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1 0 0 1 94.01 759.92 cm
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/F53 11.96 Tf 0 0 Td[(THE)-376(COEFFICIENTS)-374(OF)-374(A)-375(FIBONA)31(CCI)-375(PO)31(WER)-374(SERIES)]TJ 136.15 -23.91 Td[(F)93(ederico)-375(Ardila)]TJ/F4 6.97 Tf -72.69 -8.96 Td[(Departmen)30(t)-384(of)-386(Mathematics,)-384(Massac)31(h)31(usetts)-385(Institute)-384(of)-385(T)92(ec)30(hnology)]TJ 8.97 -8.97 Td[(77)-385(Massac)31(h)31(usetts)-385(Av)31(en)31(ue,)-386(Ro)-30(om)-385(2-333,)-384(Cam)31(bridge,)-384(MA)-385(02139)]TJ 57.84 -8.96 Td[(e-mail:)-507([email protected])]TJ/F40 6.97 Tf -54.44 -8.97 Td[(\050Submitte)59(d)-427(Septemb)60(er)-427(2001-Final)-426(R)59(evision)-428(F)88(ebruary)-426(2002\051)]TJ/F1 9.96 Tf -77.92 -23.91 Td[(Consider)-333(the)-332(in\014nite)-332(pro)-27(duct)]TJ/F7 9.96 Tf 32.68 -25.9 Td[(A)]TJ/F1 9.96 Tf 7.47 0 Td[(\050)]TJ/F7 9.96 Tf 3.87 0 Td[(x)]TJ/F1 9.96 Tf 5.7 0 Td[(\051)-277(=)]TJ/F19 9.96 Tf 18.09 9.46 Td[(Y)]TJ/F10 6.97 Tf -0.94 -21.45 Td[(k)]TJ/F16 6.97 Tf 4.41 0 Td[()]TJ/F4 6.97 Tf 6.22 0 Td[(2)]TJ/F1 9.96 Tf 3.98 11.99 Td[(\0501)]TJ/F13 9.96 Tf 11.06 0 Td[( )]TJ/F7 9.96 Tf 9.97 0 Td[(x)]TJ/F10 6.97 Tf 5.69 4.11 Td[(F)]TJ/F12 4.98 Tf 5.05 -1.06 Td[(k)]TJ/F1 9.96 Tf 4.86 -3.05 Td[(\051)-278(=)-277(\0501)]TJ/F13 9.96 Tf 28.22 0 Td[( )]TJ/F7 9.96 Tf 9.97 0 Td[(x)]TJ/F1 9.96 Tf 5.69 0 Td[(\051\0501)]TJ/F13 9.96 Tf 14.94 0 Td[( )]TJ/F7 9.96 Tf 9.97 0 Td[(x)]TJ/F4 6.97 Tf 5.69 4.11 Td[(2)]TJ/F1 9.96 Tf 4.47 -4.11 Td[(\051\0501)]TJ/F13 9.96 Tf 14.94 0 Td[( )]TJ/F7 9.96 Tf 9.97 0 Td[(x)]TJ/F4 6.97 Tf 5.69 4.11 Td[(3)]TJ/F1 9.96 Tf 4.47 -4.11 Td[(\051\0501)]TJ/F13 9.96 Tf 14.94 0 Td[( )]TJ/F7 9.96 Tf 9.97 0 Td[(x)]TJ/F4 6.97 Tf 5.69 4.11 Td[(5)]TJ/F1 9.96 Tf 4.47 -4.11 Td[(\051\0501)]TJ/F13 9.96 Tf 14.94 0 Td[( )]TJ/F7 9.96 Tf 9.97 0 Td[(x)]TJ/F4 6.97 Tf 5.69 4.11 Td[(8)]TJ/F1 9.96 Tf 4.47 -4.11 Td[(\051)]TJ/F7 9.96 Tf 5.53 0 Td[(:)-167(:)-167(:)]TJ/F1 9.96 Tf -251.44 -37.83 Td[(=)-278(1)]TJ/F13 9.96 Tf 17.71 0 Td[( )]TJ/F7 9.96 Tf 9.96 0 Td[(x)]TJ/F13 9.96 Tf 7.91 0 Td[( )]TJ/F7 9.96 Tf 9.96 0 Td[(x)]TJ/F4 6.97 Tf 5.7 4.12 Td[(2)]TJ/F1 9.96 Tf 6.68 -4.12 Td[(+)]TJ/F7 9.96 Tf 9.96 0 Td[(x)]TJ/F4 6.97 Tf 5.7 4.12 Td[(4)]TJ/F1 9.96 Tf 6.68 -4.12 Td[(+)]TJ/F7 9.96 Tf 9.96 0 Td[(x)]TJ/F4 6.97 Tf 5.69 4.12 Td[(7)]TJ/F13 9.96 Tf 6.69 -4.12 Td[( )]TJ/F7 9.96 Tf 9.96 0 Td[(x)]TJ/F4 6.97 Tf 5.69 4.12 Td[(8)]TJ/F1 9.96 Tf 6.69 -4.12 Td[(+)]TJ/F7 9.96 Tf 9.96 0 Td[(x)]TJ/F4 6.97 Tf 5.69 4.12 Td[(11)]TJ/F13 9.96 Tf 10.66 -4.12 Td[( )]TJ/F7 9.96 Tf 9.96 0 Td[(x)]TJ/F4 6.97 Tf 5.7 4.12 Td[(12)]TJ/F13 9.96 Tf 10.65 -4.12 Td[( )]TJ/F7 9.96 Tf 9.96 0 Td[(x)]TJ/F4 6.97 Tf 5.7 4.12 Td[(13)]TJ/F1 9.96 Tf 10.65 -4.12 Td[(+)]TJ/F7 9.96 Tf 9.96 0 Td[(x)]TJ/F4 6.97 Tf 5.7 4.12 Td[(14)]TJ/F1 9.96 Tf 10.65 -4.12 Td[(+)]TJ/F7 9.96 Tf 9.96 0 Td[(x)]TJ/F4 6.97 Tf 5.7 4.12 Td[(18)]TJ/F1 9.96 Tf 10.65 -4.12 Td[(+)]TJ/F7 9.96 Tf 9.96 0 Td[(:)-167(:)-167(:)]TJ/F1 9.96 Tf -342.73 -23.91 Td[(regarded)-345(as)-346(a)-347(formal)-346(p)-28(o)28(w)27(er)-346(series.)-482(In)-347([4],)-349(N.)-347(Robbins)-346(pro)28(v)28(ed)-346(that)-346(the)-346(co)-28(ecien)30(ts)-346(of)]TJ/F7 9.96 Tf 369.27 0 Td[(A)]TJ/F1 9.96 Tf 7.48 0 Td[(\050)]TJ/F7 9.96 Tf 3.87 0 Td[(x)]TJ/F1 9.96 Tf 5.69 0 Td[(\051)-347(are)]TJ -386.31 -11.95 Td[(all)-389(equal)-389(to)]TJ/F13 9.96 Tf 53.98 0 Td[( )]TJ/F1 9.96 Tf 7.75 0 Td[(1)]TJ/F7 9.96 Tf 4.98 0 Td[(;)]TJ/F1 9.96 Tf 4.43 0 Td[(0)-390(or)-389(1.)-613(W)83(e)-389(shall)-388(giv)27(e)-389(a)-389(short)-389(pro)-27(of)-390(of)-389(this)-389(fact,)-402(and)-390(a)-389(v)28(ery)-389(simple)-388(recursiv)30(e)]TJ -71.14 -11.96 Td[(description)-331(of)-333(the)-333(co)-27(ecien)30(ts)-333(of)]TJ/F7 9.96 Tf 141.28 0 Td[(A)]TJ/F1 9.96 Tf 7.47 0 Td[(\050)]TJ/F7 9.96 Tf 3.87 0 Td[(x)]TJ/F1 9.96 Tf 5.7 0 Td[(\051.)]TJ -138.4 -11.95 Td[(F)83(ollo)28(wing)-387(the)-385(notation)-386(of)-386([4],)-399(let)]TJ/F7 9.96 Tf 147.34 0 Td[(a)]TJ/F1 9.96 Tf 5.26 0 Td[(\050)]TJ/F7 9.96 Tf 3.88 0 Td[(m)]TJ/F1 9.96 Tf 8.74 0 Td[(\051)-387(b)-27(e)-386(the)-385(co)-28(ecien)30(t)-386(of)]TJ/F7 9.96 Tf 98.39 0 Td[(x)]TJ/F10 6.97 Tf 5.69 3.61 Td[(m)]TJ/F1 9.96 Tf 11.42 -3.61 Td[(in)]TJ/F7 9.96 Tf 12.14 0 Td[(A)]TJ/F1 9.96 Tf 7.48 0 Td[(\050)]TJ/F7 9.96 Tf 3.87 0 Td[(x)]TJ/F1 9.96 Tf 5.69 0 Td[(\051.)