Sess\303\243o de demonstra\303\247\303\243o para o tutorial STACS2018
sobre estruturas combinat\303\263rias recursivas Para ser lido em conjunto com os slides dispon\303\255veis em http://perso.ens-lyon.fr/bruno.salvy/talks/STACS2018.pdf
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">I. Primeiras especifica\303\247\303\265es</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2">Contando</Text-field>
Usando esta gram\303\241tica, pode-se contar
GFSeries(bintrees,unlabelled,z,21);
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Por exemplo, o coeficiente de z ^ 20 nesta s\303\251rie de pot\303\252ncias \303\251 o n\303\272mero de \303\241rvores bin\303\241rias com 20 n\303\263s internos.
V\303\241rias variantes de \303\241rvores bin\303\241rias s\303\243o poss\303\255veis, dependendo do tamanho. No exemplo acima, os n\303\263s internos t\303\252m tamanho 1 e as folhas t\303\252m tamanho 0. O contr\303\241rio tamb\303\251m \303\251 poss\303\255vel
GFSeries({B=Union(Z,Prod(B,B))},unlabelled,z,11);
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Como uma \303\241rvore bin\303\241ria com n folhas possui n-1 n\303\263s internos, as seq\303\274\303\252ncias s\303\243o apenas deslocadas em 1.
Pode-se tamb\303\251m colocar pesos nos n\303\263s internos e externos:
GFSeries({B=Union(Z,Prod(Z,B,B))},unlabelled,z,11);
NyQvSSJaRzYiKydJInpHRiUiIiJGKC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YoIiM2L0kiQkdGJSsvRidGKEYoRigiIiQiIiMiIiZGNSIiKCIjOSIiKkYpRi8=
O tamanho \303\251 necessariamente \303\255mpar e a mesma sequ\303\252ncia (os n\303\272meros catal\303\243es) conta essas \303\241rvores.
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">Involu\303\247\303\265es</Font></Text-field>
Gram\303\241tica:
inv:={Inv=Set(Cycle(Z,card<=2))};
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Enumera\303\247\303\243o nos universos rotulados e n\303\243o rotulados:
GFSeries(inv,unlabelled,z,11);
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GFSeries(inv,labelled,z,11);
NyQvSSJaRzYiKydJInpHRiUiIiJGKC1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0YoIiM2L0kkSW52R0YlKztGJ0YoIiIhRihGKEYoIiIjI0Y0IiIkRjYjIiImIiM3IiIlIyIjOCIjZ0Y4IyIjPiIkIT0iIicjIiNIIiRJJyIiKCMiJCI+IiYhMzUiIikjIiRKIiImVyI9IiIqIyIlKD0iIicrT1giIzVGKUYv
Formul\303\241rios fechados para estas fun\303\247\303\265es geradoras:
combstruct[gfeqns](inv,labelled,z);
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combstruct[gfeqns](inv,unlabelled,z);
NyQvLUkkSW52RzYiNiNJInpHRiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2Iy1JJFN1bUdGKzYkKiYsJilGKCwkKiYiIiMiIiImSSJqRzYjL0knX2xvY2FsR0YsSS1fbjQ1MzY1MTA3ODRHRiY2I0Y4RjhGOEY4KUYoRjlGOEY4RjkhIiIvRjk7RjhJKWluZmluaXR5R0YsLy1JIlpHRiZGJ0Yo
O \303\272ltimo pode ser simplificado:
simplify(value(%)) assuming z>0,z<1;
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<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">Outras especifica\303\247\303\265es</Font></Text-field>
Estes ser\303\243o usados em exemplos posteriores:
Permuta\303\247\303\265es:
perm:={P=Set(Cycle(Z))};
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\303\201rvores Cayley:
cayley:={T=Prod(Z,Set(T))};
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Gr\303\241ficos funcionais:
func_graphs:={F=Set(Cycle(T)),T=Prod(Z,Set(T))};
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Gr\303\241ficos paralelos em s\303\251rie:
spgraphs:={G=Union(Z,S,P),S=Sequence(Union(Z,P),card>1),P=Set(Union(Z,S),card>1)};
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<Text-field style="Heading 2" layout="Heading 2">Parametros</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">Usando \303\241tomos nomeados de tamanho 0:</Font></Text-field>
Contar permuta\303\247\303\265es pelo n\303\272mero de ciclos:
