminRechtaskedthefifth author to describe the geometry of low-rank semidefinite programming (see Sec- tion5).Inparticular,heaskedwhytheaugmentedLagrangianalgorithmforsolving this problem [BM05] almost always recovers the correct optimum despite the ex- istenceofmultiplelocalminima.Itquicklybecameclearthattoevencomputethe numberoflocalextrema,oneneedstoknowthedegreeoftheorthogonalgroup.In Section5wefindaformulaforthenumberofcriticalpointsoflow-ranksemidefi- niteprogramming(seeTheorem5.2). The rest of this article is organized as follows. In Section 2 we give the reader abriefintroductiontoalgebraicgroupsandstateKazarnovskij’sTheorem.Section 3provesTheorem1.1byapplyingKazarnovskij’sTheoremandsimplifyingthere- sultingexpressions.Aftersimplification,weareleftwithadeterminantofbinomial coefficients which can be interpreted combinatorially using the celebrated Gessel- ViennotlemmawhichwedescribeinSection4.Therelationshipbetweenthedegree ofSO(n)andthedegreeoflow-ranksemidefiniteprogrammingiselaboratedupon in Section 5. Section 6 contains descriptions of the symbolic and numerical tech- niques involved in the explicit computation of degSO(n). Finally, in Section 7 we explorequestionsinvolvingtherealpointsonSO(n). 2 Background Inthissectionweprovidethereaderwiththenecessarylanguagetounderstandthe statement of Kazarnovskij’s Theorem (see Theorem 2.4), our main tool for deter- miningthedegreeofSO(n).WeinvitethosewhoalreadyarefamiliarwithLiethe- orytoskiptothestatementofTheorem2.4andcontinuetoSection3forourmain result.Wenote,thatasidefromapplyingTheorem2.4,nounderstandingofthema- terial in this section is necessary for understanding the remainder of the proof of Theorem1.1.Amorethoroughtreatmentofthetheoryofalgebraicgroupscanbe foundin[DK02,FH,Hum92]. An algebraic group G is a variety equipped with a group structure such that multiplicationandinversionarebothregularmapsonG.Whentheunipotentradical ofGistrivialandGisoveranalgebraicallyclosedfield,wesaythatGisareductive 4 MadelineBrandt,DJBruce,TaylorBrysiewicz,RobertKrone,ElinaRobeva group. Throughout this section, we let G denote a connected reductive algebraic groupoveranalgebraicallyclosedfieldk.LetG denotethemultiplicativegroup m ofk,soasaset,G =k\{0}.LetT denoteafixedmaximaltorusofG.Bymaximal m torus, we mean a subgroup of G isomorphic to Gr and which is maximal with m respecttoinclusion.Thenumberriswell-definedandiscalledtherankofG.After fixing T, we define the Weyl group of G, denotedW(G), to be the quotient of the normalizer of T by its centralizer,W(G)=N (T)/Z (T). Like r,W(G) does not G G dependonthechoiceofT uptoisomorphism. Example2.1.WecanparametrizeSO(2,C)byG viathemap m 1(cid:18) t+t−1 −i(t−t−1)(cid:19) R(t):= , 2 i(t−t−1) t+t−1 which is in fact a group isomorphism. (Note that R(eiθ) is the rotation matrix by angleθ.)ThereforeSO(2)hasrank1. Fixr∈N.Then    R(t ) 0 0··· 0 T2r:= 00...1 R(0...t2)00... ··.