-603(It)-387(is)-385(clear)-386(that)]TJ/F7 9.96 Tf -329.82 -11.96 Td[(a)]TJ/F1 9.96 Tf 5.27 0 Td[(\050)]TJ/F7 9.96 Tf 3.87 0 Td[(m)]TJ/F1 9.96 Tf 8.75 0 Td[(\051)-302(=)]TJ/F7 9.96 Tf 17.64 0 Td[(r)]TJ/F10 6.97 Tf 4.49 -1.49 Td[(E)]TJ/F1 9.96 Tf 6.73 1.49 Td[(\050)]TJ/F7 9.96 Tf 3.88 0 Td[(m)]TJ/F1 9.96 Tf 8.75 0 Td[(\051)]TJ/F13 9.96 Tf 6.18 0 Td[( )]TJ/F7 9.96 Tf 10.06 0 Td[(r)]TJ/F10 6.97 Tf 4.49 -1.49 Td[(O)]TJ/F1 9.96 Tf 6.78 1.49 Td[(\050)]TJ/F7 9.96 Tf 3.87 0 Td[(m)]TJ/F1 9.96 Tf 8.75 0 Td[(\051,)-351(where)]TJ/F7 9.96 Tf 39.1 0 Td[(r)]TJ/F10 6.97 Tf 4.49 -1.49 Td[(E)]TJ/F1 9.96 Tf 6.73 1.49 Td[(\050)]TJ/F7 9.96 Tf 3.88 0 Td[(m)]TJ/F1 9.96 Tf 8.74 0 Td[(\051)-348(is)-348(equal)-347(to)-348(the)-347(n)28(um)28(b)-27(er)-347(of)-348(partitions)-347(of)]TJ/F7 9.96 Tf 178.59 0 Td[(m)]TJ/F1 9.96 Tf 12.21 0 Td[(in)28(to)-348(an)-348(ev)28(en)]TJ -353.25 -11.95 Td[(n)28(um)29(b)-28(er)-367(of)-368(distinct)-366(p)-28(ositiv)29(e)-368(Fib)-27(onacci)-367(n)28(um)28(b)-27(ers,)-375(and)]TJ/F7 9.96 Tf 230.89 0 Td[(r)]TJ/F10 6.97 Tf 4.5 -1.5 Td[(O)]TJ/F1 9.96 Tf 6.77 1.5 Td[(\050)]TJ/F7 9.96 Tf 3.87 0 Td[(m)]TJ/F1 9.96 Tf 8.75 0 Td[(\051)-368(is)-368(equal)-367(to)-368(the)-367(n)28(um)28(b)-27(er)-368(of)]TJ/F7 9.96 Tf 122.88 0 Td[(m)]TJ/F1 9.96 Tf 12.41 0 Td[(in)28(to)]TJ -390.07 -11.96 Td[(an)-388(o)-28(dd)-388(n)28(um)29(b)-28(er)-388(of)-388(distinct)-387(p)-27(ositiv)28(e)-388(Fib)-27(onacci)-388(n)29(um)28(b)-27(ers.)-609(W)83(e)-388(call)-388(these)-386(partitions)-388(\134ev)29(en")-388(and)]TJ 0 -11.95 Td[(\134o)-28(dd")-333(resp)-26(ectiv)29(ely)84(.)]TJ/F24 9.96 Tf 0 -14.95 Td[(Prop)-32(osition)-378(1)]TJ/F1 9.96 Tf 67.62 0 Td[(:)-442(Let)]TJ/F7 9.96 Tf 24.98 0 Td[(n)]TJ/F13 9.96 Tf 8.75 0 Td[()]TJ/F1 9.96 Tf 10.52 0 Td[(5)-329(b)-27(e)-329(an)-329(in)28(teger.)-442(Consider)-328(the)-328(co)-27(ecien)30(ts)]TJ/F7 9.96 Tf 181.43 0 Td[(a)]TJ/F1 9.96 Tf 5.27 0 Td[(\050)]TJ/F7 9.96 Tf 3.87 0 Td[(m)]TJ/F1 9.96 Tf 8.75 0 Td[(\051)-329(for)]TJ/F7 9.96 Tf 22.36 0 Td[(m)]TJ/F1 9.96 Tf 12.02 0 Td[(in)-329(the)-329(in)29(terv)56(al)]TJ -345.57 -11.95 Td[([)]TJ/F7 9.96 Tf 2.77 0 Td[(F)]TJ/F10 6.97 Tf 6.4 -1.5 Td[(n)]TJ/F7 9.96 Tf 5.43 1.5 Td[(;)-166(F)]TJ/F10 6.97 Tf 10.83 -1.5 Td[(n)]TJ/F4 6.97 Tf 4.93 0 Td[(+1)]TJ/F1 9.96 Tf 10.58 1.5 Td[(\051.)-645(Split)-399(this)-400(in)29(terv)56(al)-400(in)28(to)-400(the)-400(three)-399(subin)30(terv)56(als)-400([)]TJ/F7 9.96 Tf 217.5 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F7 9.96 Tf 5.42 1.5 Td[(;)-167(F)]TJ/F10 6.97 Tf 10.84 -1.5 Td[(n)]TJ/F1 9.96 Tf 8.08 1.5 Td[(+)]TJ/F7 9.96 Tf 10.4 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F16 6.97 Tf 4.92 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(3)]TJ/F13 9.96 Tf 7.13 1.5 Td[( )]TJ/F1 9.96 Tf 10.4 0 Td[(2])]TJ/F7 9.96 Tf 7.75 0 Td[(;)]TJ/F1 9.96 Tf 4.43 0 Td[([)]TJ/F7 9.96 Tf 2.77 0 Td[(F)]TJ/F10 6.97 Tf 6.4 -1.5 Td[(n)]TJ/F1 9.96 Tf 8.08 1.5 Td[(+)]TJ/F7 9.96 Tf 10.41 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F16 6.97 Tf 4.92 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(3)]TJ/F13 9.96 Tf 7.13 1.5 Td[( )]TJ/F1 9.96 Tf -399.21 -11.96 Td[(1)]TJ/F7 9.96 Tf 4.98 0 Td[(;)-167(F)]TJ/F10 6.97 Tf 10.84 -1.49 Td[(n)]TJ/F1 9.96 Tf 7.63 1.49 Td[(+)]TJ/F7 9.96 Tf 9.96 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.49 Td[(n)]TJ/F16 6.97 Tf 4.93 0 Td[( )]TJ/F4 6.97 Tf 6.22 0 Td[(2)]TJ/F13 9.96 Tf 6.69 1.49 Td[( )]TJ/F1 9.96 Tf 9.96 0 Td[(1])-333(and)-333([)]TJ/F7 9.96 Tf 33.21 0 Td[(F)]TJ/F10 6.97 Tf 6.4 -1.49 Td[(n)]TJ/F1 9.96 Tf 7.64 1.49 Td[(+)]TJ/F7 9.96 Tf 9.96 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.49 Td[(n)]TJ/F16 6.97 Tf 4.92 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(2)]TJ/F7 9.96 Tf 4.47 1.49 Td[(;)-167(F)]TJ/F10 6.97 Tf 10.83 -1.49 Td[(n)]TJ/F4 6.97 Tf 4.93 0 Td[(+1)]TJ/F13 9.96 Tf 12.8 1.49 Td[( )]TJ/F1 9.96 Tf 9.96 0 Td[(1].)]TJ -178.19 -11.95 Td[(1.)