perm2:={P=Set(Prod(U,Cycle(Z))),U=Epsilon};
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Nesta especifica\303\247\303\243o, uma marca U de tamanho 0 \303\251 anexada a cada ciclo na permuta\303\247\303\243o. Essa especifica\303\247\303\243o pode ser traduzida em fun\303\247\303\265es geradoras bivariadas
combstruct[gfeqns](perm2,labelled,z,[[u,U]]);
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perm_biv:=subs(%,P(z,u));
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSlwZXJtX2JpdkdGKC1JJGV4cEdGJTYjKiZJInVHRigiIiItSSNsbkdGJTYjKiQsJkYzRjNJInpHRighIiJGOkYzNyNGLg==
A partir da\303\255, os primeiros valores seguem
map(series,series(perm_biv,u),z);
Ky9JInVHNiIrJUkiekdGJCIiIiIiIUYoKy9GJkYnRicjRiciIiNGKyNGJyIiJEYtI0YnIiIlRi8jRiciIiZGMS1JIk9HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YnIiInRicrLUYmRipGK0YqRi0jIiM2IiNDRi8jRjEiIzdGMUYyRjhGKysrRiYjRidGOEYtRi5GLyMiIihGPEYxRjJGOEYtKylGJiNGJ0Y8Ri8jRidGPkYxRjJGOEYvKydGJiNGJyIkPyJGMUYyRjhGMQ==
Assim, por exemplo, o coeficiente 7/24 de z^5 e^3 significa que existem 35 = 5! x 7/24 permuta\303\247\303\265es de (1,2,3,4,5) com 3 ciclos.
Agora tamb\303\251m podemos calcular o n\303\272mero m\303\251dio de ciclos em uma permuta\303\247\303\243o:
subs(u=1,diff(perm_biv,u));
KiYtSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjKiQsJiIiIkYsSSJ6R0YoISIiRi5GLC1JJGV4cEdGJTYjRiNGLA==
simplify(%);
LCQqJi1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMsJCokLCYiIiIhIiJJInpHRilGLkYvRi9GLkYtRi9GLw==
series(%,z,15);
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O coeficiente de z ^ n \303\251 o en\303\251simo n\303\272mero harm\303\264nico (1 + 1/2 + ... + 1 / n), mostrando que o n\303\272mero m\303\251dio de ciclos em uma permuta\303\247\303\243o de (1, ..., n) \303\251 assintoticamente log n.
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">Usando gram\303\241ticas de atributo</Font></Text-field>
Lembre-se da gram\303\241tica para \303\241rvores bin\303\241rias
bintrees;
PCMvSSJCRzYiLUkmVW5pb25HNiI2JEkoRXBzaWxvbkc2Ii1JJVByb2RHNiI2JUkiWkc2IkkiQkc2IkkiQkc2Ig==
O atributo que calcula o comprimento do caminho interno das \303\241rvores bin\303\241rias (a soma das dist\303\242ncias de cada n\303\263 \303\240 raiz) \303\251 definido recursivamente em cima disso:
att:={pl(B)=Union(0,Prod(0,size(B)+pl(B),size(B)+pl(B)))};
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRhdHRHRig8Iy8tSSNwbEdGKDYjSSJCR0YoLUkmVW5pb25HRig2JCIiIS1JJVByb2RHRig2JUY3LCYtSSVzaXplR0YoRjIiIiJGMEY+Rjs3I0Yu
A partir da\303\255, fun\303\247\303\265es geradoras bivariadas podem ser derivadas automaticamente
combstruct[agfseries](bintrees, att, unlabeled, z, [[u, pl]]);
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O coeficiente de u ^ kz ^ n \303\251 o n\303\272mero de \303\241rvores bin\303\241rias com n n\303\263s internos e comprimento do caminho k.
As equa\303\247\303\265es tamb\303\251m podem ser derivadas automaticamente
combstruct[agfeqns](bintrees, att, unlabeled, z, [[u, pl]]);
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eqB:=op(1,%);
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Em u = 1, \303\251 o mesmo de antes: a fun\303\247\303\243o geradora dos n\303\272meros catal\303\243es.
Essa equa\303\247\303\243o pode ser diferenciada e avaliada em u = 1:
map(diff,eqB,u);
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convert(%,D);
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eval(%,u=1);
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gfpl:=solve(%,op(1,%));
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Nesta express\303\243o da fun\303\247\303\243o geradora, B (z, 1) \303\251 a fun\303\247\303\243o geradora do n\303\272mero de \303\241rvores bin\303\241rias e D_1 (B) (z, 1) \303\251 sua derivada.
