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{SECT 0 {PARA 256 "" 0 "" {TEXT 256 18 "PATTERNS IN WORDS\n" }}{PARA 
257 "" 0 "" {TEXT 260 11 "Bruno Salvy" }}{PARA 258 "" 0 "" {TEXT -1 
29 "(Version of February 7, 1997)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 23 "This worksheet applies " }{HYPERLNK 17 "c
ombstruct" 2 "combstruct" "" }{TEXT -1 5 " and " }{HYPERLNK 17 "gfun" 
2 "gfun" "" }{TEXT -1 766 " to a simple combinatorial model of a probl
em from computational biology and the study of DNA sequences. The DNA \+
can be viewed as a long text on an alphabet of  four letters (A,C,G,T)
. Large fragments of this text are tabulated. In particular, there are
 huge bases of genes, a gene being a few thousand letters long. Given \+
a short word, it is interesting to determine whether its number of occ
urrences in a gene (or a virus) is far away from the most probable num
ber of occurrences. If this number of occurrences is very improbable, \+
then this particular word may have a biological function. \n\nThe comb
inatorial model is rational. The text is a word on the alphabet (a,b,c
,d) where all words of the same length are equiprobable. The probabili
ty that a pattern occurs " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 72 
" times in the text depends on the way the pattern overlaps with itsel
f. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "libname:=`/net/blagny
/algo/maple/5.4`,libname:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
29 "with(combstruct): with(gfun):" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
264 70 "Specification and univariate generating functions for the patt
ern abab" }}{PARA 0 "" 0 "" {TEXT -1 190 "Working over the alphabet (a
,b,c,d), we first concentrate on a specific pattern (abab). To attack \+
problems related to occurrences of this pattern in words using combstr
uct, we first write a " }{TEXT 261 7 "grammar" }{TEXT -1 293 " which d
escribes a corresponding automaton.This grammar recognizes all the wor
ds on (a,b,c,d). It is written in such a way that a mark (named Mark) \+
is present in a word everytime the pattern abab occurs. Then counting \+
the number of marks in a word gives the number of occurences of abab i
n it." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 371 "G:=\{w=Union(Epsil
on,Prod(a,wa),Prod(b,w),\n            Prod(c,w),Prod(d,w)),\n    wa=Un
ion(Epsilon,Prod(a,wa),Prod(b,wab),\n            Prod(c,w),Prod(d,w)),
\n    wab=Union(Epsilon,Prod(a,waba),Prod(b,w),\n            Prod(c,w)
,Prod(d,w)),\n    waba=Union(Epsilon,Prod(a,wa),Prod(b,Prod(Mark,w)),
\n            Prod(c,w),Prod(d,w)),\n    Mark=Epsilon,a=Atom,b=Atom,c=
Atom,d=Atom\}:" }}}{PARA 0 "" 0 "" {TEXT -1 20 "We use the function " 
}{HYPERLNK 17 "combstruct[count]" 2 "combstruct[count]" "" }{TEXT -1 
45 " to check that the number of words of length " }{XPPEDIT 18 0 "n" 
"I\"nG6\"" }{TEXT -1 5 "  is " }{XPPEDIT 18 0 "4^n" ")\"\"%%\"nG" }
{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "count([w,G,
unlabelled],size=10),4^10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"(w&[5F
#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "count([w,G,unlabelled]
,size=20),4^20;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\".wxi6&*4\"F#" }}}
{PARA 0 "" 0 "" {TEXT -1 23 "It is also possible to " }{TEXT 257 5 "pr
ove" }{TEXT -1 64 " this by computing the generating function of the l
anguage with " }{HYPERLNK 17 "combstruct[gfsolve]" 2 "combstruct[gfsol
ve]" "" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "gf
solve(G,unlabelled,z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<+/-%$wabG6#
%\"zG,$*$,&F(\"\"%!\"\"\"\"\"F-F-/-%#waGF'F)/-%\"wGF'F)/-%%wabaGF'F)/-
%\"cGF'F(/-%\"dGF'F(/-%\"aGF'F(/-%\"bGF'F(/-%%MarkGF'F." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(\",w(z));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*$,&%\"zG\"\"%!\"\"\"\"\"F(F(" }}}{PARA 0 "" 0 "" 
{TEXT -1 19 "The coefficient of " }{XPPEDIT 18 0 "z^n" ")%\"zG%\"nG" }
{TEXT -1 67 " in the Taylor expansion of this generating function is t
he number " }{XPPEDIT 18 0 "4^n" ")\"\"%%\"nG" }{TEXT -1 20 " of words
 of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 127 " in the lang
uage, which confirms the correctness of our grammar.\n\nHere are a few
 typical words of the language obtained by the " }{TEXT 262 24 "unifor
m random generator" }{TEXT -1 13 " provided by " }{HYPERLNK 17 "combst
ruct[draw]" 2 "combstruct[draw]" "" }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 76 "to 20 do eval(subs(Prod=proc() args end,draw
([w,G,unlabelled],size=30))) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%
\"cG%\"dG%\"aGF$F$F%%\"bGF#F&F$F#F&F%F$F&F%F#F&F#F$F#F%F%F#F%F%F$F&F%F
#%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"bGF#%\"cGF$%\"dG%\"a
GF#F#F#F#F$F&F#F$F&F#F%F$F&F&F$F%F$F%F&F$F$F&F&F&%(EpsilonG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6A%\"dGF#%\"cG%\"aG%\"bGF&F%F$F$F$F&F&F$F%F#F&
F$F&F%F&F#F&F#F$F#F%F&F%F$F%%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6A%\"bG%\"aG%\"cGF$%\"dGF%F$F$F$F&F#F#F#F%F%F&F$F&F#F%F#F$F#F#F$F%
F#F%F&F#%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"dG%\"aG%\"cGF
#F%F#%\"bGF#F&F$F%F#F#F&F#F&F%F#F$F%F#F#F%F$F#F$F#F$F#F#%(EpsilonG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"cG%\"bGF$F$F$F#%\"aGF#F%F$%\"dGF$F$
F$F%F$F$F%F$F#F%F#F$F#F%F%F$F$F$F%%(EpsilonG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6A%\"bG%\"cGF#%\"dGF%%\"aGF$F$F&F$F$F&F%F#F$F$F&F%F$F#F%F
&F&F#F$F&F&F&F&F%%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"cG%
\"aG%\"bGF#%\"dGF#F%F#F$F$F%F%F$F#F$F&F#F&F&F&F%F&F&F&F#F&F%F$F&F%%(Ep
silonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"cG%\"dGF$F$%\"bGF$F%%\"aG
F$F#F&F$F$F%F%F&F#F$F#F&F%F%F#F&F$F&F&F%F%F%%(EpsilonG" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6A%\"aGF#%\"bGF$%\"dGF#F#F$F#F%F%F%F%F#F#F#F#%\"cGF%
F$F&F$F$F$F#F#F&F#F#F%%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%
\"aG%\"bG%\"cG%\"dGF#F$F$F%F&F%F&F%F#F%F%F%F&F%F&F#F&F#F#F&F&F#F$F#F%F
#%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"aG%\"dG%\"bGF%F$F#F$
F$F#F%F$F#F#F$%\"cGF#F$F&F$F$F$F%F$F&F#F$F&F%F%F$%(EpsilonG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6B%\"dG%\"bGF$%\"aGF$F%F#F$F%F$F%F$%%MarkGF#F$
F%%\"cGF$F#F'F#F#F$F#F#F'F%F'F'F$F#%(EpsilonG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6A%\"bG%\"dGF#%\"aGF%F#F$F$%\"cGF#F&F#F&F#F%F%F&F#F#F$F$F
$F&F#F%F%F#F$F#F#%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"bGF#
F#%\"cG%\"aGF$F$F%F#%\"dGF&F$F#F%F%F#F#F#F$F&F%F%F%F%F&F&F$F#F$F$%(Eps
ilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"aG%\"bGF#%\"cGF$F%%\"dGF$F
$F$F#F%F&F$F$F&F$F$F&F%F#F#F$F%F$F%F%F$F&F%%(EpsilonG" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6A%\"aG%\"cG%\"dG%\"bGF$F#F#F&F&F&F&F&F&F&F#F&F$F&F&
F#F#F#F$F%F#F&F%F%F&F%%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6B%
\"bG%\"dGF#%\"aGF%F#F$F#F#%\"cGF&F#F&F&F&F&F&F$F&F&F$F%F&F%F#F%F#%%Mar
kGF&F%F#%(EpsilonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"dG%\"bGF$%\"a
G%\"cGF&F#F#F%F$F&F&F&F&F&F&F%F$F&F&F%F$F$F%F&F&F&F$F$F&%(EpsilonG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6A%\"bG%\"cGF$F$F$F$F#%\"dGF$F%F#F#F$F%F
%F%F$%\"aGF#F#F$F&F&F&F&F$F&F#F%F&%(EpsilonG" }}}{PARA 0 "" 0 "" 
{TEXT -1 109 "Some of these words countain the pattern abab, as indica
ted by the letter `Mark' right after its occurrence. " }}}{SECT 0 
{PARA 4 "" 0 "" {TEXT 265 30 "Bivariate generating functions" }}{PARA 
0 "" 0 "" {TEXT -1 50 "From the grammar specification above, the comma
nd " }{HYPERLNK 17 "combstruct[gfsolve]" 2 "combstruct[gfsolve]" "" }
{TEXT -1 29 "  can also be used to derive " }{TEXT 263 12 "multivariat
e" }{TEXT -1 96 " generating functions. From this, it is easy to compu
te the probability that the pattern occurs " }{XPPEDIT 18 0 "k" "I\"kG
6\"" }{TEXT -1 34 " times in a random word of length " }{XPPEDIT 18 0 
"n" "I\"nG6\"" }{TEXT -1 109 ", the expectation of the number of occur
rences of the pattern in such a word, and the corresponding variance.