-501(The)-737(n)28(um)29(b)-27(ers)]TJ/F7 9.96 Tf 81.47 0 Td[(a)]TJ/F1 9.96 Tf 5.27 0 Td[(\050)]TJ/F7 9.96 Tf 3.87 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F1 9.96 Tf 5.42 1.5 Td[(\051)]TJ/F7 9.96 Tf 3.87 0 Td[(;)-167(a)]TJ/F1 9.96 Tf 9.7 0 Td[(\050)]TJ/F7 9.96 Tf 3.87 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F1 9.96 Tf 10.32 1.5 Td[(+)-491(1\051)]TJ/F7 9.96 Tf 21.5 0 Td[(;)-166(:)-167(:)-166(:)-167(;)-167(a)]TJ/F1 9.96 Tf 27.4 0 Td[(\050)]TJ/F7 9.96 Tf 3.88 0 Td[(F)]TJ/F10 6.97 Tf 6.4 -1.5 Td[(n)]TJ/F1 9.96 Tf 10.32 1.5 Td[(+)]TJ/F7 9.96 Tf 12.65 0 Td[(F)]TJ/F10 6.97 Tf 6.4 -1.5 Td[(n)]TJ/F16 6.97 Tf 4.93 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(3)]TJ/F13 9.96 Tf 9.36 1.5 Td[( )]TJ/F1 9.96 Tf 12.65 0 Td[(2\051)-737(are)-737(equal)-736(to)-738(the)-736(n)28(um)28(b)-27(ers)]TJ -245.6 -11.96 Td[(\050)]TJ/F13 9.96 Tf 3.88 0 Td[( )]TJ/F1 9.96 Tf 7.75 0 Td[(1\051)]TJ/F10 6.97 Tf 8.85 3.62 Td[(n)]TJ/F16 6.97 Tf 4.93 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(1)]TJ/F7 9.96 Tf 4.46 -3.62 Td[(a)]TJ/F1 9.96 Tf 5.27 0 Td[(\050)]TJ/F7 9.96 Tf 3.87 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.49 Td[(n)]TJ/F16 6.97 Tf 4.93 0 Td[( )]TJ/F4 6.97 Tf 6.22 0 Td[(3)]TJ/F13 9.96 Tf 6.69 1.49 Td[( )]TJ/F1 9.96 Tf 9.96 0 Td[(2\051)]TJ/F7 9.96 Tf 8.85 0 Td[(;)]TJ/F1 9.96 Tf 4.43 0 Td[(\050)]TJ/F13 9.96 Tf 3.88 0 Td[( )]TJ/F1 9.96 Tf 7.75 0 Td[(1\051)]TJ/F10 6.97 Tf 8.85 3.62 Td[(n)]TJ/F16 6.97 Tf 4.93 0 Td[( )]TJ/F4 6.97 Tf 6.22 0 Td[(1)]TJ/F7 9.96 Tf 4.47 -3.62 Td[(a)]TJ/F1 9.96 Tf 5.27 0 Td[(\050)]TJ/F7 9.96 Tf 3.87 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.49 Td[(n)]TJ/F16 6.97 Tf 4.92 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(3)]TJ/F13 9.96 Tf 6.68 1.49 Td[( )]TJ/F1 9.96 Tf 9.97 0 Td[(3\051)]TJ/F7 9.96 Tf 8.85 0 Td[(;)-167(:)-166(:)-167(:)-166(;)]TJ/F1 9.96 Tf 22.14 0 Td[(\050)]TJ/F13 9.96 Tf 3.87 0 Td[( )]TJ/F1 9.96 Tf 7.75 0 Td[(1\051)]TJ/F10 6.97 Tf 8.86 3.62 Td[(n)]TJ/F16 6.97 Tf 4.92 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(1)]TJ/F7 9.96 Tf 4.47 -3.62 Td[(a)]TJ/F1 9.96 Tf 5.27 0 Td[(\0500\051)-333(in)-333(that)-333(order.)]TJ -257.27 -11.95 Td[(2.)-501(The)-333(n)28(um)29(b)-28(ers)]TJ/F7 9.96 Tf 73.42 0 Td[(a)]TJ/F1 9.96 Tf 5.27 0 Td[(\050)]TJ/F7 9.96 Tf 3.87 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F1 9.96 Tf 7.64 1.5 Td[(+)]TJ/F7 9.96 Tf 9.96 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F16 6.97 Tf 4.92 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(3)]TJ/F13 9.96 Tf 6.68 1.5 Td[( )]TJ/F1 9.96 Tf 9.96 0 Td[(1\051)]TJ/F7 9.96 Tf 8.86 0 Td[(;)-167(a)]TJ/F1 9.96 Tf 9.69 0 Td[(\050)]TJ/F7 9.96 Tf 3.88 0 Td[(F)]TJ/F10 6.97 Tf 6.4 -1.5 Td[(n)]TJ/F1 9.96 Tf 7.64 1.5 Td[(+)]TJ/F7 9.96 Tf 9.96 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F16 6.97 Tf 4.93 0 Td[( )]TJ/F4 6.97 Tf 6.22 0 Td[(3)]TJ/F1 9.96 Tf 4.47 1.5 Td[(\051)]TJ/F7 9.96 Tf 3.88 0 Td[(;)-166(:)-167(:)-166(:)-167(;)-166(a)]TJ/F1 9.96 Tf 27.4 0 Td[(\050)]TJ/F7 9.96 Tf 3.87 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F1 9.96 Tf 7.64 1.5 Td[(+)]TJ/F7 9.96 Tf 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Td[(No)27(w)-335(w)27(e)-334(use)-334(a)-335(similar)-333(analysis)-333(to)-335(pro)28(v)28(e)-335(Prop)-27(osition)-334(1.3.)-448(Let)]TJ/F7 9.96 Tf 263.72 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.49 Td[(n)]TJ/F1 9.96 Tf 7.64 1.49 Td[(+)]TJ/F7 9.96 Tf 9.97 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.49 Td[(n)]TJ/F16 6.97 Tf 4.92 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(2)]TJ/F13 9.96 Tf 7.26 1.49 Td[()]TJ/F7 9.96 Tf 10.54 0 Td[(m)]TJ/F13 9.96 Tf 11.54 0 Td[()]TJ/F7 9.96 Tf 10.53 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.49 Td[(n)]TJ/F4 6.97 Tf 4.93 0 Td[(+1)]TJ/F13 9.96 Tf 12.8 1.49 Td[( )]TJ/F1 9.96 Tf 9.98 0 Td[(1.)]TJ -399.21 -11.95 Td[(As)-401(b)-28(efore,)-417(the)-401(largest)-401(part)-401(of)-402(a)-402(partition)-401(of)]TJ/F7 9.96 Tf 197.19 0 Td[(m)]TJ/F1 9.96 Tf 12.75 0 Td[(m)28(ust)-401(b)-27(e)]TJ/F7 9.96 Tf 39.61 0 Td[(F)]TJ/F10 6.97 Tf 6.4 -1.5 Td[(n)]TJ/F1 9.96 Tf 9.43 1.5 Td[(or)]TJ/F7 9.96 Tf 12.88 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F16 6.97 Tf 4.92 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(1)]TJ/F1 9.96 Tf 4.47 1.5 Td[(.)