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">II. Enumera\303\247\303\243o</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">Gr\303\241ficos paralelos em s\303\251rie</Font></Text-field>
Lembre-se da gram\303\241tica
spgraphs;
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Conte at\303\251 o tamanho 20
GFSeries(spgraphs,labelled,z,21);
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O tamanho da complexidade \303\251 bom:
GFSeries(spgraphs,labelled,z,100);
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Mesmos experimentos no universo n\303\243o marcado
GFSeries(spgraphs,unlabelled,z,21);
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
O tamanho da complexidade \303\251 bom:
GFSeries(spgraphs,unlabelled,z,100);
7&/I"ZG6"+'I"zGF%"""F(-I"OG6$%*protectedGI(_syslibGF%6#F("$+"/I"PGF%+cwF'F(""#F3""$""'""%"#=""&"#kF5"$F#""("$c)"")"%!G$""*"&&)G""#5"&U8&"#6"'Wv?"#7"'')y%)"#8"(%Q(\$"#9")K6a9"#:")tT)3'"#;"*&*3[c#"#<"+\0J'3"F7"+c'GJi%"#>",\T'*f(>"#?",z`t%y%)"#@"-aekm]O"#A"..3!z#pd""#B".^U8=:$o"#C"/O&oc]u'H"#D"0(yZ1j;#H""#E"0:=#fg\Rc"#F"1#\vc!QZmC"#G"2G]IT!=%33""#H"2<E$Rj)H]u%"#I"3bt<*\ftm3#"#J"3uj`<8B(4>*"#K"4<,d7O_3V0%"#L"5/h![5DqT4z""#M"5m(Q!Qs!>D<#z"#N"6$z="='yU'>$3N"#O"7NM_-B&*y/db:"#P"7$G]rPz(eL+0p"#Q"8S++6&)p/<!HoI"#R"9pg_mL0nYRyk8"#S"9i2lc1TZflPwg"#T":<zr4v0gN48yq#"#U";>&>`n>3-[$Gs27"#V";*)H)Gp&fA(3(p/"R&"#W"<=ugPW7'4euBN3C"#X"=E'3zR-@s.Md&pw5"#Y"=[yf^4X'RC_h6q"["#Z">6l8]'H`r=tYybc@"#[">3//)f)eSDT$*p$>h'*"#\"?7=P+b'pNb2!3%p3L%"#]"@/E'4h>A#*Q&R&y&)fU>"#^"@sLJ^&yJ%pOE]B`&=()"#_"AkFR%z%4U(34JFUf^"R"#`"B[`J**\-_&eOLadP4f<"#a"Bko7<lZ4uk.$)\Pwx!z"#b"C#=$*G&Rebg6D1)3")>mb$"#c"Dy![0:"H]&>i$QyV'pS+;"#d"DQc*fVLViE-0r%okn[?("#e"E'3,A.IwC=#)H!**z6(=]C$"#f"FtV,V%RAg))*4Uhh?[o@Y""#g"FKHT&QGayB86EgK8u8"f'"#h"G:]NlDT5jj0BPS*)*)3NsH"#i"H%=Nj@K#30=(ofhsp*pW4M""#j"H^<fd,[&Q0Dr&**pW#G%Q=0'F9"I,\;c?0=b%>?LTILJBuAt#"#l"J"*z],8Q\AWB#*Gs"p"fu/SB""#m"J;^qI;'>]+2$3>UYliZ(>vb"#n"KSr-hDf=B0s4I9%fN^;2(>D"#o"LKf8U"3EQ3Qp+Sge/d3G:R6"#p"L=-QWn-E%)*[ZM["G"=7Ldr^^"#q"Mi3o#yS()4&\n:_1LHqG@i`IB"#r"NJ&pDO&Ghca7yL37_"[')3dga5"#s"Nfv&)\WlYx"os:!3$\!o<IRbOx%"#t"OZq7?<R\<a^I%[MQ9+l$Q/Sh@"#u"ON<RqX[H$Qq@woWt0Qv`5A!*y*"#v"Pd)*>\q$fJ``A.\"Q*e*)\nTwlMW"#w"Q(zt$Q7c+cWv&*ewd2z)pU**>I&4?"