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "gfsolve(G,unlabelled,z,[
[u,Mark]]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<+/-%\"cG6$%\"zG%\"uGF(
/-%\"dGF'F(/-%\"aGF'F(/-%\"bGF'F(/-%%MarkGF'F)/-%%wabaGF',$*&,.\"\"\"F
<F(!\"\"*$F(\"\"#F<*$F(\"\"$F=*&F(F<F)F<F<*&F(FAF)F<F<F<,.F=F<F(\"\"%F
>F=F@FE*$F(FEF=*&F(FEF)F<F<F=F=/-%$wabGF',$*&,&F<F<*&F(F?F)F<F<F<FDF=F
=/-%\"wGF',$*&,&F<F<F>F<F<FDF=F=/-%#waGF',$*&,*F<F<F>F<F@F=FCF<F<FDF=F
=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "sol:=subs(\",w(z,u));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG,$*&,&\"\"\"F(*$%\"zG\"\"#F
(F(,.!\"\"F(F*\"\"%F)F-*$F*\"\"$F.*$F*F.F-*&F*F.%\"uGF(F(F-F-" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The coefficient of " }{XPPEDIT 18 
0 "z^n*u^k" "*&)%\"zG%\"nG\"\"\")%\"uG%\"kGF&" }{TEXT -1 44 " in the T
aylor series of this expression at " }{XPPEDIT 18 0 "z=0" "/%\"zG\"\"!
" }{TEXT -1 34 " is the number of words of length " }{XPPEDIT 18 0 "n
" "I\"nG6\"" }{TEXT -1 31 " where the pattern abab occurs " }{XPPEDIT 
18 0 "k" "I\"kG6\"" }{TEXT -1 18 " times. Replacing " }{XPPEDIT 18 0 "
z" "I\"zG6\"" }{TEXT -1 4 " by " }{XPPEDIT 18 0 "z/4" "*&%\"zG\"\"\"\"
\"%!\"\"" }{TEXT -1 21 " directly yields the " }{TEXT 258 31 "probabil
ity generating function" }{TEXT -1 30 " under the uniform model (see \+
" }{HYPERLNK 17 "below" 1 "" "biased" }{TEXT -1 21 " for a biased mode
l):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "GF:=normal(subs(z=z/4,sol));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#GFG,$*&,&\"#;\"\"\"*$%\"zG\"\"#
F)F),.!$c#F)F+\"$c#F*!#;*$F+\"\"$F(*$F+\"\"%!\"\"*&F+F4%\"uGF)F)F5F0" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Here are the first coefficients
:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "S:=map(normal,series(GF,u));" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"SG+1%\"uG,$*&,&\"#;\"\"\"*$%\"zG
\"\"#F+F+,,\"$c#F+F-!$c#F,F**$F-\"\"$!#;*$F-\"\"%F+!\"\"F*\"\"!,$*(F)F
+F-F6F/!\"#F*\"\"\",$*(F)F+F-\"\")F/!\"$F*\"\"#,$*(F)F+F-\"#7F/!\"%F*
\"\"$,$*(F)F+F-F*F/!\"&F*\"\"%,$*(F)F+F-\"#?F/!\"'F*\"\"&-%\"OG6#F+\"
\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "For instance, the coeffici
ent of " }{XPPEDIT 18 0 "u^0" "*$%\"uG\"\"!" }{TEXT -1 77 " in this se
ries gives the probabilities that the pattern abab does not occur:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "series(coeff(S,u,0),z,31);
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+]o%\"zG\"\"\"\"\"!F%\"\"\"F%\"\"#
F%\"\"$#\"$b#\"$c#\"\"%#\"$F\"\"$G\"\"\"&#\"%\\S\"%'4%\"\"'#\"%<?\"%[?
\"\"(#\"%>SF4\"\")#\"&lS'\"&Ob'\"\"*#\"(J7-\"\"(w&[5\"#5#\"'fVD\"'W@E
\"#6#\")\"[=i\"\");sx;\"#7#\")6#eh\"FK\"#8#\")l\")4;FK\"#9#\"*PnIG\"\"
*Gx@M\"\"#:#\"+Ryb!4%\"+'Hn\\H%\"#;#\"+(Ryw.#\"+[O[Z@\"#<#\",4eRj\\'\"
,OnZ>(o\"#=#\",Xe\\!=;\",%=p)zr\"\"#>#\",r!o9[kF[o\"#?#\".rZapy-\"\".w
xi6&*4\"\"#@#\"/rac(z%Q;\"/;W/'=#f<\"#A#\".23p)[S?\"._bDB!*>#\"#B#\"0D
]3k>@g#\"0c1rw\\Z\"G\"#C#\"0p#zD)\\Cf#Fbp\"#D#\"0pD]%f\"Ge#Fbp\"#E#\"1
,^+YrGH5\"1CE%o!***e7\"\"#F#\"2R'o%*ez&Hc'\"2Oz#z.%fd?(\"#G#\"2N]Xnd%G
pK\"2oR'*=qzGg$\"#H#\"4TyP,oN$GU5\"4wp%og/:#H:\"\"#I-%\"OG6#F%\"#J" }}
}{PARA 0 "" 0 "" {TEXT -1 144 "Thus the random draws of words of lengt
h 30 that we did before are typical: the probability that the pattern \+
does not occur in such a word being" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "coeff(\",z,30)=evalf(coeff(\",z,30));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/#\"4TyP,oN$GU5\"4wp%og/:#H:\"$\"+QqOS!*!#5" }}}
{PARA 0 "" 0 "" {TEXT -1 135 "The expected number of occurrences is ob
tained very directly from the bivariate generating function GF. Here i
s its generating function" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 
"mom1:=factor(subs(u=1,diff(GF,u)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%%mom1G,$*(%\"zG\"\"%,&F'\"\"\"!\"\"F*!\"#,&\"#;F**$F'\"\"#F*F+#F*F
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "smom1:=series(\",z,31)
;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&smom1G+en%\"zG#\"\"\"\"$c#\"\"
%#F(\"$G\"\"\"&#\"#Z\"%'4%\"\"'#\"#J\"%[?\"\"(#\"%L7\"&Ob'\"\")#\"$P(
\"&oF$\"\"*#\"&Ru#\"(w&[5\"#5#\"&Zc\"\"')GC&\"#6#\"'&Qi&\");sx;\"#7#\"
'L?J\"(3')Q)\"#8#\")^>(4\"\"*caVo#\"#9#\"(B%zf\"*Gx@M\"\"#:#\"*d=82#\"
+'Hn\\H%\"#;#\"**3h96\"+[O[Z@\"#<#\"+$)**R>Q\",OnZ>(o\"#=#\"+fD-O?\",o
$Q(fV$\"#>#\",\\S/&>p\".wxi6&*4\"\"#?#\",0Jo=m$\"-))Q\"ev\\&\"#@#\".NR
,vkB\"\"/;W/'=#f<\"#A#\"-bU3w0l\".3A-$4'z)\"#B#\"/h$p\"pK&=#\"0c1rw\\Z
\"G\"#C#\"/\"G@=0W9\"\"0G`N)[P29\"#D#\"02A&)4qw#Q\"1'\\qti*f.X\"#E#\"0
6<r!=i'*>\"1[_o8)*z^A\"#F#\"1$>9zS2Tl'\"2Oz#z.%fd?(\"#G#\"1<o_;D^fM\"2
oR'*=qzGg$\"#H#\"3*>*3JD3V\\6\"4wp%og/:#H:\"\"#I-%\"OG6#F(\"#J" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(\");" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#+en%\"zG$\"+++D1R!#7\"\"%$\"+++]7yF'\"\"&$\"+Q4YZ6!#6
\"\"'$\"+v=n8:F.\"\"($\"+\"p39)=F.\"\")$\"+3b9\\AF.\"\"*$\"+dpy;EF.\"#
5$\"+1%GW)HF.\"#6$\"+:e2_LF.\"#7$\"+DKs>PF.\"#8$\"+i-P(3%F.\"#9$\"+)H<
]X%F.\"#:$\"+fVmA[F.\"#;$\"+>9J!>&F.\"#<$\"+y%ezb&F.\"#=$\"+ObgDfF.\"#
>$\"+&f_KH'F.\"#?$\"+a'**3m'F.\"#@$\"+8naGqF.\"#A$\"+sP>'R(F.\"#B$\"+I
3%Qw(F.\"#C$\"+*)y[J\")F.\"#D$\"+[\\8*\\)F.\"#E$\"+2?ym))F.\"#F$\"+m!H
WB*F.\"#G$\"+Dh2-'*F.\"#H$\"+$=B(p**F.\"#I-%\"OG6#\"\"\"\"#J" }}}