-649(If)-402(it)-401(is)]TJ/F7 9.96 Tf 41.22 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F1 9.96 Tf 5.42 1.5 Td[(,)-419(the)-401(second)]TJ -353.34 -11.96 Td[(largest)-382(part)-382(cannot)-382(b)-27(e)]TJ/F7 9.96 Tf 101.98 0 Td[(F)]TJ/F10 6.97 Tf 6.4 -1.49 Td[(n)]TJ/F16 6.97 Tf 4.93 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(1)]TJ/F1 9.96 Tf 4.46 1.49 Td[(.)-593(If,)-394(on)-383(the)-382(other)-382(hand,)-394(it)-383(is)]TJ/F7 9.96 Tf 129.78 0 Td[(F)]TJ/F10 6.97 Tf 6.4 -1.49 Td[(n)]TJ/F16 6.97 Tf 4.93 0 Td[( )]TJ/F4 6.97 Tf 6.22 0 Td[(1)]TJ/F1 9.96 Tf 4.47 1.49 Td[(,)-395(then)-382(the)-382(second)-381(largest)-381(part)]TJ -275.8 -11.95 Td[(m)28(ust)-332(b)-28(e)]TJ/F7 9.96 Tf 38.25 0 Td[(F)]TJ/F10 6.97 Tf 6.4 -1.5 Td[(n)]TJ/F16 6.97 Tf 4.93 0 Td[( )]TJ/F4 6.97 Tf 6.22 0 Td[(2)]TJ/F1 9.96 Tf 4.47 1.5 Td[(.)]TJ -40.35 -11.96 Td[(Again,)-498(w)28(e)-464(can)-464(split)]TJ/F37 9.96 Tf 92.5 0 Td[(a)-479(subset)-478(of)-479(the)-479(set)-479(of)-478(p)51(artitions)]TJ/F1 9.96 Tf 146.89 0 Td[(in)28(to)-465(pairs.)-836(Eac)28(h)-464(pair)-464(consists)-462(of)]TJ -259.31 -11.95 Td[(t)28(w)27(o)-478(partitions)-477(of)-479(the)-477(form)]TJ/F7 9.96 Tf 123.56 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F1 9.96 Tf 8.6 1.5 Td[(+)]TJ/F7 9.96 Tf 10.92 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(a)]TJ/F1 9.96 Tf 7.99 1.5 Td[(+)]TJ/F7 9.96 Tf 10.93 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(b)]TJ/F1 9.96 Tf 7.17 1.5 Td[(+)]TJ/F7 9.96 Tf 10.93 0 Td[(:)-167(:)-166(:)]TJ/F1 9.96 Tf 18.05 0 Td[(and)]TJ/F7 9.96 Tf 20.81 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F16 6.97 Tf 4.93 0 Td[( )]TJ/F4 6.97 Tf 6.22 0 Td[(1)]TJ/F1 9.96 Tf 7.65 1.5 Td[(+)]TJ/F7 9.96 Tf 10.92 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F16 6.97 Tf 4.92 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(2)]TJ/F1 9.96 Tf 7.65 1.5 Td[(+)]TJ/F7 9.96 Tf 10.92 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(a)]TJ/F1 9.96 Tf 7.99 1.5 Td[(+)]TJ/F7 9.96 Tf 10.93 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(b)]TJ/F1 9.96 Tf 7.17 1.5 Td[(+)]TJ/F7 9.96 Tf 10.93 0 Td[(:)-166(:)-167(:)]TJ/F1 9.96 Tf 13.28 0 Td[(,)-515(where)]TJ/F7 9.96 Tf -373.57 -11.96 Td[(n)]TJ/F13 9.96 Tf 8.19 0 Td[( )]TJ/F1 9.96 Tf 9.97 0 Td[(3)]TJ/F13 9.96 Tf 7.74 0 Td[()]TJ/F7 9.96 Tf 10.52 0 Td[(a)-278(>)-277(b)-279(>)-277(:)-167(:)-167(:)]TJ/F1 9.96 Tf 49.39 0 Td[(.)-445(In)-333(eac)29(h)-334(pair)-332(there)-332(is)-333(an)-334(ev)29(en)-333(and)-333(an)-333(o)-28(dd)-333(partition.)]TJ -65.89 -11.95 Td[(The)-340(remaining)-338(partitions)-338(are)-339(of)-340(the)-338(form)]TJ/F7 9.96 Tf 181.59 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F1 9.96 Tf 7.68 1.5 Td[(+)]TJ/F7 9.96 Tf 10 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F16 6.97 Tf 4.92 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(2)]TJ/F1 9.96 Tf 6.72 1.5 Td[(+)]TJ/F7 9.96 Tf 10.01 0 Td[(F)]TJ/F10 6.97 Tf 6.4 -1.5 Td[(a)]TJ/F1 9.96 Tf 7.08 1.5 Td[(+)]TJ/F7 9.96 Tf 10 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(b)]TJ/F1 9.96 Tf 6.26 1.5 Td[(+)]TJ/F7 9.96 Tf 10 0 Td[(:)-166(:)-167(:)]TJ/F1 9.96 Tf 13.28 0 Td[(,)-341(where)]TJ/F7 9.96 Tf 35.04 0 Td[(n)]TJ/F13 9.96 Tf 8.23 0 Td[( )]TJ/F1 9.96 Tf 10.01 0 Td[(3)]TJ/F13 9.96 Tf 7.85 0 Td[()]TJ/F7 9.96 Tf 10.62 0 Td[(a)-288(>)]TJ -391.07 -11.96 Td[(b)-368(>)-368(:)-167(:)-167(:)]TJ/F1 9.96 Tf 32.64 0 Td[(.)-607(T)83(o)-388(eac)29(h)-387(one)-387(of)-388(these)-386(partitions)-386(w)28(e)-388(can)-387(assign)-386(a)-388(partition)-386(of)]TJ/F7 9.96 Tf 276.53 0 Td[(m)]TJ/F16 6.97 Tf 8.74 3.62 Td[(0)]TJ/F1 9.96 Tf 6.47 -3.62 Td[(=)]TJ/F7 9.96 Tf 11.41 0 Td[(m)]TJ/F13 9.96 Tf 11.32 0 Td[( )]TJ/F7 9.96 Tf 10.32 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.49 Td[(n)]TJ/F13 9.96 Tf 8 1.49 Td[( )]TJ/F7 9.96 Tf 10.32 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.49 Td[(n)]TJ/F16 6.97 Tf 4.92 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(2)]TJ/F1 9.96 Tf 4.47 1.49 Td[(,)]TJ -404.19 -11.95 Td[(b)28(y)-386(just)-385(remo)28(ving)-385(the)-386(parts)]TJ/F7 9.96 Tf 122.07 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F1 9.96 Tf 9.27 1.5 Td[(and)]TJ/F7 9.96 Tf 19.9 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F16 6.97 Tf 4.92 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(2)]TJ/F1 9.96 Tf 4.47 1.5 Td[(.)