#x"QkGk8X9H`Bp+f#3#y&**GVFW\$3"*"#y"Re"*QO6@A&[k^?d#pD:5M$e-ZXHT"#z"S:lIg0h"4$4'*3a%>ew/DS'ev_is="#!)"STM609HPC,p*3bWi!RIOEklo)R\)"#")"Tw-vDFH!>jYO(3+?T44zL,vA]O&Q"##)"U3[OX#o;'o-">:I#38,uU#Qy@Yh([<"#$)"U")[Cf&ew=k&=J=>#ptMPE2TYD1v$z"#%)"VG]pVB0'3)*Q1K!)H*e7Zq8Ti?**\Ng$"#&)"W!H-P^$[p.+%QCwVV=Gmr1#\%[O=jj""#')"WvrZD8)>!*oKKQeNz9V;OJd-P@&zJu"#()"XM0&olv-]"46[Rg/-(4$[y**))Hn1IgP$"#))"Yri3zZm=;I-MVHUykv$>***Qb-')H#R`""#*)"Y&*e[$3aN$G/1e%[Pz3SjhSdC[7W:3(p"#!*"ZU'\yT_n8)*Ge_fp=R9O$p*)*>'Gt@L%oJ"#"*"ens6m$\D1g&y%yl:h]#=8/2E`h*>XE0/W""##*"en9A.<?">3T"GQ8)G,'fuvrJrZ$)fV<T\l"#$*"fn124H&)y*Q0(3e))4-**3OQRve8'f\!z'[yH"#%*"gn&3-%=>Dxj+KTD#)y1Z$*)ye"QkP@:RFwa8"#&*"gnu&y?2^CL5"[2(4Y/)H.@-H=HGfvS)fJ;'"#'*"hnsyN!en@Apwxh5/&)G?(>'G&=1%>,+3rB/G"#(*"in>*oqnb?tG/(\G$)e.`5!>-M4<A%Q2XA8w7"#)*"in%eL-_"=4(e)e70"\:^Er9BG&[gW^jfTC3e"#**F)F//I"SGF%+cwF'F(F3F4F4F?F6FboF8"$."F5"$v$F;"%+9F="%!Q&F?"&t5#FA"&]R)FC"'y)Q$FE"(wNQ"FG"(&[-dFI")"3'pBFK")BL;**FM"*_Kb<%FO"+?s#yw"F7"++h'4_(FR",&ebf8KFT"-C/R\y8FV"-&o#p2MfFX".e+y&piDFZ"/]Sa1)*46Ffn"/]FcOh?[Fhn"0.j-lo()4#Fjn"0,$Rd4Se"*F\o"1]W4(pB\+%F^o"2*y5fU2ya<F`o"2?%H77rw-xFbo"3U/bv!p$)pQ$Fdo"4>z$G%*pYn"\"Ffo"4$*pE5&pKSzlFho"5M'e_i1U-h!HFjo"6.snYr%R0K&G"F\p"63?FR1&[R(=p&F^p"7@Sd(o]dA_N_#F`p"8*>ZK&*=W"*\4?6Fbp"8mmZbO.")Q$)o(\Fdp"9!\Ph6*)Qm,(e8AFfp"9'=G\qK`Bqg[&)*Fhp":SI6tSL4,=f8R%Fjp";UFe@')*yNei)\e>F\q";@!)))H$fetz1.>u)F^q"<\269PlO]pH)30RF`q"=]suVbuc.vzEvX<Fbq"=]tey`ZD+"oNM*4yFdq">"fYg"HQ5AmnDBj\$Ffq"?r%oM`"\)))yqKQcic"Fhq"?$GC8ih&>8;0pr%3-(Fjq"@v:'y)>@(R8ntlu0\JF\r"AU8u6*3[229*Q`=F89F^r"AV'[j#\_.,>F<q-AYjF`r"Bj)GP(ov&oS!36#)zy7&GFbr"C5ftqk"[#>o#)))f*e7<G"Fdr"C8At=l%eaZJMf"ReZkdFfr"D*HmiumW#[jQ&G3sv"Qf#Fhr"Eb;*ff>P=-8I:o+Jvw;"Fjr"E/gnWm%[]!=_JmnB=(*e_F\s"FW%H-Lc`7pbj)H$*=Bn&pBF^s"G*4W/3*fX<VbS:m9&z?"o5F`s"GZG/H#4ivjD#*pEV"y<p;[Fbs"HD">\;)*))o%o!=YD^Z'GXH<#Fds"H56Wg0B6Qc^'>]"R&Hg^1)*F9"I,i>P6mWMrcdmk'f">jKFWFgs"J")H"Hx7'*\%4;fmq&e5iH^**>Fis"JZOK!e?6%[R*4j(*odxeEaL!*F[t"Kx&pA,m0pCcm;EN()H(frh#3%F]t"LPP05w6.?)fo<'HJxqY--d%=F_t"LhGh@'pnz$[+)HF^%fJeA`oM)Fat"Mvpq_eqm@rO,"p2pB`G#3)ex$Fct"NTWL)*z"[/O"p**HPR(G_i,ah3<Fet"Nc!=K@bAdgYfc*)G:Q'>l@xQt(Fgt"ORo)o+4Bc`d9hA()R#G%)R&ed;]$Fit"P*e3e3#[H"H(o7^"p^$Q6&>N&zee"F[u"PoP5?k+(fH!Gs3H5\+e!)4L'G%=(F]u"Q^'4+Z#)3Om`U*\FyMS0Hzn"\aD$F_u"Rdu[G2wIN&zl\d9^b$Q(>)oeLbZ"Fau"R*RLcp,@:'Q3VaPqOsgG$Hse_*o'Fcu"S5O!4aI#*z+@S,W[ajVBdT\F?N.