{PARA 0 "" 0 "" {TEXT -1 106 "Thus in a sequence of 20 draws as above,
 we can expect the following number of occurrences of the pattern:" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "20*coeff(\",z,30);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"+PY%R*>!\"*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 57 "The variances are computed as easily as the expectations:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "mom2:=factor(subs(u=1,d
iff(u*diff(GF,u),u)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%mom2G,$**
%\"zG\"\"%,,\"$G\"\"\"\"F'!$G\"*$F'\"\"#\"\")*$F'\"\"$!\")*$F'F(F+F+,&
F'F+!\"\"F+!\"$,&\"#;F+F-F+!\"##F5F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "evalf(series(mom2-add(coeff(smom1,z,i)^2*z^i,i=0..30)
,z,31));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+en%\"zG$\"+67*4*Q!#7\"\"%
$\"+W[Y^xF'\"\"&$\"+rUHM6!#6\"\"'$\"+])f2\\\"F.\"\"($\"+jM1\\=F.\"\")$
\"+EUr2AF.\"\"*$\"+%HSic#F.\"#5$\"+YMtCHF.\"#6$\"+/]B$G$F.\"#7$\"+@$R<
k$F.\"#8$\"+uIC+SF.\"#9$\"+6mueVF.\"#:$\"+'=]sr%F.\"#;$\"+xPvv]F.\"#<$
\"+ltDMaF.\"#=$\"+_4w#z&F.\"#>$\"+SXE^hF.\"#?$\"+F\"o(4lF.\"#@$\"+:<Fo
oF.\"#A$\"+-`xEsF.\"#B$\"+*))y_e(F.\"#C$\"+xCyVzF.\"#D$\"+kgG-$)F.\"#E
$\"+_'*yg')F.\"#F$\"+RKH>!*F.\"#G$\"+Fozx$*F.\"#H$\"+9/IO(*F.\"#I-%\"O
G6#\"\"\"\"#J" }}}{PARA 0 "" 0 "" {TEXT -1 19 "The coefficient of " }
{XPPEDIT 18 0 "z^n" ")%\"zG%\"nG" }{TEXT -1 96 " in this series is the
 variance of the number of occurrences of the pattern in a word of len
gth " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 1 "." }}}{SECT 0 {PARA 
4 "" 0 "" {TEXT 266 27 "Fixed number of occurrences" }}{PARA 0 "" 0 "
" {TEXT -1 51 "We now consider the probabilities that abab occurs " }
{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 12 " times, for " }{XPPEDIT 18 
0 "k=0..5" "/%\"kG;\"\"!\"\"&" }{TEXT -1 8 ". Using " }{HYPERLNK 17 "g
fun" 2 "gfun" "" }{TEXT -1 64 ", we can compute these probabilities fo
r random words of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 7 "
, with " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 23 " up to a few thou
sands." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "maxnb:=5:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "for i from 0 to maxnb do pro
ba[i]:=coeff(S,u,i) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&probaG6
#\"\"!,$*&,&\"#;\"\"\"*$%\"zG\"\"#F,F,,,\"$c#F,F.!$c#F-F+*$F.\"\"$!#;*
$F.\"\"%F,!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&probaG6#\"\"
\",$*(,&\"#;F'*$%\"zG\"\"#F'F'F-\"\"%,,\"$c#F'F-!$c#F,F+*$F-\"\"$!#;*$
F-F/F'!\"#F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&probaG6#\"\"#,$*(,
&\"#;\"\"\"*$%\"zGF'F,F,F.\"\"),,\"$c#F,F.!$c#F-F+*$F.\"\"$!#;*$F.\"\"
%F,!\"$F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&probaG6#\"\"$,$*(,&\"
#;\"\"\"*$%\"zG\"\"#F,F,F.\"#7,,\"$c#F,F.!$c#F-F+*$F.F'!#;*$F.\"\"%F,!
\"%F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&probaG6#\"\"%,$*(,&\"#;\"
\"\"*$%\"zG\"\"#F,F,F.F+,,\"$c#F,F.!$c#F-F+*$F.\"\"$!#;*$F.F'F,!\"&F+
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&probaG6#\"\"&,$*(,&\"#;\"\"\"*
$%\"zG\"\"#F,F,F.\"#?,,\"$c#F,F.!$c#F-F+*$F.\"\"$!#;*$F.\"\"%F,!\"'F+
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 231 "Since we want to investigate
 these probabilities for texts of large size (a typical gene is a few \+
thousand letters long), we need the Taylor expansions of these rationa
l functions for very large orders. These can be computed using " }
{HYPERLNK 17 "gfun" 2 "gfun" "" }{TEXT -1 31 ", which will first compu
te the " }{TEXT 259 6 "linear" }{TEXT -1 52 " recurrence satisfied by \+
these Taylor coefficients (" }{HYPERLNK 17 "gfun[diffeqtorec]" 2 "gfun
[diffeqtorec]" "" }{TEXT -1 73 "), and then exploit these recurrences \+
to compute the series efficiently (" }{HYPERLNK 17 "gfun[rectoproc]" 
2 "gfun[rectoproc]" "" }{TEXT -1 2 "):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 216 "for i from 0 to maxnb do   \n   rec:=diffeqtorec(y(z
)-proba[i],y(z),u(n));\n   print(i,rec);\n   rec:=select(has,rec,n) un
ion \{seq(op(1,i)=evalf(op(2,i)),i=remove(has,rec,n))\};\n   P[i]:=rec
toproc(rec,u(n),remember) \nod:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"
\"!<',,-%\"uG6#%\"nG\"\"\"-F'6#,&F)F*F*F*!#;-F'6#,&F)F*\"\"#F*\"#;-F'6
#,&F)F*\"\"$F*!$c#-F'6#,&F)F*\"\"%F*\"$c#/-F'6#F2F*/-F'6#F7F*/-F'6#F#F
*/-F'6#F*F*" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$\"\"\"<+/-%\"uG6#F#\"\"
!/-F'6#\"\"$F)/-F'6#\"\"#F),4-F'6#%\"nGF#-F'6#,&F5F#F#F#!#K-F'6#,&F5F#
F1F#\"$)G-F'6#,&F5F#F-F#!%C5-F'6#,&F5F#\"\"%F#\"%g*)-F'6#,&F5F#\"\"&F#
!&%Q;-F'6#,&F5F#\"\"'F#\"&GP(-F'6#,&F5F#\"\"(F#!'s58-F'6#,&F5F#\"\")F#
\"&Ob'/-F'6#F)F)/-F'6#FE#F#\"$c#/-F'6#FJ#F#\"$G\"/-F'6#FO#\"#Z\"%'4%/-
F'6#FT#\"#J\"%[?" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$\"\"#<//-%\"uG6#\"
\"\"\"\"!/-F'6#\"\"$F*/-F'6#F#F*/-F'6#F*F*/-F'6#\"\"%F*/-F'6#\"\"&F*/-
F'6#\"\"'F*/-F'6#\"\"(F*/-F'6#\"#6#\"#x\"')GC&/-F'6#\"\")#F)\"&Ob'/-F'
6#\"\"*#F.FQ/-F'6#\"#5#\"#ZFK,<-F'6#%\"nGF)-F'6#,&FjnF)F)F)!#[-F'6#,&F
jnF)F#F)\"$;)-F'6#,&FjnF)F.F)!%+k-F'6#,&FjnF)F8F)\"&+%Q-F'6#,&FjnF)F<F
)!'[!e#-F'6#,&FjnF)F@F)\"'/^\")-F'6#,&FjnF)FDF)!(o(GT-F'6#,&FjnF)FOF)
\"(+/$)*-F'6#,&FjnF)FUF)!)+W@E-F'6#,&FjnF)FZF)\")wtZ`-F'6#,&FjnF)FHF)!