-603(This)-386(is)-385(in)-386(fact)-386(a)-386(bijection.)-601(Since)]TJ/F7 9.96 Tf 156.42 0 Td[(m)]TJ/F16 6.97 Tf 8.75 3.61 Td[(0)]TJ/F7 9.96 Tf 6.44 -3.61 Td[(<)-366(F)]TJ/F10 6.97 Tf 17.8 -1.5 Td[(n)]TJ/F16 6.97 Tf 4.92 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(2)]TJ/F1 9.96 Tf 4.47 1.5 Td[(,)-399(an)28(y)]TJ -384.71 -11.96 Td[(partition)-291(of)]TJ/F7 9.96 Tf 52.06 0 Td[(m)]TJ/F16 6.97 Tf 8.75 3.62 Td[(0)]TJ/F1 9.96 Tf 5.71 -3.62 Td[(has)-291(largest)-292(part)-292(less)-290(than)-292(or)-292(equal)-292(to)]TJ/F7 9.96 Tf 160.58 0 Td[(F)]TJ/F10 6.97 Tf 6.4 -1.49 Td[(n)]TJ/F16 6.97 Tf 4.93 0 Td[( )]TJ/F4 6.97 Tf 6.22 0 Td[(3)]TJ/F1 9.96 Tf 4.47 1.49 Td[(;)-306(therefore)-290(it)-292(can)-292(b)-27(e)-292(obtained)-291(in)-292(that)]TJ -249.12 -11.95 Td[(w)27(a)28(y)-333(from)-333(a)-333(partition)-333(of)]TJ/F7 9.96 Tf 104.94 0 Td[(m)]TJ/F1 9.96 Tf 8.75 0 Td[(.)]TJ -93.76 -11.96 Td[(It)-365(is)-364(clear)-365(that,)-372(under)-364(this)-364(bijection,)-371(o)-28(dd)-365(partitions)-364(of)]TJ/F7 9.96 Tf 239.9 0 Td[(m)]TJ/F1 9.96 Tf 12.39 0 Td[(go)-365(to)-366(o)-27(dd)-365(partitions)-364(of)]TJ/F7 9.96 Tf 103.5 0 Td[(m)]TJ/F16 6.97 Tf 8.75 3.62 Td[(0)]TJ/F1 9.96 Tf 6.43 -3.62 Td[(and)]TJ -390.9 -11.95 Td[(ev)28(en)-316(partitions)-316(of)]TJ/F7 9.96 Tf 79.02 0 Td[(m)]TJ/F1 9.96 Tf 11.91 0 Td[(go)-317(to)-317(ev)28(en)-316(partitions)-316(of)]TJ/F7 9.96 Tf 104.15 0 Td[(m)]TJ/F16 6.97 Tf 8.75 3.61 Td[(0)]TJ/F1 9.96 Tf 2.79 -3.61 Td[(.)-439(It)-318(follo)29(ws)-318(that)]TJ/F7 9.96 Tf 71.75 0 Td[(a)]TJ/F1 9.96 Tf 5.26 0 Td[(\050)]TJ/F7 9.96 Tf 3.88 0 Td[(m)]TJ/F1 9.96 Tf 8.74 0 Td[(\051)-278(=)]TJ/F7 9.96 Tf 17.16 0 Td[(a)]TJ/F1 9.96 Tf 5.27 0 Td[(\050)]TJ/F7 9.96 Tf 3.87 0 Td[(m)]TJ/F13 9.96 Tf 10.64 0 Td[( )]TJ/F7 9.96 Tf 9.64 0 Td[(F)]TJ/F10 6.97 Tf 6.4 -1.5 Td[(n)]TJ/F13 9.96 Tf 7.32 1.5 Td[( )]TJ/F7 9.96 Tf 9.63 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F16 6.97 Tf 4.92 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(2)]TJ/F1 9.96 Tf 4.47 1.5 Td[(\051,)-320(as)]TJ -388.21 -11.96 Td[(claimed.)]TJ 19.93 -11.95 Td[(Finally)-403(w)27(e)-404(pro)28(v)28(e)-404(Prop)-28(osition)-403(1.1.)-659(Consider)]TJ/F7 9.96 Tf 196.26 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F13 9.96 Tf 9.37 1.5 Td[()]TJ/F7 9.96 Tf 11.7 0 Td[(m)]TJ/F13 9.96 Tf 12.7 0 Td[()]TJ/F7 9.96 Tf 11.7 0 Td[(F)]TJ/F10 6.97 Tf 6.4 -1.5 Td[(n)]TJ/F1 9.96 Tf 8.11 1.5 Td[(+)]TJ/F7 9.96 Tf 10.44 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F16 6.97 Tf 4.92 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(3)]TJ/F13 9.96 Tf 7.15 1.5 Td[( )]TJ/F1 9.96 Tf 10.44 0 Td[(2.)-658(The)-405(parts)-403(of)-405(a)]TJ -328.17 -11.96 Td[(partition)-285(of)]TJ/F7 9.96 Tf 51.94 0 Td[(m)]TJ/F1 9.96 Tf 11.6 0 Td[(come)-284(from)-285(the)-286(list)]TJ/F7 9.96 Tf 80.93 0 Td[(F)]TJ/F4 6.97 Tf 6.41 -1.49 Td[(2)]TJ/F7 9.96 Tf 4.47 1.49 Td[(;)-166(F)]TJ/F4 6.97 Tf 10.83 -1.49 Td[(3)]TJ/F7 9.96 Tf 4.47 1.49 Td[(;)-167(:)-166(:)-167(:)-166(;)-167(F)]TJ/F10 6.97 Tf 28.55 -1.49 Td[(n)]TJ/F1 9.96 Tf 5.42 1.49 Td[(.)-429(T)83(o)-286(eac)29(h)-286(partition)]TJ/F7 9.96 Tf 84.24 0 Td[()]TJ/F1 9.96 Tf 8.89 0 Td[(of)]TJ/F7 9.96 Tf 10.87 0 Td[(m)]TJ/F1 9.96 Tf 8.75 0 Td[(,)-295(assign)-285(the)-285(partition)]TJ/F7 9.96 Tf -317.37 -11.95 Td[()]TJ/F16 6.97 Tf 6.04 3.61 Td[(0)]TJ/F1 9.96 Tf 5.96 -3.61 Td[(of)]TJ/F7 9.96 Tf 11.2 0 Td[(m)]TJ/F16 6.97 Tf 8.75 3.61 Td[(0)]TJ/F1 9.96 Tf 5.56 -3.61 Td[(=)]TJ/F7 9.96 Tf 10.52 0 Td[(F)]TJ/F10 6.97 Tf 6.4 -1.5 Td[(n)]TJ/F4 6.97 Tf 4.93 0 Td[(+2)]TJ/F13 9.96 Tf 12.5 1.5 Td[( )]TJ/F1 9.96 Tf 9.67 0 Td[(2)]TJ/F13 9.96 Tf 6.89 0 Td[( )]TJ/F7 9.96 Tf 9.67 0 Td[(m)]TJ/F1 9.96 Tf 11.92 0 Td[(consisting)-316(of)-319(all)-318(the)-318(n)28(um)29(b)-28(ers)-317(on)-318(the)-318(ab)-28(o)28(v)28(e)-318(list)-318(that)-318(do)-318(not)-319(app)-27(ear)-318(in)]TJ/F7 9.96 Tf -110.01 -11.96 Td[()]TJ/F1 9.96 Tf 6.04 0 Td[(.)