$Feu"Tg];$\1(GO#>&pUR%)o5&[z#HBGXfP"Fgu"TKt)pR\))[$fAfbd(z@,g(>@=XdCC'Fiu"U?!*GF^M!\enCj8W"QYa7BS]=nuKGF[v"V0F*G(4dtK'=k#)>***3xJi"*3'4wiu&G"F]v"Ve@6x:nvra$4g_')p8?X._`C0O3PeF_v"WOu"zMy^3B-"\(G:$4'\!RO_u3dz\]EFav"XcJozxr++[W*oFa_*oZ$pA:6H(*>y.7Fcv"X+?i4z7y`o&4s+zdc"eo;'[*e@TL$oaFev"Y*z^hjQ$e4()[0bvEJ')*y')4gn7sOXX[#Fgv"Zab6GzG$G')**)33*)RZ.;zQC,T08Os!H6Fiv"Z%GrmWw&[eue2">/@As)>?<">!)3dS*=8&F[w"enEuL1j<<5?DT/<+mr!p')=%HGuQg**)HL#F]w"fn!QV?0C]#yIM,Y1(os!ev.1%>$)H!>s!yg5F_w"fn5d_<nc2Twi#))ew#=]Z3\"zh#y8uMxS#[Faw"gn0!4!y\TeCV8,CWp)>HVr?f'Hy)Q_%\?%>#Fcw"gn=J>ISZ"*[c7(p)[kU(zsM:0!\RwyS8*=)**Few"hnVcqQIfrHAP#f\5)o2:?GQfQ*y&oh38<a%Fgw"intBi"=g^*>cl=eg9nn@a<a'*e=P%=)G(*ym?Fiw"in\?sC+]0$zT13ZYXf.\=V8<!og;*H%\y1%*F[xF)F//I"GGF%+ewF'F(F(F3F3F8F4FKF6FfqF8"$n"F5"$-'F;"%cAF="%g')F?"&eR$FA"'#HN"FC"'AkaFE"(i9B#FG"(p)*>*FI")8sBQFK"*'\Z+;FM"*ZT.u'FO"+px8aGF7",cZ4W@"FR",M(>f*=&FT"-.e7MEAFV"-R&QVZe*FX".h3oB'RTFZ"/,$yYKJz"Ffn"/'GJAk!)y(Fhn"0!4uc\$4R$Fjn"1;6m,(*yz9F\o"1U*pF](RrkF^o"2<e@naAc$GF`o"3P?;b(pzZC"Fbo"3(zFZdGdOZ&Fdo"4$Hu.E,>x5CFfo"55r$G7$z6Pj5Fho"5QZ1I<BT/(p%Fjo"6pfr%QaeI\x?F\p"6,3Rd#H"f$>+#*F^p"7cu4!*Hq/F7zSF`p"8#[(RI$)>tK&f5=Fbp"81nZm@t&eN<X!)Fdp"9fNm#[U4L'4PyNFfp":[*y:OVF=EP7$f"Fhp":d4$Ge"RpOFs"*4(Fjp";hA!pH=(yjg9AmJF\q"<55xA]Xe%)Q+&H89F^q"<n"=<:yF8`r1W8jF`q"=welTz%)yS:`#[C#GFbq">)>&=Ij#>UM?(f%pi7Fdq">-J=mDrD4)\.6)Gl&Ffq"?z)3:8!3H9\?`dPKDFhq"@&4'p@r_wm"fqd;<N6Fjq"@zT#))fJ%>BDwU/c;4&F\r"A9Z0ju)z,ux"*o<F^G#F^r"B29u?(>c%))4Q!Hpz8E5F`r"B6U/t=yP#*pTadbs.h%Fbr"CuF'y)HHM$Hj=(4F-\s?Fdr"C&RD;ZI9gj#o*R+l&4@$*Ffr"DxV"o*yvuV&[P7YOXA%>%Fhr"E$H()>.`!3)G:N'Gvu?;))=Fjr"E!4xonwCv)R]MlZN0*R])F\s"F<QCtdfFxbXG"\&RrN<$QF^s"GJq&eYF5`bn;!=#zk`Mss"F`s"GiyR%z@m1+#GA/t.oE/*y(Fbs"H4VD"QIr>ly'eqQsh)**)Q^$Fds"IhG.=2r'>pSO:]QydWNee"F9"I-6O$>8F'*e'e*)zoHH_0gfrFgs"Js4UuS**[n`R^&HuF-3x^LKFis"Kj(G5@#3V`4IR&>L7/N,u3Y"F[t"K<nHt&e"4qnPw"pwYl[KCBg'F]t"Lp'*=_ds&Q!zz$=OtJ7CL[&[)HF_t"MzI*fO'zAAQvUcFtS`*e*o&)\8Fat"MPyQNmWls?/<V$QiEST/<k5'Fct"NsR!4O.hq"o"yPc9&R/!\56AjFFet"O:c2j'4*=$y9Kspfk=t`4E`2D"Fgt"O')Q,F2q6`H(>/r@y'HMwB!eIm&Fit"PC.?cm'*euw&[(QQ^#*=lKdu"ykDF[u"QDOIpM%H"HQ`/*R%[Q'pblsR%*=;"F]u"Q[MQ3PWh>"3+*3/OU>/ctn$z\E&F_u"R@.8U_!*f))=F(R$GKLJGIc6`ojQ#Fau"SdD-KGJuY$[#[EjRi(3Fm([d!)