)[;L]-F'6#,&FjnF)\"#7F)\");sx;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$\"\"
$<3/-%\"uG6#\"\"\"\"\"!/-F'6#F#F*/-F'6#\"\"#F*/-F'6#F*F*/-F'6#\"\"*F*/
-F'6#\"#5F*/-F'6#\"#6F*/-F'6#\"\")F*,D-F'6#%\"nGF)-F'6#,&FHF)F)F)!#k-F
'6#,&FHF)F1F)\"%+;-F'6#,&FHF)F#F)!&![?-F'6#,&FHF)\"\"%F)\"'+k;-F'6#,&F
HF)\"\"&F)!(Kg>\"-F'6#,&FHF)\"\"'F)\"(obS(-F'6#,&FHF)\"\"(F)!)+!oF$-F'
6#,&FHF)FDF)\"*SI#[;-F'6#,&FHF)F8F)!*+!)GC&-F'6#,&FHF)F<F)\"+3a#e*=-F'
6#,&FHF)F@F)!+sq%*)*[-F'6#,&FHF)\"#7F)\",+/>04\"-F'6#,&FHF)\"#8F)!,![O
[Z@-F'6#,&FHF)\"#9F)\",+caVo#-F'6#,&FHF)\"#:F)!,%=p)zr\"-F'6#,&FHF)\"#
;F)\"+'Hn\\H%/-F'6#FXF*/-F'6#FgnF*/-F'6#F\\oF*/-F'6#FaoF*/-F'6#F`q#\"$
d\"\"*caVo#/-F'6#Feq#\"#x\")k)3r'/-F'6#Ffp#F)\");sx;/-F'6#F[q#F)\"(/V>
%" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$\"\"%<7/-%\"uG6#\"\"\"\"\"!/-F'6#
\"\"$F*/-F'6#\"\"#F*/-F'6#F*F*/-F'6#\"\"*F*/-F'6#\"#5F*/-F'6#\"#6F*/-F
'6#\"\")F*/-F'6#\"#8F*/-F'6#\"#9F*/-F'6#\"#:F*/-F'6#\"#7F*,L-F'6#%\"nG
F)-F'6#,&FYF)F)F)!#!)-F'6#,&FYF)F2F)\"%SE-F'6#,&FYF)F.F)!&gt%-F'6#,&FY
F)F#F)\"'?j`-F'6#,&FYF)\"\"&F)!('47Y-F'6#,&FYF)\"\"'F)\")gp&[$-F'6#,&F
YF)\"\"(F)!*!)[B=#-F'6#,&FYF)FEF)\"+?FQc6-F'6#,&FYF)F9F)!+SUv()e-F'6#,
&FYF)F=F)\",ca.1K#-F'6#,&FYF)FAF)!,Sy1/U*-F'6#,&FYF)FUF)\"-?j(R.'H-F'6
#,&FYF)FIF)!-![o+*Q*)-F'6#,&FYF)FMF)\".g0t&Q%G#-F'6#,&FYF)FQF)!.'HvJ8O
[-F'6#,&FYF)\"#;F)\".?^[cz**)-F'6#,&FYF)\"#<F)!/gh>.Jr7-F'6#,&FYF)\"#=
F)\"/S9m8(Q8\"-F'6#,&FYF)\"#>F)!.!))Q\"ev\\&-F'6#,&FYF)\"#?F)\".wxi6&*
4\"/-F'6#F#F*/-F'6#FgoF*/-F'6#F\\pF*/-F'6#FapF*/-F'6#Ffr#F)\"+'Hn\\H%/
-F'6#F[s#FgoF\\u/-F'6#F`s#\"#f\",%=p)zr\"/-F'6#Fes#\"$N\"Ffu" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6$\"\"&<;/-%\"uG6#\"\"\"\"\"!/-F'6#\"\"$F*/-F'
6#\"\"#F*/-F'6#F*F*/-F'6#\"\"*F*/-F'6#\"#5F*/-F'6#\"#6F*/-F'6#\"\")F*/
-F'6#\"#8F*/-F'6#\"#9F*/-F'6#\"#:F*/-F'6#\"#7F*,T-F'6#,&%\"nGF)FIF)!/;
s$*3de9-F'6#,&FZF)FMF)\"/O6*o:!Q^-F'6#,&FZF)FQF)!0'*[E&[*eb\"-F'6#,&FZ
F)\"#;F)\"0;wW9oaF%-F'6#,&FZF)F)F)!#'*-F'6#,&FZF)F2F)\"%OR-F'6#,&FZF)F
.F)!&O6*-F'6#,&FZF)\"\"%F)\"(cqN\"-F'6#,&FZF)\"#=F)\"1c5Y3&o:=#-F'6#,&
FZF)\"#>F)!1wp#4e`=$R-F'6#,&FZF)\"#?F)\"1w0%)Q6^Ge-F'6#,&FZF)\"#<F)!1c
qJUrpZ5-F'6#,&FZF)\"#CF)\"0c1rw\\Z\"G-F'6#,&FZF)\"#BF)!1OREg)\\))o\"-F
'6#,&FZF)\"#@F)!1'47=B=GE'-F'6#,&FZF)\"#AF)\"1Oj#pwxwK%-F'6#,&FZF)FAF)
!-w&3og6*-F'6#,&FZF)FUF)\".;3p%*y(R-F'6#,&FZF)F9F)!,wTr&)z$-F'6#,&FZF)
F=F)\"-cyt.2?-F'6#FZF)-F'6#,&FZF)\"\"'F)\"*;;.I\"-F'6#,&FZF)F#F)!)'HZY
\"-F'6#,&FZF)FEF)\"+cm%Q_'-F'6#,&FZF)\"\"(F)!*c=;***/-F'6#FaoF*/-F'6#F
fqF*/-F'6#FgpF*/-F'6#F\\qF*/-F'6#FbpF*/-F'6#F#F*/-F'6#FatF*/-F'6#F^uF*
/-F'6#Faq#F)\".wxi6&*4\"/-F'6#Fer#F.\"-))Q\"ev\\&/-F'6#Fjr#\"$J$\"/;W/
'=#f</-F'6#F`r#\"$L%\".3A-$4'z)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
101 "We now compute the probability that the pattern abab occurs 0,1,.
..,5 times in a word of length 1000:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "Digits:=30:for i from 0 to maxnb do i,P[i](1000) od;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"!$\"?H)f4f^R0U>$R8EUC!#J" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"\"$\"?gier+tR,%\\+Vs;=*!#J" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"#$\"?C=5L0q8+%e<-kTr\"!#I" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"$$\"?k@mY![0hd;Y0$))=@!#I" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"%$\"?]_Mx7(R*[%*oXK%3&>!#I" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"&$\"?CEykKv=B\">0`epU\"!#I" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "The following picture then shows h
ow these probabilities evolve with the length of the word:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "plots[display](\{seq(plot([seq([10*
i,P[j](10*i)],i=1..100)]),j=0..maxnb)\});" }}{PARA 13 "" 1 "" 
{INLPLOT "6(-%'CURVESG6$7`q7$$\"#5\"\"!$\"+u&QX'*)!#97$$\"#?F*$\"?%Re)
zu!of-B1RtNC\"!#K7$$\"#IF*$\"?l3;/S#)3\">=fBD$)f$F37$$\"#SF*$\">/N&eJB
H3Fn\"o#e<q!#J7$$\"#]F*$\"?UbyD@nV)==B(R^P6F>7$$\"#gF*$\"?%\\?0.=^m:9'
4ZUb;F>7$$\"#qF*$\"?$Hd;mTp^[)[=HqWAF>7$$\"#!)F*$\"?6&fqK$fMl=$GJ$R&*G
F>7$$\"#!*F*$\"?;1vjj@b/\\#o%3K)f$F>7$$\"$+\"F*$\"?vF(*f(eO*f%pldT]M%F
>7$$\"$5\"F*$\">`.>e3t;F\"zoizF^!#I7$$\"$?\"F*$\">9P$yt:6P^B9mYRfF\\o7
$$\"$I\"F*$\">*pSG&RDPWv.\"R`tnF\\o7$$\"$S\"F*$\">`v>$3M1;_t%3SSi(F\\o
7$$\"$]\"F*$\">'[%Qm_5?9hk$4b&[)F\\o7$$\"$g\"F*$\">#Q.,0T9c[&)z97`$*F
\\o7$$\"$q\"F*$\"?h3m!eP8]fKzKEA-\"F\\o7$$\"$!=F*$\"?$o+hCym0IsqO\"*)3
6F\\o7$$\"$!>F*$\"?L@\"G58*=1G#pnS\\>\"F\\o7$$\"$+#F*$\"?>qw['*zXUz*oh
W+G\"F\\o7$$\"$5#F*$\"?.Wc)*4c89HVCz!RO\"F\\o7$$\"$?#F*$\"?yK$*4gk'Ge:
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g%F*$\"?\"*pI'o<u!RUo#Gdzm#F\\o7$$\"$q%F*$\"?&)>4**)4+(>]]T.-&o#F\\o7$
$\"$![F*$\"?REMnrZ7+\"f1Oz'*p#F\\o7$$\"$!\\F*$\"?!\\^\"))=&\\%f,N>$**>
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RFF\\o7$$\"$S&F*$\"?+?d\"HC6!=$)Q\"=()3u#F\\o7$$\"$]&F*$\"?-*\\?v2*[_8
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rtvjXt#F\\o7$$\"$!eF*$\"?SwBd:eF]N)R9O*GFF\\o7$$\"$!fF*$\"?@h#etC;*>A2
f`m@FF\\o7$$\"$+'F*$\"?13W%GA.(zsFG1#Gr#F\\o7$$\"$5'F*$\"?%zY2[9g\"zc2
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dp#\\*G`tk#F\\o7$$\"$g'F*$\"?Hf&GG&zIL5(f:q/j#F\\o7$$\"$q'F*$\"?=n;\\m
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h-Ls[:W9m(*f@F\\o7$$\"$g)F*$\"?d@>]hvm#)otXKmI@F\\o7$$\"$q)F*$\"?UIZOg
:\"z6'4J4<,@F\\o7$$\"$!))F*$\"?7M9))z0D(QYCJJ:2#F\\o7$$\"$!*)F*$\"?Qh!