··.··. R(0...t )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ti∈Gm∼=SO(2)r⊂SO(2r) r    R(t ) 0 0··· 0 0 T2r+1:= 0...1 R(...t2)0... ·.·.·. 0... 0... (cid:12)(cid:12)(cid:12)(cid:12)ti∈Gm∼=SO(2)r⊂SO(2r+1)  00 00 00······ R(0tr)01 (cid:12)  are maximal tori of rank r of their respective groups. Therefore, rank(SO(2r))= rank(SO(2r+1))=randweseethattherankofSO(n)dependsfundamentallyon theparityofn. ThecharactergroupM(T)isthesetofalgebraicgrouphomomorphismsfromT toG ,i.e.grouphomomorphismsdefinedbypolynomialmaps, m M(T):=Hom (T,G ). AlgGrp m SinceT isisomorphictoGr ,allsuchhomomorphismsmustbeoftheform m (t ,...,t )(cid:55)→ta1···tar 1 r 1 r with a ,...,a integers. This character group is isomorphic to Zr and for this rea- 1 r sonitisoftencalledthecharacterlattice.Dualtothisisthegroupof1-parameter subgroups N(T):=Hom (G ,T), AlgGrp m ThedegreeofSO(n) 5 which is also isomorphic to Zr. Indeed, each 1-parameter subgroup is of the form t (cid:55)→(tb1,...,tbr) for integers b1,...,br. There exists a natural bilinear pairing be- tweenN(T)andM(T),givenby M(T)×N(T)→Hom (G ,G )∼=Z AlgGrp m m (cid:104)χ,σ(cid:105)(cid:55)→χ◦σ. Nowifρ:G→ GL(V)isarepresentationofGweattachtoitspecialcharacters calledweights.Aweightoftherepresentationρ isacharacterχ ∈M(T)suchthat theset (cid:92) V := ker(ρ(s)−χ(s)Id ) χ V s∈T is non-trivial. This condition is equivalent to saying that all of the matrices in {ρ(s)|s∈T}haveasimultaneouseigenvectorv∈V suchthattheassociatedeigen- valueforρ(s)is χ(s).WewilluseC todenotetheconvexhulloftheweightsof V therepresentationρ. Example2.2.An example that will be important for us later will be the standard representation coming from the natural embedding ρ :SO(n)→GL(Cn). For any t∈G ,thematrixR(t)∈SO(2)haseigenvectorse +ie ande −ie witheigen- m 1 2 1 2 valuest andt−1 respectively.FromtheexplicitdescriptionofT inExample2.1we seethattheeigenvectorsofρ(t ,...,t )areallvectorsoftheforme ±ie with 1 r 2j−1 2j 1≤ j≤r andtheeigenvaluesaret±1,...,t±1.Theseeigenvalues,viewedaschar- 1 r acters,aretheweightsofρ.Additionallywhenn=2r+1,wehavethate isan 2r+1 eigenvectorwitheigenvalue1,correspondingtothetrivialcharacter. Another representation of a matrix group G⊆End(V) is the adjoint representation,Ad:G→GL(End(V)),withAd(g)thelinearmapdefinedbyA(cid:55)→gAg−1. The roots of G are the weights of the adjoint representation. Given a linear func- tional(cid:96)onM(T),wedefinethepositiverootsofGwithrespectto(cid:96)tobetheroots χ suchthat(cid:96)(χ)>0.WedenotethepositiverootsofGbyα ,...,α.Forthealge- 1 l braicgroupsinthispaper,wecanchoose(cid:96)tobetheinnerproductwiththevector (r,r−1,...,1)sothatarootoftheforme −e ispositiveifandonlyif j<k.To j k eachrootα,weassociateacorootαˇ,definedtobethelinearfunctionαˇ(x):= 2(cid:104)x,α(cid:105) (cid:104)α,α(cid:105) where(cid:104),(cid:105)mustbeW(G)-invariant.Throughoutthispaper,wefixthistobethestan- dardinnerproduct. Example2.3.We