*=3"Fcu"SD,@,6%3*Q>)HU*yE,%[[(z_]b91\Feu"T,&y#)*y*f1O4#f$\)3v\:Ja$*)oRMDAFgu"U3wtA@9zmD(GVwv"f@4b`A$zw5'45Fiu"UGQlsL,_`yP%yVE7v%Gb0Co!=3:e%F[v"V'eP@`H7YF/wl6@fW_gU'>g:!p\z?F]v"V'=23#Rsh_Wd@Hj"fR"*\SjxI(fL1W*F_v"WE(>;'=maMA%H*\!fOzxcNIPK>K;oG%Fav"XJ.;0"*p-*[xE2')*=V+7bSDPh'=:'p%>Fcv"XMDZmNSyox1pY]#yE"*o^fQ))))yMOW))Fev"Yq!Q_T.qds6&R)\!p4^lh!4]@QKNo%=SFgv"Z\9g6?%o6HgpERO`V+b\%)pM-V0R:E=Fiv"ZE4_k)G`)RkTL'ytRh'[`*o5RmTusK+$)F[w"en)f)***z,yh')*4*4'G1"**Q5dzE)*Q2\A&RtPF]w"fn%fv!pg$p!*[C'Rf%**poE8bx`'z"GE'*=s:<F_w"fn;khY_N(\p92ud(H<f$o%)om(Ru4z8kD!yFaw"gn!46k*omN)QaC%\E[0RE.&zSSf,"R%on*[NFcw"hn#prA5DRAv1YS)44BFJob!)=ynNa"=^]9;Few"hn:N1>1w$>#*[,@g9t0r)R9"zZM)poT>]fM(Fgw"in#H"pee@F2*f$o'QM22AV%R%**)HSzA*Q(>#HM$Fiw"jnLc&\a"o9!QIKfd&41,.Kj;C]G0oi-"H]@:F[xF)F/
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">\303\201rvores enraizadas rotuladas e n\303\243o identificadas</Font></Text-field>
Lembre-se de suas especifica\303\247\303\265es:
cayley;
PCMvSSJURzYiLUklUHJvZEc2IjYkSSJaRzYiLUkkU2V0RzYiNiNJIlRHNiI=
Enumerar no universo rotulado
GFSeries(cayley,labelled,z,41);
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
o n\303\272mero \303\251 obtido multiplicando por um fatorial:
coeff(subs(%,T),z,40)*40!;
ImpuKysrKysrKysrKysrKysrKysrK1Nhd09IZE8hXFhKQUk=
Esses n\303\272meros s\303\243o conhecidos:
40^39;
ImpuKysrKysrKysrKysrKysrKysrK1Nhd09IZE8hXFhKQUk=
No caso n\303\243o marcado, uma f\303\263rmula t\303\243o boa n\303\243o \303\251 conhecida, mas \303\251 igualmente f\303\241cil calcular valores para \303\255ndices altos
GFSeries(cayley,unlabelled,z,41);
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
Este tamb\303\251m \303\251 r\303\241pido:
GFSeries(cayley,unlabelled,z,101);
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
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">III. Gera\303\247\303\243o aleat\303\263ria</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">M\303\251todo recursivo</Font></Text-field>
Desenhe uma \303\241rvore bin\303\241ria aleat\303\263ria de tamanho 30:
combstruct[draw]([B,bintrees,unlabelled],size=30);
LUklUHJvZEc2IjYlSSJaR0YkSShFcHNpbG9uR0YkLUYjNiVGJkYnLUYjNiVGJi1GIzYlRiZGJy1GIzYlRiYtRiM2JUYmLUYjNiVGJi1GIzYlRiYtRiM2JUYmLUYjNiVGJi1GIzYlRiYtRiM2JUYmLUYjNiVGJi1GIzYlRiZGJy1GIzYlRiZGJ0YnRidGJ0YnRkJGQEYnLUYjNiVGJi1GIzYlRiZGJy1GIzYlRiYtRiM2JUYmRkJGQi1GIzYlRiYtRiM2JUYmLUYjNiVGJkYnRkBGJ0YnRidGJ0ZCRic=