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t?EVXY$RD?)>F\\o7$$\"$?*F*$\"?98#f/<I$f$G%ep3_>F\\o7$$\"$I*F*$\"?#)\\i
pWEvIgyc,9A>F\\o7$$\"$S*F*$\"?+\"oa\">c\"33K!>\"4A*=F\\o7$$\"$]*F*$\"?
a/-O&p=g\"f*[#oJi=F\\o7$$\"$g*F*$\"?KS/o$*p[B5BL^[K=F\\o7$$\"$q*F*$\"?
D#e[(>3md-17[t-=F\\o7$$\"$!)*F*$\"?w_CKR<F7lpob3t<F\\o7$$\"$!**F*$\"?I
cO!)Rd[P[')zgbV<F\\o7$$\"%+5F*$\"?C=5L0q8+%e<-kTr\"F\\o-%'COLOURG6&%$R
GBG$F)!\"\"F*F*-F$6$7`q7$F(F*7$F/$\"?+++++DJ?3%4wdHa\"!#P7$F5$\">x`e96
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\"?#Qy^Dnf\\Sq*)*********F-7$F9$\"?_vu(>lRJ+[k*********F-7$F>$\"??B*z.
r(yfN$R(********F-7$FC$\"?)zJ^3E*>3Cxk)*******F-7$FH$\"?mK)QB$y#R3AW]*
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***F-7$Fgr$\"?p>Pi_X9*RKqE2z***F-7$F\\s$\"?JfDJCbUet%HA_t***F-7$Fas$\"
?yKuQs(o#3tG$p(o'***F-7$Ffs$\"?AQe^XT?PG\"\\C**e***F-7$F[t$\"?dkrHOFRd
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**F-7$Fjt$\"?vo(z&3#>vajX)f=\"***F-7$F_u$\"?'[$zveox85i:.`*)**F-7$Fdu$
\"?QTDWP+M^&R(yfk()**F-7$Fiu$\"?**e4XaM9g'e,)G^&)**F-7$F^v$\"?K.0&yP?4
0b'3.6$)**F-7$Fcv$\"?Krf3&\\Q3(QV$*pT!)**F-7$Fhv$\"?8\"fOwJws3uU;6u(**
F-7$F]w$\"?V@_!*Hqf\"y2NhqS(**F-7$Fbw$\"?pkmWX7u1&G%fFPq**F-7$Fgw$\"?@
NZ)o-]i%3u5ZHm**F-7$F\\x$\"?*QY['*y*G^:oJL\"='**F-7$Fax$\"?l\"H2L%H&[$
=Y2`!p&**F-7$Ffx$\"?\\Au].]>4.q4sa^**F-7$F[y$\"?#>_%zvX\"QVO:c:d%**F-7
$F`y$\"?*p'=^:Y&Q*\\8-pQR**F-7$Fey$\"?'oyHbo7U(GJ]y`K**F-7$Fjy$\"?KkOA
,Xxw+/n^9D**F-7$F_z$\"?4#>Fj%pE6ue;e=<**F-7$Fdz$\"?Y0S$y%>+dM6Eqj3**F-
7$Fiz$\"?&Gb\")H'HxUJ]VjZ**)*F-7$F^[l$\"?J%H)f!\\$4S&*G$p\"o*))*F-7$Fc
[l$\"?>lr:]THR\"40VJ#z)*F-7$Fh[l$\"?\\N1`d#y&>*eBR/\"o)*F-7$F]\\l$\"?D
(**Q`A))\\)3aZ*zi&)*F-7$Fb\\l$\"?(e\"\\;ffQ!Q'>V!QP%)*F-7$Fg\\l$\"?<0t
6M))*fR[$\\#f/$)*F-7$F\\]l$\"?:v=vNk$[>&y+[U;)*F-7$Fa]l$\"?*[+-2fw/%HL
Omh,)*F-7$Ff]l$\"?7j:,jh!ebyaV<gy*F-7$F[^l$\"?@`5SynJ/oJ_1hp(*F-7$F`^l
$\"?aP'3q'pVdv!oa!Q_(*F-7$Fe^l$\"?S\"f]')Hd8SSL@7Vt*F-7$Fj^l$\"?0/()H:
s#e:9mg\"R:(*F-7$F__l$\"?(==5LOR'R&[[c0cp*F-7$Fd_l$\"?:H*o7(p))RTNI=%
\\n*F-7$Fi_l$\"?wUQh:z#Q/&\\n!*Q`'*F-7$F^`l$\"?RyCg8u]%*f3yo$4j*F-7$Fc
`l$\"?y'*z;YE(pvsU\"ed2'*F-7$Fh`l$\"?[]N'))pR]b]*)Q(H$e*F-7$F]al$\"?I[
8eu%QU)4y$3%4e&*F-7$Fbal$\"?]&ze>p:1vs[Nf>`*F-7$Fgal$\"?\"[A8!3X!R&f>O
w)[]*F-7$F\\bl$\"?*f5'Q1(*)QX.3MuoZ*F-7$Fabl$\"?J]\"yt\"G[dh$*fe\"zW*F
-7$Ffbl$\"?T**>H%GUz'=4g&4!=%*F-7$F[cl$\"?fDH%3PeNnR\"yP:(Q*F-7$F`cl$
\"?tl[7/dVZPc9yMb$*F-7$Fecl$\"?C2.t[H%)3>#\\#>fA$*F-7$Fjcl$\"?)Gsj2(=f
&Q0'4t)))G*F-7$F_dl$\"?11uC(p[6%p)>5OUD*F-7$Fddl$\"?o#)f,#p_T\"\\/b8k=
#*F-7$Fidl$\"?yh\"Gr.Rby]l-2@=*F-7$F^el$\"?<YA8qFkiw!H'zjW\"*F-7$Fcel$
\"?w))yLNc&GdE<))Ri5*F-7$Fhel$\"?KpiB=o>8nA`$>p1*F-7$F]fl$\"?5jF+VzJF[
a!y$oE!*F-7$Fbfl$\"?k4We7+&zpP!z8a&)*)F-7$Fgfl$\"?:)QIsLV(p.ma6]V*)F-7
$F\\gl$\"?:)R(Q?T9t6]\")Gd+*)F-7$Fagl$\"?hw:QM/&=#Hbyqwc))F-7$Ffgl$\"?
h)=5'>+7@yz&)\\47))F-7$F[hl$\"?V,htq/cn;oR&olw)F-7$F`hl$\"?K@[(y,+!e30
[.??()F-7$Fehl$\"?*z-5m2\"zpNikO+t')F-7$Fjhl$\"?fve>Oj#3CkIO#*\\i)F-7$
F_il$\"??%[g_*HEx\"G2\"4=w&)F-7$Fdil$\"?2&H%f%G:BwS@M%eE&)F-7$Fiil$\"?