now compute the roots of SO(n), starting with n even. It can be shown that the simultaneous eigenvectors of Ad(s) over all s∈T are matrices A withthefollowingstructure.Thesematricesarezerooutsidea2×2blockBinrows 2j−1,2jandcolumns2k−1,2kforsome1≤ j,k≤r.Furthermore,B=v vT with 1 2 each v equal to one of the eigenvectors of R(t), e ±ie . Indeed, suppose s∈T k 1 2 has blocks along the diagonal R(t ) witht ,...,t ∈G . Then Ad(s)(A) will also j 1 r m bezeroexceptinthesame2×2block,andthatblockwillbe R(t )BR(t )T =t±1t±1B, j k j k 6 MadelineBrandt,DJBruce,TaylorBrysiewicz,RobertKrone,ElinaRobeva wherethesignsintheexponentsdependonthechoicesofv andv .Thustheroots 1 2 ofSO(2r)arethecharactersoftheformt±1t±1for1≤ j,k≤r. j k Inthecasethatnisodd,Ahasanextrarowandcolumn.ConsiderAwithsupport onlyinthelastcolumn.Thenfors∈T,Ad(s)(A)=sAs−1 buts−1 actstriviallyon theleft,whilesactsonthelastcolumnasanelementofGL(Cn)asinthestandard representation.AsinExample2.2wegetweightst±1,...,t±1,1.Thesameweights 1 r appearforAwithsupportinthelastrow. AssociatedtoGisaLiealgebrag,whichcomesequippedwithaLiebracket[, ]. ACartansubalgebrahisanilpotentsubalgebraofgthatisself-normalizing,mean- ingif[x,y]∈hforallx∈h,theny∈h.LetS(h∗)betheringofpolynomialfunctions on h. The Weyl groupW(G) acts on h, and this extends to an action ofW(G) on S(h∗).ThespaceS(h∗)W(G) ofpolynomialswhichareinvariantuptotheactionof W(G)isgeneratedbyrhomogeneouspolynomialswhosedegrees,c +1,...,c +1, 1 r areuniquelydetermined.Thevaluesc ,...,c arecalledCoxeterexponents. 1 r WearenowpreparedtostateKazarnovskij’stheorem. Theorem2.4(Kazarnovskij’s Theorem, Prop 4.7.18 [DK02]). Let G be a connected reductive group of dimension m and rank r over an algebraically closed field.Ifρ :G→ GL(V)isarepresentationwithfinitekernelthen, m! (cid:90) degρ(G)= (αˇ αˇ ···αˇ )2dv. |W(G)|(c1!c2!···cr!)2|ker(ρ)| CV 1 2 l whereW(G) is the Weyl group, c are Coxeter exponents,C is the convex hull of i V theweights,andαˇ arethecoroots. i Ifρ isthestandardrepresentationforanalgebraicgroupG,thenitfollowsthat degρ(G)=degG. Therefore, in order to compute degSO(n), all we must do is apply this theorem for the standard representation of SO(n). The relevant data for thistheoremisgiveninTable2belowforSO(n)andSp(n). Table2 Group DimensionRank PositiveRoots Weights |W(G)| CoxeterExponents SO(2r) (cid:0)22r(cid:1) r {ei±ej}i<j {±ei} r!2r−1 1,3,...,2r−3,r−1 SO(2r+1) (cid:0)2r2+1(cid:1) r {ei±ej}i<j∪{ei} {±ei} r!2r 1,3,5,...,2r−1 Sp(r) (cid:0)22r(cid:1) r {ei±ej}i<j∪{2ei} {±ei} r!2r 1,3,5,...,2r−1 ThedegreeofSO(n) 7 3 MainResult:TheDegreeofSO(n) Wenowproveourmainresult,Theorem1.1.Attheendofthissectionweusethe samemethodtoobtainaformulaforthedegreeofthesymplecticgroup. WebeginbydirectlyapplyingTheorem2.4toSO(2r)andSO(2r+1)toobtain (cid:18) (cid:19) 2r ! (cid:32) (cid:33) 2 (cid:90) degSO(2r)= ∏ (x2−x2)2 dv, (1) r!2r−1(r−1)!2r∏−1(2k−1)!2 