-440(An)28(y)-322(partition)-321(of)]TJ/F7 9.96 Tf 81 0 Td[(m)]TJ/F16 6.97 Tf 8.75 3.62 Td[(0)]TJ/F1 9.96 Tf 6 -3.62 Td[(can)-321(b)-28(e)-321(obtained)-321(in)-321(suc)29(h)-322(a)-322(w)27(a)28(y)-322(from)-321(a)-322(partition)-321(of)]TJ/F7 9.96 Tf 218.86 0 Td[(m)]TJ/F1 9.96 Tf 8.74 0 Td[(:)-439(the)-321(partitions)-321(of)]TJ/F7 9.96 Tf -329.39 -11.95 Td[(m)]TJ/F16 6.97 Tf 8.75 3.61 Td[(0)]TJ/F1 9.96 Tf 6.04 -3.61 Td[(also)-326(ha)28(v)28(e)-325(all)-326(their)-325(parts)-325(less)-325(than)-326(or)-325(equal)-326(to)]TJ/F7 9.96 Tf 197.71 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F1 9.96 Tf 5.42 1.5 Td[(,)-328(b)-27(ecause)-324(it)-326(is)-326(easily)-325(seen)-324(that)]TJ/F7 9.96 Tf 133.11 0 Td[(m)]TJ/F16 6.97 Tf 8.75 3.61 Td[(0)]TJ/F7 9.96 Tf 5.56 -3.61 Td[(<)-278(F)]TJ/F10 6.97 Tf 16.93 -1.5 Td[(n)]TJ/F4 6.97 Tf 4.92 0 Td[(+1)]TJ/F1 9.96 Tf 10.59 1.5 Td[(.)]TJ -384.26 -11.96 Td[(So)-401(the)-401(partitions)-400(of)]TJ/F7 9.96 Tf 90.52 0 Td[(m)]TJ/F1 9.96 Tf 12.74 0 Td[(are)-401(in)-401(bijection)-400(with)-401(the)-401(partitions)-400(of)]TJ/F7 9.96 Tf 170.35 0 Td[(m)]TJ/F16 6.97 Tf 8.75 3.62 Td[(0)]TJ/F1 9.96 Tf 2.79 -3.62 Td[(.)-649(If)-401(a)-402(partition)]TJ/F7 9.96 Tf 71.07 0 Td[()]TJ/F1 9.96 Tf 10.03 0 Td[(of)]TJ/F7 9.96 Tf 12.03 0 Td[(m)]TJ/F1 9.96 Tf -398.21 -11.95 Td[(has)]TJ/F7 9.96 Tf 17.96 0 Td[(k)]TJ/F1 9.96 Tf 9.02 0 Td[(parts,)-357(the)-353(corresp)-25(onding)-353(partition)]TJ/F7 9.96 Tf 151.88 0 Td[()]TJ/F16 6.97 Tf 6.04 3.61 Td[(0)]TJ/F1 9.96 Tf 6.31 -3.61 Td[(of)]TJ/F7 9.96 Tf 11.54 0 Td[(m)]TJ/F16 6.97 Tf 8.75 3.61 Td[(0)]TJ/F1 9.96 Tf 6.31 -3.61 Td[(has)]TJ/F7 9.96 Tf 17.97 0 Td[(n)]TJ/F13 9.96 Tf 8.32 0 Td[( )]TJ/F1 9.96 Tf 10.1 0 Td[(1)]TJ/F13 9.96 Tf 7.32 0 Td[( )]TJ/F7 9.96 Tf 10.1 0 Td[(k)]TJ/F1 9.96 Tf 9.01 0 Td[(parts.)-503(Therefore,)-356(if)]TJ/F7 9.96 Tf 87.52 0 Td[(n)]TJ/F1 9.96 Tf 9.49 0 Td[(is)-353(o)-28(dd,)]TJ -377.64 -11.96 Td[(the)-293(bijection)-291(tak)28(es)-293(o)-28(dd)-293(partitions)-292(to)-293(o)-28(dd)-293(partitions)-292(and)-293(ev)28(en)-292(partitions)-292(to)-294(ev)28(en)-292(partitions,)-300(and)]TJ/F7 9.96 Tf 0 -11.95 Td[(a)]TJ/F1 9.96 Tf 5.27 0 Td[(\050)]TJ/F7 9.96 Tf 3.87 0 Td[(m)]TJ/F1 9.96 Tf 8.75 0 Td[(\051)-405(=)]TJ/F7 9.96 Tf 19.69 0 Td[(a)]TJ/F1 9.96 Tf 5.27 0 Td[(\050)]TJ/F7 9.96 Tf 3.87 0 Td[(m)]TJ/F16 6.97 Tf 8.75 3.61 Td[(0)]TJ/F1 9.96 Tf 2.8 -3.61 Td[(\051.)-673(If)]TJ/F7 9.96 Tf 24.07 0 Td[(n)]TJ/F1 9.96 Tf 10.06 0 Td[(is)-410(ev)29(en,)-428(the)-409(bijection)-408(tak)27(es)-408(o)-28(dd)-410(partitions)-408(to)-410(ev)28(en)-409(partitions,)-427(and)-410(ev)29(en)]TJ -92.4 -11.96 Td[(partitions)-332(to)-334(o)-28(dd)-332(partitions,)-332(and)]TJ/F7 9.96 Tf 144.9 0 Td[(a)]TJ/F1 9.96 Tf 5.27 0 Td[(\050)]TJ/F7 9.96 Tf 3.87 0 Td[(m)]TJ/F1 9.96 Tf 8.75 0 Td[(\051)-278(=)]TJ/F13 9.96 Tf 17.16 0 Td[( )]TJ/F7 9.96 Tf 7.74 0 Td[(a)]TJ/F1 9.96 Tf 5.27 0 Td[(\050)]TJ/F7 9.96 Tf 3.88 0 Td[(m)]TJ/F16 6.97 Tf 8.74 3.62 Td[(0)]TJ/F1 9.96 Tf 2.8 -3.62 Td[(\051.)-444(In)-334(an)28(y)-333(case,)]TJ/F7 9.96 Tf 66.19 0 Td[(a)]TJ/F1 9.96 Tf 5.27 0 Td[(\050)]TJ/F7 9.96 Tf 3.87 0 Td[(m)]TJ/F1 9.96 Tf 8.75 0 Td[(\051)-278(=)-278(\050)]TJ/F13 9.96 Tf 21.03 0 Td[( )]TJ/F1 9.96 Tf 7.75 0 Td[(1\051)]TJ/F10 6.97 Tf 8.86 3.62 Td[(n)]TJ/F16 6.97 Tf 4.92 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(1)]TJ/F7 9.96 Tf 4.47 -3.62 Td[(a)]TJ/F1 9.96 Tf 5.26 0 Td[(\050)]TJ/F7 9.96 Tf 3.88 0 Td[(m)]TJ/F16 6.97 Tf 8.75 3.62 Td[(0)]TJ/F1 9.96 Tf 2.79 -3.62 Td[(\051.)]TJ -346.48 -11.95 Td[(No)27(w,)-301(it)-292(is)-291(easily)-291(seen)-290(that)]TJ/F7 9.96 Tf 113.73 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F1 9.96 Tf 6.81 1.5 Td[(+)]TJ/F7 9.96 Tf 9.14 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F16 6.97 Tf 4.92 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(2)]TJ/F13 9.96 Tf 7.24 1.5 Td[()]TJ/F7 9.96 Tf 10.51 0 Td[(m)]TJ/F16 6.97 Tf 8.75 3.61 Td[(0)]TJ/F13 9.96 Tf 5.56 -3.61 Td[()]TJ/F7 9.96 Tf 10.52 0 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9.96 