Th)e@)[hTLFK#=iZ)F-7$F^jl$\"?wI(49Vm*Hanp')4D%)F-7$Fcjl$\"?rk9)GTj0V?&
HACt$)F-Fgjl" 2 236 216 216 2 0 1 0 2 9 0 4 2 1.000000 45.000000 
45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 17 0 0 0 0 0 
0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "The complexity of these compu
tations grows only linearly with the number " }{XPPEDIT 18 0 "k" "I\"k
G6\"" }{TEXT -1 95 " of occurrences under study. Other kinds of constr
aints like number of occurrences larger than " }{XPPEDIT 18 0 "k" "I\"
kG6\"" }{TEXT -1 11 " for fixed " }{XPPEDIT 18 0 "k" "I\"kG6\"" }
{TEXT -1 12 " or between " }{XPPEDIT 18 0 "k[1]" "&%\"kG6#\"\"\"" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "k[2]" "&%\"kG6#\"\"#" }{TEXT -1 
141 " also give rise to rational generating functions that can be extr
acted from the generating function GF and thus can be treated the same
 way. " }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 267 14 "Other patterns" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 241 "All the above computation was der
ived from the grammar describing the language, a mark being appended t
o every occurrence of the pattern. It is actually easy to write a Mapl
e procedure taking as input a word, and producing the corresponding " 
}{HYPERLNK 17 "combstruct grammar" 2 "combstruct[specification]" "" }
{TEXT -1 94 ". Then the whole computation above can be reproduced for \+
any pattern completely automatically." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 561 "gengram:=proc(pattern::list(\{identical(a),identical
(b),identical(c),identical(d)\}))local i, eq, letter, state, j;\nfor i
 to nops(pattern) do for letter in [a,b,c,d] do for j from 0 to i-1 do
 if [op(1..i-j,pattern)]=[op(j+1..i-1,pattern),letter] then state[lett
er]:=cat(w,op(1..i-j,pattern)); break fi od; if j=i then state[letter]
:=w fi od;eq[i]:=cat(w,op(1..i-1,pattern))=Union(Epsilon,seq(Prod(lett
er,state[letter]),letter=[a,b,c,d])) od; subs(cat(w,op(pattern))=Prod(
Mark,w),\{seq(eq[i],i=1..nops(pattern)),seq(letter=Atom,letter=[a,b,c,
d]),Mark=Epsilon\}) end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(gengram
G:6#'%(patternG-%%listG6#<&-%*identicalG6#%\"aG-F.6#%\"bG-F.6#%\"cG-F.
6#%\"dG6'%\"iG%#eqG%'letterG%&stateG%\"jG6\"F@C$?(8$\"\"\"FD-%%nopsG6#
9$%%trueGC$?&8&7&F0F3F6F9FIC$?(8(\"\"!FD,&FCFD!\"\"FDFI@$/7#-%#opG6$;F
D,&FCFDFPFSFH7$-FX6$;,&FPFDFDFDFRFHFLC$>&8'6#FL-%$catG6$%\"wGFW%&break
G@$/FPFC>F]oFco>&8%6#FC/-Fao6$Fco-FX6$;FDFRFH-%&UnionG6$%(EpsilonG-%$s
eqG6$-%%ProdG6$FLF]o/FLFM-%%subsG6$/-Fao6$Fco-FXFG-Fjp6$%%MarkGFco<%/F
fqFep-Fgp6$/FL%%AtomGF\\q-Fgp6$Fio/FC;FDFEF@F@" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 120 "Given a pattern, this procedure outputs the corresp
onding combstruct grammar. Thus for instance, using the pattern abab,
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "gengram([a,b,a,b]);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#<+/%\"wG-%&UnionG6'%(EpsilonG-%%ProdG6
$%\"aG%#waG-F+6$%\"bGF%-F+6$%\"cGF%-F+6$%\"dGF%/F.-F'6'F)F*-F+6$F1%$wa
bGF2F5/F=-F'6'F)-F+6$F-%%wabaGF/F2F5/FC-F'6'F)F*-F+6$F1-F+6$%%MarkGF%F
2F5/FKF)/F-%%AtomG/F1FN/F4FN/F7FN" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
211 "we obtain the grammar we started with. It is now possible to stud
y longer patterns easily. Here are the different states leading to the
 probability that the pattern abacab occurs twice  in a word of length
 5000:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "First the grammar is ge
nerated" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "G:=gengram([a,b,
a,c,a,b]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"GG<-/%\"wG-%&UnionG6
'%(EpsilonG-%%ProdG6$%\"aG%#waG-F-6$%\"bGF'-F-6$%\"cGF'-F-6$%\"dGF'/F0
-F)6'F+F,-F-6$F3%$wabGF4F7/F?-F)6'F+-F-6$F/%%wabaGF1F4F7/%%MarkGF+/F/%
%AtomG/F3FI/F6FI/F9FI/FE-F)6'F+F,F=-F-6$F6%&wabacGF7/FR-F)6'F+-F-6$F/%
'wabacaGF1F4F7/FX-F)6'F+F,-F-6$F3-F-6$FGF'F4F7" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 51 "then the bivariate generating functions are derived
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "gfsolve(G,unlabelled,z,
[[u,Mark]]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<-/-%\"cG6$%\"zG%\"uGF
(/-%\"dGF'F(/-%\"aGF'F(/-%\"bGF'F(/-%%MarkGF'F)/-%\"wGF',$*&,&\"\"\"F<
*$F(\"\"%F<F<,.!\"\"F<F(F>F=F@*$F(\"\"&F>*$F(\"\"'F@*&F(FDF)F<F<F@F@/-
%#waGF',$*&,*F<F<F=F<FAF@*&F(FBF)F<F<F<F?F@F@/-%&wabacGF',$*&,*F<F<F=F
<*$F(\"\"#F@*&F(FTF)F<F<F<F?F@F@/-%'wabacaGF',$*&,.F<F<F(F@F=F<FAF@*&F
(F<F)F<F<FLF<F<F?F@F@/-%%wabaGF',$*&,.*$F(\"\"$F@*&F(F^oF)F<F<F<F<F=F<
FAF@FLF<F<F?F@F@/-%$wabGF',$*&,&F<F<*&F(F>F)F<F<F<F?F@F@" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 105 "the generating function is then converte
d into a probability generating function by a simple substitution" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "normal(subs(\",z=z/4,w(z,u))
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&\"$c#\"\"\"*$%\"zG\"\"%F'F
',.!%'4%F'F)\"%'4%F(!#;*$F)\"\"&\"#;*$F)\"\"'!\"\"*&F)F3%\"uGF'F'F4F.
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "extracting a coefficient we \+
get the probability generating function of words with 2 occurrences of
 the pattern" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "coeff(serie
s(\",u,3),u,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**,&\"$c#\"\"\"*$
%\"zG\"\"%F'F'F)\"#7,,\"%'4%F'F)!%'4%F(\"#;*$F)\"\"&!#;*$F)\"\"'F'!\"#
,,F.F'F)F-F(F2F0F/F3!\"\"F7F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "
this gives rise to a linear recurrence satisfied by the Taylor coeffic
ients" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "diffeqtorec(y(z)-
\",y(z),u(n));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<5/-%\"uG6#\"\"\"\"
\"!/-F&6#\"\"$F)/-F&6#\"\"#F)/-F&6#F)F)/-F&6#\"\"*F)/-F&6#\"#5F)/-F&6#
\"#6F)/-F&6#\"\")F),H-F&6#%\"nGF(-F&6#,&FHF(F(F(!#[-F&6#,&FHF(F1F(\"$;
)-F&6#,&FHF(F-F(!%Kc-F&6#,&FHF(\"\"%F(\"&cI\"-F&6#,&FHF(\"\"&F(!&wX#-F
&6#,&FHF(\"\"'F(\"'+'4%-F&6#,&FHF(\"\"(F(!(g@$R-F&6#,&FHF(FDF(\"(+/$)*
-F&6#,&FHF(F8F(!(%=P%*-F&6#,&FHF(F<F(\")wtZ`-F&6#,&FHF(F@F(!*k'pf!*-F&
6#,&FHF(\"#7F(\"+_2DmC-F&6#,&FHF(\"#8F(!+/\">fT#-F&6#,&FHF(\"#9F(\"*oj
I0)-F&6#,&FHF(\"#:F(!,OnZ>(o-F&6#,&FHF(\"#;F(\"-3-Veh?-F&6#,&FHF(\"#<F
(!-3-Veh?-F&6#,&FHF(\"#=F(\",OnZ>(o/-F&6#FXF)/-F&6#FgnF)/-F&6#F\\oF)/-
F&6#FaoF)/-F&6#F_r#\"%&o#\"+[O[Z@/-F&6#Fjq#\"%>>Fgs/-F&6#Feq#Fgn\"(3')
Q)/-F&6#F[q#F-\");sx;/-F&6#F`q#F-Fat/-F&6#Ffp#F(Fft" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 110 "As before, in order to speed up the computatio
n, we change the initial conditions into floating point numbers:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=50:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "select(has,\"\",n) union \{seq(op(1
,i)=evalf(op(2,i)),i=remove(has,\"\",n))\}:" }}}{EXCHG {PARA 0 "" 0 "
" {HYPERLNK 17 "gfun[rectoproc]" 2 "gfun[rectoproc]" "" }{TEXT -1 64 "
 then produces a procedure to evaluate this sequence efficiently" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "p:=rectoproc(\",u(n),remembe
r):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "for i from 0 by 500 \+
to 10000 do i,p(i) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"!F#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$\"$+&$\"R#oj..Hv'*=N0T!zP<%p+!oVs6]J'!