CV 1≤i<j≤r i j k=1 (cid:18) (cid:19) 2r+1 ! (cid:32) (cid:33) 2 (cid:90) r degSO(2r+1)= ∏ (x2−x2)2∏(2x)2 dv. (2) r i j i r!2r∏(2k−1)!2 CV 1≤i<j≤r i=1 k=1 Thus, to compute the degree of SO(n) it suffices to find formulas for the integrals above. We do this by first expanding the integrand into monomials, and then in- tegrating the result. We use the well-known expression for the determinant of the Vandermondematrix, (cid:32) (cid:33) r ∏ (y −y)= ∑ sgn(σ)∏yσ(i)−1 . j i i 1≤i<j≤r σ∈Sr i=1 Substitutingy =x2andsquaringtheentireexpressionyields i i (cid:32) (cid:33) r ∏ (x2−x2)2= ∑ sgn(στ)∏x2σ(i)+2τ(i)−4 . (3) i j i 1≤i<j≤r σ,τ∈Sr i=1 Additionally,wepointoutthateveryvariableintheintegrandisbeingraisedtoan evenpowerandC istheconvexhullofweights,{±e}.Becauseofthissymmetry, V i theintegralsoverC are2rtimesthesameintegralsover∆ ,thestandardr-simplex. V r Wehavenowreducedthecomputationofthisintegraltounderstandingtheintegral of any monomial over the standard simplex. The following proposition provides a formulaforthis. Proposition3.1(Lemma4.23[Mil14]).Let∆ ⊂Rr bethestandardr-simplex.If r a=(a ,...,a )∈Zr then 1 r >0 (cid:90) (cid:90) 1 r xadx= xa1xa2···xardx dx ···dx = ∏a!. ∆r ∆r 1 2 r 1 2 r (r+∑ai)!i=1 i Wecannowgetexpressionsfortheintegralsin(1)and(2)directlybyapplying (3)andProposition3.1. 8 MadelineBrandt,DJBruce,TaylorBrysiewicz,RobertKrone,ElinaRobeva Proposition3.2.LetI (r)andI (r)denotetheintegralsin(1)and(2)respec- even odd tively.Then, r!2r I (r)= det((2i+2j−4)!) . even (cid:0)2r(cid:1)! 1≤i,j≤r 2 r!23r I (r)= det((2i+2j−2)!) . odd (cid:0)2r+1(cid:1)! 1≤i,j≤r 2 Proof. Asmentionedabove,wecancomputeI byconsideringtheintegrandonly odd overthesimplex.This,alongwithequation(3)givesusthat (cid:90) r I (r)=2r ∏ (x2−x2)2∏(2x)2dv odd i j i ∆r1≤i<j≤r i=1 (cid:32) (cid:33) (cid:90) r r =2r ∑ sgn(στ)∏x2σ(i)+2τ(i)−4 ∏(2x)2dv i i ∆r σ,τ∈Sr i=1 i=1 (cid:90) r =23r ∑ sgn(στ) ∏x2σ(i)+2τ(i)−2dv. i σ,τ∈Sr ∆ri=1 Astheintegrandishomogeneousofdegree4(cid:0)r(cid:1)+2r,applyingProposition3.1and 2 simplifyingyields 23r r Iodd(r)= (cid:0)4(cid:0)r(cid:1)+3r(cid:1)! ∑ sgn(στ)∏(2σ(i)+2τ(i)−2)!, 2 σ,τ∈Sr i=1 whichafterreplacingiwithσ−1(i)givesus r r ∏(2σ(i)+2τ(i)−2)!=∏(2i+2τσ−1(i)−2)!. i=1 i=1 Letρ=τσ−1.Overallpairsσ,τ∈S ,thepermutationρ appearsaseachpermuta- r tioninS exactlyr!times,andsgn(στ)=sgn(ρ).Therefore,wehavethat r r!23r r Iodd(r)= (cid:0)4(cid:0)r(cid:1)+3r(cid:1)! ∑ sgn(ρ)∏(2i+2ρ(i)−2)! 2 ρ∈Sr i=1 r!23r = det((2i+2j−2)!) . (cid:0)2r+1(cid:1)! 1≤i,j≤r 2 ThederivationofI followspreciselythesamesteps. (cid:116)(cid:117) even Theorem1.1nowfollowsdirectlyfromthesubsequentsimplification. ThedegreeofSO(n) 9 22r degSO(2r+1)= det((2i+2j−2)!) (1!3!···(2r−1)!)2 22r (cid:18)(2i+2j−2)!(cid:19) = det (1!2!···(2r−1)!) (2i−1)! (cid:18) (cid:19) (2i+2j−2)! =22rdet (2i−1)!(2j−1)! (cid:18)(cid:18) (cid:19)(cid:19) 2i+2j−2 =22rdet . 