e da mesma forma um gr\303\241fico paralelo em s\303\251rie rotulado aleatoriamente:
combstruct[draw]([G,spgraphs,labelled],size=30);
LUkkU2V0RzYiNiQtSSlTZXF1ZW5jZUdGJDYkJkkiWkdGJDYjIiM7LUYjNiQtRic2Jy1GIzYlLUYnNiUmRio2IyIiJyZGKjYjIiIoLUYjNiQtRic2KCZGKjYjIiM+LUYjNiUtRic2JCZGKjYjIiM3LUYjNiQtRic2JSZGKjYjIiNFJkYqNiMiIzUtRiM2JCZGKjYjIiM5JkYqNiMiIzgmRio2IyIiKiZGKjYjIiM6JkYqNiMiI0YmRio2IyIjQyZGKjYjIiNAJkYqNiMiI0gmRio2IyIiJiZGKjYjIiIlLUYnNiQmRio2IyIjQiZGKjYjIiNBLUYnNiQtRiM2JCZGKjYjIiNELUYnNiQmRio2IyIiJCZGKjYjIiIpJkYqNiMiIz0mRio2IyIiIyZGKjYjIiNJJkYqNiMiIz8mRio2IyIiIi1GJzYkJkYqNiMiI0cmRio2IyIjPCZGKjYjIiM2
Hora do tamanho 500
time(combstruct[draw]([G,spgraphs,labelled],size=500));
JCIlKW8jISIk
O pr\303\263ximo \303\251 muito mais r\303\241pido, j\303\241 que a pr\303\251-computa\303\247\303\243o das seq\303\274\303\252ncias de enumera\303\247\303\243o foi feita
time(combstruct[draw]([G,spgraphs,labelled],size=500));
JCIjZCEiJA==
Mas aumentar o tamanho se torna caro:
time(combstruct[draw]([G,spgraphs,labelled],size=1000));
JCImMVciISIk
<Text-field style="Heading 2" layout="Heading 2">Amostragem de Boltzmann</Text-field>
Encontre o valor do par\303\242metro x que leva a um tamanho esperado de 10000 para gr\303\241ficos paralelos em s\303\251rie rotulados
val:=BoltzmannParameter(spgraphs,labelled,G,10000);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR2YWxHRigkIittJVE5WCMhIzU3I0Yu
Calcular os valores das fun\303\247\303\265es geradoras nesse ponto
pp:=NumericalNewtonIteration(spgraphs,labelled);
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
O que foi calculado \303\251 um procedimento que recebe como entrada um n\303\272mero real (menor que o raio de converg\303\252ncia das fun\303\247\303\265es geradoras) e retorna os valores das fun\303\247\303\265es geradoras nesse ponto:
pp(%%);
NyUvSSJHRzYiJCIrNmUnKXpoISM1L0kiUEdGJSQiK2cwL284RigvSSJTR0YlJCIrJnknUWdCRig=
Podemos verificar o tamanho esperado para essa aproxima\303\247\303\243o
BoltzmannExpectedSize(spgraphs,labelled)(val);
NyYvLUkiRUc2IjYjSSJHR0YmJCIrKkciPSF6KiEiJy8tRiU2I0kiUEdGJiQiK3ZnPCpvIiEiJi8tRiU2I0kiU0dGJiQiKytCNSVlIkYyLy1GJTYjSSJaR0YmJCIrKysrKzUhIio=