#^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+5$\"S<s0?zhJlpF[w;V$4))[(Qxm
pc%G#!#^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+:$\"S$*4`)RH#\\Rz#[!Q7
Xm]e!3^sZ)>$e%!#^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+?$\"R1/Lv5ZU>
29z5o8>)=/i&QSa#Rs!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+D$\"S*G.l
yI5$onRiZhRsv*)o)3XnuN+\"!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+I$
\"S$[&*fl-_#eE#fZSF$)*)3hE?yxv7G\"!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6$\"%+N$\"S-$z#\\*)o87w/ty/y#\\8R$GFAHeX:!#]" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"%+S$\"SWxl&*H2Vo>x$\\l1B\\@3AO/9U')y\"!#]" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$\"%+X$\"S(=A15$4Z$ziuB/N^SARc9-B5a+#!#]" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+]$\"S.ZskEY])4(fA0&HkiCO$*>9#z,$>#
!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+b$\"S'[tBBA\"QI'4KfLfiLlP-s
V2`-N#!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+g$\"SNIud*Q`*>#GqT\"o
Dh$3+>AG]VrZ#!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+l$\"S/Z%)yRJ-v
6?q(=)z`Etf;3fAiuD!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+q$\"S<\\n
pmo)R/Sz*yG/-wrgdOs4EWE!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+v$\"
S)[N*GAL6<s$)3>0[Lu7&)\\>k\"f!)o#!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6$\"%+!)$\"Sr%H,g.9m0ffd^^BkA)*\\z!R%z#3F!#]" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"%+&)$\"SM-I%f!>(Hcv%ft#RHq+@pkw6?tq#!#]" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6$\"%+!*$\"S:48c+o=d#)=j()e#y#)\\o%oU)G=wo#!#]" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+&*$\"SAa(Hqm+*Q`\"RQm8f%eyo2tTud^E
!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"&++\"$\"SCEWCN4uk-#zc*RJ$\\Uv
KA)\\[^,E!#]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "The following pic
ture shows the evolution of this probability with the length of the wo
rd" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot([seq([100*i,p(10
0*i)],i=0..100)]);" }}{PARA 13 "" 1 "" {INLPLOT "6#-%'CURVESG6$7aq7$\"
\"!F(7$$\"$+\"F($\"SYX.5hLk[*z&)eKd_4G4(z.;D)yK#!#`7$$\"$+#F($\"SsRs4l
03(>v%ziq$H*\\7[:rJ<G=5!#_7$$\"$+$F($\"S,kuRj+_2T6,**RHCUDfG(oi:#>BF47
$$\"$+%F($\"SIFF3G/kNRuu)pN_T_FLF\\HAq4%F47$$\"$+&F($\"R#oj..Hv'*=N0T!
zP<%p+!oVs6]J'!#^7$$\"$+'F($\"RS&\\W#y)=]6zb+(>%=C\"4q/*3eHQ*)FD7$$\"$
+(F($\"SeZQ>@xl&>lYktPD5mw(4UO:P$>\"FD7$$\"$+)F($\"S#G^gxJg*Q9Gjs=5eif
!RF%zp(p_\"FD7$$\"$+*F($\"Sx%oXA<kObWN(4v]p&yzTx<$\\l\"*=FD7$$\"%+5F($
\"S<s0?zhJlpF[w;V$4))[(Qxmpc%G#FD7$$\"%+6F($\"SQ'QqbK>)GoVc!>07J'*>_e!
=4-.FFD7$$\"%+7F($\"SxmFPOv\")y)[U9()R-w-%>;L>[YWJFD7$$\"%+8F($\"SYQ@^
)e[([S[.N'\\@\\w9vFq1\"[1OFD7$$\"%+9F($\"SYQs-^2i#oQD%>GUU%=oaU$)z!y'3
%FD7$$\"%+:F($\"S$*4`)RH#\\Rz#[!Q7Xm]e!3^sZ)>$e%FD7$$\"%+;F($\"R([)Rb8
<gp%[Bk!44[>TC(R\"o'o$4&!#]7$$\"%+<F($\"Rp!z\"p]>@<jL5]\"*)G_$3VkH36jh
&Ffp7$$\"%+=F($\"RX.j,9pP\"[[+(RMQTkCuhohX#\\hFfp7$$\"%+>F($\"R4kI7+YW
<Yv+T'*[iZVyw(Hyw!p'Ffp7$$\"%+?F($\"R1/Lv5ZU>29z5o8>)=/i&QSa#RsFfp7$$
\"%+@F($\"R&QEY1twJf0#Gj\"=jV8T'y*[o<$z(Ffp7$$\"%+AF($\"R6'oh$eFiL5!)z
)3J\\Vi!*=ydt4^$)Ffp7$$\"%+BF($\"Rlx\"=DCbv'oVgafZUf'[qz<[m6*)Ffp7$$\"
%+CF($\"Rnrd5A>EBXxln%*=-a[OF_(4ht%*Ffp7$$\"%+DF($\"S*G.lyI5$onRiZhRsv
*)o)3XnuN+\"Ffp7$$\"%+EF($\"S5&\\DH?*[QDQtD%z_Mb<>gi'epf5Ffp7$$\"%+FF(
$\"S%)*3Y\"*3!o&)yj'>7Cd822f*3Z0i:6Ffp7$$\"%+GF($\"S&H<5*))4\\s!Gy-d(z
^bZ3U-%[^7<\"Ffp7$$\"%+HF($\"SR%>YL@n,n'=5\"=Z_&H&='*>9-)\\E7Ffp7$$\"%
+IF($\"S$[&*fl-_#eE#fZSF$)*)3hE?yxv7G\"Ffp7$$\"%+JF($\"S#eP3'f!H.:B`fM
k:ZG6!piaj]N8Ffp7$$\"%+KF($\"S>c+$>z.'4o#e:NAKuk'o_;cq6*Q\"Ffp7$$\"%+L
F($\"SIX7,!3&e$>z'3\"Qece1<:S2iS?W\"Ffp7$$\"%+MF($\"S'p8pz`rMb_F,?tdSA
!4:Hi\\@%\\\"Ffp7$$\"%+NF($\"S-$z#\\*)o87w/ty/y#\\8R$GFAHeX:Ffp7$$\"%+
OF($\"SJ4*Rli?/z6PQFn0co\"*=(QJ?4'f\"Ffp7$$\"%+PF($\"Sg1>kjK%y2[]aO\\*
oQ$G@G3G%pX;Ffp7$$\"%+QF($\"SxV'=@()Q<0%GS1VhHB+%=&o-fM%p\"Ffp7$$\"%+R
F($\"S.PR3$4d=6.Js9sTf8jl#=fr+U<Ffp7$$\"%+SF($\"SWxl&*H2Vo>x$\\l1B\\@3
AO/9U')y\"Ffp7$$\"%+TF($\"Som5X')3*py0@y=>(e[bgmTq&=U$=Ffp7$$\"%+UF($
\"S1(4w'[cOz%px#f^E]dO<^#3i2(y=Ffp7$$\"%+VF($\"S-v/$*f3+e0<K0n#zj\"4=>
fKP3A>Ffp7$$\"%+WF($\"SUqm+sVX692*HOp!4Y!>T@ZWCV'>Ffp7$$\"%+XF($\"S(=A
15$4Z$ziuB/N^SARc9-B5a+#Ffp7$$\"%+YF($\"S0fW>_?aBjDY18!=)fNT,\")QVKX?F
fp7$$\"%+ZF($\"SPly#\\/KxK&[#Q6E\\)*=q7%HYE0%3#Ffp7$$\"%+[F($\"So6a6:H
:O6eAjvM@u**4b85Ne@@Ffp7$$\"%+\\F($\"S4!e0MVp[+`A_SNX/rM%[@Jw!z:#Ffp7$
$\"%+]F($\"S.ZskEY])4(fA0&HkiCO$*>9#z,$>#Ffp7$$\"%+^F($\"SdDKGtQ/`81+A
-&**3j&R2*oP4pA#Ffp7$$\"%+_F($\"SXJ5]Y9z$)4s%fQQ-Wv,..n&*y&fAFfp7$$\"%
+`F($\"S?n!H4eTO1tUNSkV>ixc(yma-\"H#Ffp7$$\"%+aF($\"SNMO8sfk#yk]]J<Z)z
\\x&)))[%\\7K#Ffp7$$\"%+bF($\"S'[tBBA\"QI'4KfLfiLlP-sV2`-N#Ffp7$$\"%+c
F($\"S86TR7-GnU-LY&4\"*)e;c2qU+/yBFfp7$$\"%+dF($\"SN-_kp.dZW\"z%\\5=f^
Z;**R%[:YS#Ffp7$$\"%+eF($\"S)Q5,$z6+q]_^fr\\n7j>@Moe)*HCFfp7$$\"%+fF($
\"S]\"pv:$QuK\"p%y3x![U4N%y=8*eTX#Ffp7$$\"%+gF($\"SNIud*Q`*>#GqT\"oDh$
3+>AG]VrZ#Ffp7$$\"%+hF($\"SjwB@66^t*f?)z1i;x%*>Zj0'\\*)\\#Ffp7$$\"%+iF
($\"S$z9RTI#G&yZsHFFp-ZBU%\\(>)e>DFfp7$$\"%+jF($\"S&3eQ!e.(*)Q$*H#Ha;7
#[01x#)=r!RDFfp7$$\"%+kF($\"SE-X\\r`;q_o)4E>*\\5!*[r!=N6ub#Ffp7$$\"%+l
F($\"S/Z%)yRJ-v6?q(=)z`Etf;3fAiuDFfp7$$\"%+mF($\"S(*R\"fI0`]Q_$HZFB26G
Ysk9#=2f#Ffp7$$\"%+nF($\"SB(fyo@fj)p%yE\"fJ_YC)H2g>9dg#Ffp7$$\"%+oF($
\"S]gd)))>6))HatY]d&G8[F&RfzD'>EFfp7$$\"%+pF($\"SmV'[LH?!>VKJy@KI+%p,4
w;pCj#Ffp7$$\"%+qF($\"S<\\npmo)R/Sz*yG/-wrgdOs4EWEFfp7$$\"%+rF($\"SG?>
Y3MphuY^wQ!3\"=o\"o>,L=]l#Ffp7$$\"%+sF($\"ST$=.)41)fa!)Q))y15-/(R**=s(
eZm#Ffp7$$\"%+tF($\"S(o,hsKi(o4m`LIaAESU!Hl>+Nn#Ffp7$$\"%+uF($\"Sy\\mf
0#euXGg0KTVXua%eAD3E\"o#Ffp7$$\"%+vF($\"S)[N*GAL6<s$)3>0[Lu7&)\\>k\"f!