2i−1 1≤i,j≤r Reversing the order of the rows and columns of the final matrix and reindexing producestheformulagiveninTheorem1.1.Similarly,fortheevencase,wehave 2 degSO(2r)= det((2i+2j−4)!) (1!3!···(2r−3)!(r−1)!)2 2·(2r−1)2 = det((2i+2j−4)!) (1!3!···(2r−3)!2·4···(2r−2))2 (cid:18) (cid:19) (2i+2j−4)! =22r−1det (2i−2)!(2j−2)! (cid:18)(cid:18) (cid:19)(cid:19) 4r−2i−2j =22r−1det . 2r−2i 1≤i,j≤r ThisfinishestheproofofTheorem1.1. Since the orthogonal group O(n) has two components that are isomorphic to SO(n),weimmediatelygetaformulaforthedegreeofO(n). Corollary3.3.ThedegreeofO(n)isgivenby (cid:18)(cid:18) (cid:19)(cid:19) 2n−2i−2j degO(n)=2ndet . n−2i 1≤i,j≤(cid:98)n(cid:99) 2 Furthermore,asmentionedintheintroduction,thereisnoreason,apriori,thatthe stepstakeninthissectionareparticulartoSO(n).Wenowapplythesemethodsto findthedegreeofSp(r),thegroupof(complex)symplecticmatrices. RecallthesymplecticgroupoverCisdefinedtobe Sp(r):=Sp(r,C)={M∈Mat (C)|MTΩM=Ω}, 2r,2r where (cid:18) (cid:19) 0 I Ω = r . −I 0 r Theorem3.4.ThedegreeofSp(r)isgivenby (cid:18)(cid:18) (cid:19)(cid:19) 2i+2j−2 degSp(r)=det . 2i−1 1≤i,j≤r 10 MadelineBrandt,DJBruce,TaylorBrysiewicz,RobertKrone,ElinaRobeva For 1≤r≤5 the values of degSp(r) are 2,24,1744,769408,2063048448,.... Thiswasverifiedusingbothnumericalandsymbolictechniquesuptor=3. Proof. This is an application of Kazarnovskij’s result which is completely analo- goustothecomputationforthespecialorthogonalgroup.Theintegralisthesame astheoneforSO(2r+1)uptofactorsof2,soitisevaluatedinthesameway,and thentheexpressioncanbesimplified (cid:32) (cid:33) (r(2r+1))! (cid:90) r deg(Sp(r))= ∏ (x −x )2(x +x )2∏x2 dv r!2r(1!3!···(2r−1)!)2 CV 1≤i<j≤r i j i j i=1 i 1 = det((2i+2j−2)!) (1!3!···(2r−1)!)2 1≤i,j≤r (cid:18)(cid:18) (cid:19)(cid:19) 2i+2j−2 =det . 2i−1 1≤i,j≤r (cid:116)(cid:117) WeremarkthatourformulafordegSp(r)isparticularlyinterestingbecausethe determinant in Theorem 3.4 is the same as the determinant in Theorem 1.1 when n=2r+1. Corollary3.5. degSO(2r+1)=22rdegSp(r) Proof. Sendingthe(i,j)entryofthematrixinTheorem3.4tothe(r−i+1,r−j+1) entrydoesnotchangethedeterminantandgivesusthat (cid:18)(cid:18) (cid:19)(cid:19) 4r+2−2i−2j degSp(r)=det . 2r+1−2i 1≤i,j≤r Whenn=2r+1,thisisthematrixappearinginTheorem1.1andallthatisdifferent isthecoefficientinfront.Accountingforthiscoefficientfinishestheproof. (cid:116)(cid:117) 4 Non-IntersectingLatticePaths TheformulasgivenintheprevioussectionforthedegreesofSO(n),O(n),andSp(r) canbeinterpretedasacountofnon-intersectinglatticepathsviatheGessel-Viennot Lemma[GV85]. Lemma4.1(Gessel-Viennot(WeakVersion)).LetA={a ,...,a },B={b ,...,b } 1 r 1 r be collections of lattice points in Z2. Let M be the number of lattice paths from i,j a tob usingonlyunitstepsineithertheNorthorEastdirection.Iftheonlyway i j thatasystemoftheselatticepathsfromA→Bdonotcrosseachotherisbysend- inga (cid:55)→b,thenthedeterminantofM equalsthenumberofsuchnon-intersecting i i latticepaths.