Isso \303\251 um pouco menor que 10000, devido \303\240 precis\303\243o usada para calcular o valor val acima. Se uma expectativa mais precisa for necess\303\241ria, \303\251 poss\303\255vel calcular esse valor com maior precis\303\243o:
Digits:=20:
val:=BoltzmannParameter(spgraphs,labelled,G,10000);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSR2YWxHRigkIjVaJEgjb1FtJVE5WCMhIz83I0Yu
ent\303\243o a obten\303\247\303\243o dos valores ainda \303\251 r\303\241pida
pp(val);
NyUvSSJHRzYiJCI1L3JiW1dpYSgpemghIz8vSSJQR0YlJCI1LCE0KW9WWVUvbzhGKC9JIlNHRiUkIjVjKD06QCdcRlJnQkYo
e as expectativas se comportam melhor
BoltzmannExpectedSize(spgraphs,labelled)(val);
NyYvLUkiRUc2IjYjSSJHR0YmJCI1RjomW1oqKioqKioqKioqKiEjOy8tRiU2I0kiUEdGJiQiNTIiKSpwJDMiXHhgcyIhIzovLUYlNiNJIlNHRiYkIjUoUUM4NChHPDA9O0YyLy1GJTYjSSJaR0YmJCI1KysrKysrKysrNSEjPg==
Observe que o primeiro valor de val foi perfeito por sua precis\303\243o, portanto n\303\243o houve erro de arredondamento, mas uma alta depend\303\252ncia da precis\303\243o t\303\243o pr\303\263xima da singularidade.
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">IV. Assint\303\263ticos</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2">Comprimento do percurso</Text-field>
Lembre-se de que a fun\303\247\303\243o geradora para o comprimento esperado do caminho foi
gfpl;
LCQqLCIiIyIiIi1JIkJHNiI2JEkiekdGKEYlRiUtLSZJIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRig2I0YlNiNGJ0YpRiUpRipGJEYlLCYqKEYkRiVGKkYlRiZGJUYlRiUhIiJGN0Y3
em que B (z, 1) denota a fun\303\247\303\243o geradora de \303\241rvores bin\303\241rias e D [1] (B) (z, 1) sua derivada. Ou seja, temos
gfpl:=subs(B(z,1)=solB,D[1](B)(z,1)=diff(solB,z),gfpl);
LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSVnZnBsR0YoLCQqKiwmIiIiISIiKiQpLCZGMUYxKiYiIiVGMUkiekdGKEYxRjIjRjEiIiNGMUYxRjFGOEYxLCYqJkY0RjJGOEYyRjEqKEY5RjFGMEYxKUY4RjpGMkYxRjFGNEYyRjI3I0Yu
A partir da\303\255, o comportamento assint\303\263tico segue
equivalent(%,z,n,3);
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e a propor\303\247\303\243o dividida por n fornece a dist\303\242ncia m\303\251dia de um n\303\263 \303\240 sua raiz:
asympt(%/as_B/n,n,3);
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