Tf 8.74 0 Td[(\051)-278(=)]TJ/F7 9.96 Tf 17.16 0 Td[(r)]TJ/F10 6.97 Tf 4.5 -1.5 Td[(O)]TJ/F1 9.96 Tf 6.77 1.5 Td[(\050)]TJ/F7 9.96 Tf 3.87 0 Td[(m)]TJ/F1 9.96 Tf 8.75 0 Td[(\051.)]TJ -345.14 -11.96 Td[(Then)-333(lim)]TJ/F10 6.97 Tf 39.86 -1.49 Td[(n)]TJ/F16 6.97 Tf 4.92 0 Td[(!1)]TJ/F7 9.96 Tf 18.04 1.49 Td[(p)]TJ/F10 6.97 Tf 5.02 -1.49 Td[(n)]TJ/F1 9.96 Tf 8.19 1.49 Td[(=)-277(1.)]TJ/F24 9.96 Tf -76.03 -14.94 Td[(Pro)-33(of)]TJ/F1 9.96 Tf 27.83 0 Td[(:)-699(Let)]TJ/F7 9.96 Tf 28.85 0 Td[()]TJ/F10 6.97 Tf 6.38 -1.5 Td[(n)]TJ/F1 9.96 Tf 10.01 1.5 Td[(b)-27(e)-461(the)-460(n)28(um)29(b)-28(er)-460(of)-461(non-zero)-460(co)-27(ecien)30(ts)-460(among)-461(the)-460(\014rst)]TJ/F7 9.96 Tf 250.44 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F1 9.96 Tf 10.01 1.5 Td[(co)-28(ecien)30(ts)]TJ/F7 9.96 Tf -359.85 -11.96 Td[(a)]TJ/F1 9.96 Tf 5.27 0 Td[(\0500\051)]TJ/F7 9.96 Tf 12.73 0 Td[(;)-166(a)]TJ/F1 9.96 Tf 9.69 0 Td[(\0501\051)]TJ/F7 9.96 Tf 12.73 0 Td[(;)-167(:)-166(:)-167(:)-166(;)-167(a)]TJ/F1 9.96 Tf 27.4 0 Td[(\050)]TJ/F7 9.96 Tf 3.88 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.49 Td[(n)]TJ/F13 9.96 Tf 7.68 1.49 Td[( )]TJ/F1 9.96 Tf 10.01 0 Td[(1\051,)-342(so)-340(that)]TJ/F7 9.96 Tf 48.99 0 Td[(p)]TJ/F4 6.97 Tf 5.01 -1.8 Td[(\050)]TJ/F10 6.97 Tf 3.11 0 Td[(F)]TJ/F12 4.98 Tf 5.06 -0.99 Td[(n)]TJ/F16 6.97 Tf 4.88 0.99 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(1\051)]TJ/F1 9.96 Tf 10.47 1.8 Td[(=)-289(1)]TJ/F13 9.96 Tf 17.87 0 Td[( )]TJ/F7 9.96 Tf 10.01 0 Td[()]TJ/F10 6.97 Tf 6.37 -1.49 Td[(n)]TJ/F7 9.96 Tf 5.43 1.49 Td[(=F)]TJ/F10 6.97 Tf 11.38 -1.49 Td[(n)]TJ/F1 9.96 Tf 5.43 1.49 Td[(.)-465(Notice)-340(that)-340(for)]TJ/F7 9.96 Tf 75.72 0 Td[(F)]TJ/F10 6.97 Tf 6.4 -1.49 Td[(n)]TJ/F16 6.97 Tf 4.93 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(1)]TJ/F13 9.96 Tf 7.35 1.49 Td[()]TJ/F7 9.96 Tf 10.63 0 Td[(m)-290(<)-290(F)]TJ/F10 6.97 Tf 28.67 -1.49 Td[(n)]TJ/F1 9.96 Tf 8.82 1.49 Td[(there)]TJ -384.79 -11.95 Td[(are)-387(at)-387(most)]TJ/F7 9.96 Tf 54.82 0 Td[()]TJ/F10 6.97 Tf 6.38 -1.5 Td[(n)]TJ/F1 9.96 Tf 9.27 1.5 Td[(non-zero)-387(co)-27(ecien)30(ts)-387(among)]TJ/F7 9.96 Tf 124.57 0 Td[(a)]TJ/F1 9.96 Tf 5.26 0 Td[(\0500\051)]TJ/F7 9.96 Tf 12.73 0 Td[(;)-167(a)]TJ/F1 9.96 Tf 9.7 0 Td[(\0501\051)]TJ/F7 9.96 Tf 12.73 0 Td[(;)-166(:)-167(:)-166(:)-167(;)-166(a)]TJ/F1 9.96 Tf 27.4 0 Td[(\050)]TJ/F7 9.96 Tf 3.88 0 Td[(m)]TJ/F1 9.96 Tf 8.74 0 Td[(\051,)-401(so)]TJ/F7 9.96 Tf 23.4 0 Td[(p)]TJ/F10 6.97 Tf 5.01 -1.5 Td[(m)]TJ/F13 9.96 Tf 11.23 1.5 Td[()]TJ/F1 9.96 Tf 11.41 0 Td[(1)]TJ/F13 9.96 Tf 7.55 0 Td[( )]TJ/F7 9.96 Tf 10.32 0 Td[()]TJ/F10 6.97 Tf 6.37 -1.5 Td[(n)]TJ/F7 9.96 Tf 5.43 1.5 Td[(=)]TJ/F1 9.96 Tf 4.98 0 Td[(\050)]TJ/F7 9.96 Tf 3.87 0 Td[(m)]TJ/F1 9.96 Tf 11.32 0 Td[(+)-258(1\051)]TJ/F7 9.96 Tf 22.84 0 Td[(>)]TJ/F1 9.96 Tf -399.21 -11.96 Td[(1)]TJ/F13 9.96 Tf 7.19 0 Td[( )]TJ/F1 9.96 Tf 9.97 0 Td[(2)]TJ/F7 9.96 Tf 4.98 0 Td[()]TJ/F10 6.97 Tf 6.37 -1.49 Td[(n)]TJ/F7 9.96 Tf 5.42 1.49 Td[(=F)]TJ/F10 6.97 Tf 11.39 -1.49 Td[(n)]TJ/F1 9.96 Tf 5.43 1.49 Td[(.)-444(W)83(e)-333(shall)-332(no)28(w)-334(pro)28(v)28(e)-333(that)-333(lim)]TJ/F10 6.97 Tf 130.7 -1.49 Td[(n)]TJ/F16 6.97 Tf 4.92 0 Td[(!1)]TJ/F7 9.96 Tf 18.05 1.49 Td[()]TJ/F10 6.97 Tf 6.37 -1.49 Td[(n)]TJ/F7 9.96 Tf 5.42 1.49 Td[(=F)]TJ/F10 6.97 Tf 11.39 -1.49 Td[(n)]TJ/F1 9.96 Tf 8.19 1.49 Td[(=)-278(0,)-333(from)-333(whic)28(h)-333(Prop)-27(osition)-333(2)-333(follo)28(ws.)]TJ -215.87 -11.95 Td[(First)-354(w)27(e)-353(obtain)-354(a)-355(recurrence)-350(relation)-354(for)]TJ/F7 9.96 Tf 179.76 0 Td[()]TJ/F10 6.97 Tf 6.38 -1.5 Td[(n)]TJ/F1 9.96 Tf 5.42 1.5 Td[(.)-507(Consider)-353(the)-354(non-zero)-353(co)-27(ecien)30(ts)]TJ/F7 9.96 Tf 158.26 0 Td[(a)]TJ/F1 9.96 Tf 5.26 0 Td[(\050)]TJ/F7 9.96 Tf 3.88 0 Td[(m)]TJ/F1 9.96 Tf 8.74 0 Td[(\051)-355(for)]TJ/F7 9.96 Tf -387.62 -11.96 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.49 Td[(n)]TJ/F13 9.96 Tf 8.19 1.49 Td[()]TJ/F7 9.96 Tf 10.51 0 Td[(m)]TJ/F13 9.96 Tf 11.52 0 Td[()]TJ/F7 9.96 Tf 10.51 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.49 Td[(n)]TJ/F4 6.97 Tf 4.92 0 Td[(+1)]TJ/F13 9.96 Tf 11.3 1.49 Td[( )]TJ/F1 9.96 Tf 8.46 0 Td[(1.)-420(W)83(e)-257(kno)28(w)-259(that)-258(there)-256(are)]TJ/F7 9.96 Tf 115.06 0 Td[()]TJ/F10 6.97 Tf 6.37 -1.49 Td[(n)]TJ/F4 6.97 Tf 4.92 0 Td[(+1)]TJ/F13 9.96 Tf 11.3 1.49 Td[( )]TJ/F7 9.96 Tf 8.46 0 Td[()]TJ/F10 6.97 Tf 6.38 -1.49 Td[(n)]TJ/F1 9.96 Tf 7.99 1.49 Td[(suc)29(h)-258(co)-27(ecien)30(ts.)-419(No)27(w)-258(split)-257(the)-258(in)29(terv)56(al)]TJ -238.71 -11.95 Td[([)]TJ/F7 9.96 Tf 2.77 0 Td[(F)]TJ/F10 6.97 Tf 6.4 -1.5 Td[(n)]TJ/F7 9.96 Tf 5.43 1.5 Td[(;)-166(F)]TJ/F10 6.97 Tf 10.83 -1.5 Td[(n)]TJ/F4 6.97 Tf 4.93 0 Td[(+1)]TJ/F13 9.96 Tf 12.75 1.5 Td[( )]TJ/F1 9.96 Tf 9.91 0 Td[(1])-331(in)28(to)-331(the)-331(three)-329(subin)29(terv)56(als)-330([)]TJ/F7 9.96 Tf 131.51 0 Td[(F)]TJ/F10 6.97 Tf 6.4 -1.5 Td[(n)]TJ/F7 9.96 Tf 5.42 1.5 Td[(;)-167(F)]TJ/F10 6.97 Tf 10.84 -1.5 Td[(n)]TJ/F1 9.96 Tf 7.59 1.5 Td[(+)]TJ/F7 9.96 Tf 9.92 0 Td[(F)]TJ/F10 6.97 Tf 6.4 -1.5 Td[(n)]TJ/F16 6.97 Tf 4.93 0 Td[( )]TJ/F4 6.97 Tf 6.22 0 Td[(3)]TJ/F13 9.96 Tf 6.64 1.5 Td[( )]TJ/F1 9.96 Tf 9.92 0 Td[(2])]TJ/F7 9.96 Tf 7.74 0 Td[(;)]TJ/F1 9.96 Tf 4.43 0 Td[([)]TJ/F7 9.96 Tf 2.77 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F1 9.96 Tf 7.59 1.5 Td[(+)]TJ/F7 9.96 Tf 9.91 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F16 6.97 Tf 4.92 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(3)]TJ/F13 9.96 Tf 6.64 1.5 Td[( )]TJ/F1 9.96 Tf 9.91 0 Td[(1)]TJ/F7 9.96 Tf 4.98 0 Td[(;)-167(F)]TJ/F10 6.97 Tf 10.84 -1.5 Td[(n)]TJ/F1 9.96 Tf 7.59 1.5 Td[(+)]TJ/F7 9.96 Tf 9.91 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.5 Td[(n)]TJ/F16 6.97 Tf 4.93 0 Td[( )]TJ/F4 6.97 Tf 6.22 0 Td[(2)]TJ/F13 9.96 Tf 6.64 1.5 Td[( )]TJ/F1 9.96 Tf 9.92 0 Td[(1])]TJ -399.21 -11.96 Td[(and)-399([)]TJ/F7 9.96 Tf 22.8 0 Td[(F)]TJ/F10 6.97 Tf 6.4 -1.49 Td[(n)]TJ/F1 9.96 Tf 8.08 1.49 Td[(+)]TJ/F7 9.96 Tf 10.4 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.49 Td[(n)]TJ/F16 6.97 Tf 4.92 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(2)]TJ/F7 9.96 Tf 4.47 1.49 Td[(;)-166(F)]TJ/F10 6.97 Tf 10.83 -1.49 Td[(n)]TJ/F4 6.97 Tf 4.93 0 Td[(+1)]TJ/F13 9.96 Tf 13.24 1.49 Td[( )]TJ/F1 9.96 Tf 10.4 0 Td[(1].)-643(Prop)-27(osition)-399(1.2)-399(sho)28(ws)-399(that)-400(there)-398(are)-399(no)-399(non-zero)-399(co)-27(ecien)30(ts)-399(in)]TJ -109.11 -11.95 Td[(the)-317(second)-316(subin)29(terv)56(al,)-321(and)-317(Prop)-27(osition)-317(1.3)-318(sho)28(ws)-318(that)-317(there)-317(are)]TJ/F7 9.96 Tf 282.95 0 Td[()]TJ/F10 6.97 Tf 6.37 -1.5 Td[(n)]TJ/F16 6.97 Tf 4.92 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(3)]TJ/F1 9.96 Tf 7.64 1.5 Td[(non-zero)-317(co)-27(ecien)30(ts)-318(in)]TJ -308.11 -11.96 Td[(the)-362(third)-361(subin)29(terv)57(al.)-531(Because)]TJ/F7 9.96 Tf 136.8 0 Td[(a)]TJ/F1 9.96 Tf 5.27 0 Td[(\050)]TJ/F7 9.96 Tf 3.87 0 Td[(F)]TJ/F10 6.97 Tf 6.41 -1.49 Td[(n)]TJ/F16 6.97 Tf 4.92 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(3)]TJ/F13 9.96 Tf 6.88 1.49 Td[( )]TJ/F1 9.96 Tf 10.15 0 Td[(1\051)-362(is)-362(non-zero)-361(for)-362(all)]TJ/F7 9.96 Tf 93.15 0 Td[(n)]TJ/F13 9.96 Tf 9.23 0 Td[()]TJ/F1 9.96 Tf 10.99 0 Td[(5)-363(\050this)-361(follo)28(ws)-362(inductiv)30(ely)]TJ -293.9 -11.95 Td[(from)-358(Prop)-27(osition)-358(1.3\051,)-365(Prop)-28(osition)-358(1.1)-359(sho)29(ws)-359(that)-359(there)-357(are)]TJ/F7 9.96 Tf 264.23 0 Td[()]TJ/F10 6.97 Tf 6.37 -1.5 Td[(n)]TJ/F16 6.97 Tf 4.92 0 Td[( )]TJ/F4 6.97 Tf 6.23 0 Td[(3)]TJ/F13 9.96 Tf 6.85 1.5 Td[( )]TJ/F1 9.96 Tf 10.13 0 Td[(1)-360(non-zero)-357(co)-28(ecien)30(ts)-358(in)]TJ -298.73 -11.96 Td[(the)-333(\014rst)-332(subin)29(terv)57(al.)-445(W)83(e)-333(conclude)-331(that)]TJ/F7 9.96 Tf 172.3 0 Td[()]TJ/F10 6.97 Tf 6.37 -1.49 Td[(n)]TJ/F4 6.97 Tf 4.93 0 Td[(+1)]TJ/F13 9.96 Tf 12.79 1.49 Td[( )]TJ/F7 9.96 Tf 9.97 0 Td[()]TJ/F10 6.97 Tf 6.37 -1.49 Td[(n)]TJ/F1 9.96 Tf 8.19 1.49 Td[(=)-278(2)]TJ/F7 9.96 Tf 15.5 0 Td[()]TJ/F10 6.97 Tf 6.37 -1.49 Td[(n)]TJ/F16 6.97 Tf 4.93 0 Td[( )]TJ/F4 6.97 Tf 6.22 0 Td[(3)]TJ/F13 9.96 Tf 6.69 1.49 Td[( )]TJ/F1 9.96 Tf 9.96 0 Td[(1.)]TJ -74.59 -45.42 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