)o#Ffp7$$\"%+wF($\"S[&R?iD<GC,cc\\Rn)HkCO'\\'R\"Rp#Ffp7$$\"%+xF($\"S$H
:T)**)\\Pmu*y.cTyPRC'p%)=W))p#Ffp7$$\"%+yF($\"S1%GRH(Gnj$)\\A&yAyp3LO)
yq*oGq#Ffp7$$\"%+zF($\"S#GXbmzF7#)4p)\\jwg0H`%32f2gq#Ffp7$$\"%+!)F($\"
Sr%H,g.9m0ffd^^BkA)*\\z!R%z#3FFfp7$$\"%+\")F($\"ScX))eEQ#fMVGx<#QlS@vy
&)*)Rq4FFfp7$$\"%+#)F($\"S4y%o0`p/O>,Mo8gL'z)=zFx+.r#Ffp7$$\"%+$)F($\"
S^K!QrE\"\\grj!>M?/9hRw)eZ$*35FFfp7$$\"%+%)F($\"S=1wv/z>mF.;%yx\"3^>oR
wz#*34FFfp7$$\"%+&)F($\"SM-I%f!>(Hcv%ft#RHq+@pkw6?tq#Ffp7$$\"%+')F($\"
SE:AxvjF^$)*zU>GGY%eR10r8![q#Ffp7$$\"%+()F($\"S>9.^:tL'3z8p%[9o&[mP14
\\_:q#Ffp7$$\"%+))F($\"SoQ6nHC$)[@6Rt?T.#fJH<0&Gf(p#Ffp7$$\"%+*)F($\"S
Y%>N>%fl\"\\Z'=x\"4_$R7j<uF<%Hp#Ffp7$$\"%+!*F($\"S:48c+o=d#)=j()e#y#)
\\o%oU)G=wo#Ffp7$$\"%+\"*F($\"SBO$47=(\\n+lEGi6&3d_'o\"4dT;o#Ffp7$$\"%
+#*F($\"S\\xcN+<r=%*od-,q9\")*[41;ZI]n#Ffp7$$\"%+$*F($\"S6\\T-ff$z99vY
pQ2W:To)[JP!ym#Ffp7$$\"%+%*F($\"S<,5!\\#)\\RkJ^A%=z7.Vy1mB*z*fEFfp7$$
\"%+&*F($\"SAa(Hqm+*Q`\"RQm8f%eyo2tTud^EFfp7$$\"%+'*F($\"SU#HQzbrgBykU
UA#=qqT!)4*[9Ek#Ffp7$$\"%+(*F($\"S:'4Px-3^(\\BFx#z//axhc!p!4Jj#Ffp7$$
\"%+)*F($\"SRUy$)H0$fjl)3cU+%)Gu:')ov*yIi#Ffp7$$\"%+**F($\"S'**yY=&=o%
RjYsKSdcMpP3byTDh#Ffp7$$\"&++\"F($\"SCEWCN4uk-#zc*RJ$\\UvKA)\\[^,EFfp-
%'COLOURG6&%$RGBG$\"#5!\"\"F(F(" 2 236 204 204 2 0 1 0 2 9 0 4 2 
1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 
0 0 0 0 0 0 0 0 0 0 }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Extension
s" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 24 "Non uniform distribution" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 219 "The bases A,C,G,T in DNA are not \+
equiprobable. The combstruct model can be refined as follows to take i
nto account such irregularities. We use here the values A=0.25, C=0.18
, G=0.24, T=0.33 which come from a viral DNA (" }{XPPEDIT 18 0 "phi" "
I$phiG6\"" }{TEXT -1 165 "X174). The pattern is the same as above. We \+
only have to add marks in the grammar, compute a multivariate generati
ng function and substitute the probabilities in it." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 2 "G;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<-/%
\"wG-%&UnionG6'%(EpsilonG-%%ProdG6$%\"aG%#waG-F+6$%\"bGF%-F+6$%\"cGF%-
F+6$%\"dGF%/F.-F'6'F)F*-F+6$F1%$wabGF2F5/F=-F'6'F)-F+6$F-%%wabaGF/F2F5
/%%MarkGF)/F-%%AtomG/F1FG/F4FG/F7FG/FC-F'6'F)F*F;-F+6$F4%&wabacGF5/FP-
F'6'F)-F+6$F-%'wabacaGF/F2F5/FV-F'6'F)F*-F+6$F1-F+6$FEF%F2F5" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 327 "Marks are added by replacing a by
 Prod(Atom,Marka) everywhere it occurs, and similarly for the other le
tters. There Marka, Markb, Markc and Markd have size 0 and do not modi
fy the counting sequences and the related probabilities, but make it p
ossible to extract multivariate generating functions which contain mor
e information." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 194 "Gprob:=s
ubs(a=Prod(Atom,Marka),b=Prod(Atom,Markb),c=Prod(Atom,Markc),d=Prod(At
om,Markd),G minus \{a=Atom,b=Atom,c=Atom,d=Atom\}) union \{Marka=Epsil
on, Markb=Epsilon, Markc=Epsilon, Markd=Epsilon\};" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#>%&GprobG<-/%%MarkG%(EpsilonG/%&MarkaGF(/%&MarkbGF(/%
&MarkcGF(/%&MarkdGF(/%'wabacaG-%&UnionG6'F(-%%ProdG6$-F76$%%AtomGF*%#w
aG-F76$-F76$F;F,-F76$F'%\"wG-F76$-F76$F;F.FC-F76$-F76$F;F0FC/F<-F46'F(
F6-F76$F?%$wabGFDFH/%&wabacG-F46'F(-F76$F9F2-F76$F?FCFDFH/FQ-F46'F(-F7
6$F9%%wabaGFXFDFH/FC-F46'F(F6FXFDFH/Fin-F46'F(F6FO-F76$FFFSFH" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Here is the multivariate generatin
g function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "gfsolve(Gpro
b,unlabelled,z,[[u,Mark],[a,Marka],[b,Markb],[c,Markc],[d,Markd]]);" }
}{PARA 12 "" 1 "" {XPPMATH 20 "6#<-/-%#waG6(%\"zG%\"uG%\"aG%\"bG%\"cG%
\"dG,$*&,*\"\"\"F1*,F(\"\"&F*\"\"#F+F4F,F1F)F1F1**F(\"\"%F,F1F+F1F*F4F
1**F(F3F,F1F+F4F*F4!\"\"F1,:*&F(F1F*F1F1*,F(\"\"'F,F1F*\"\"$F+F4F)F1F1
*,F(F3F,F1F-F1F+F1F*F4F1**F(F3F,F4F+F1F*F4F1F7F1F5F8**F(F3F,F1F*F=F+F1
F1**F(F<F,F1F*F=F+F4F8F8F1*&F(F1F+F1F1*&F(F1F,F1F1*&F(F1F-F1F1F8F8/-%
\"wGF',$*&,&F1F1F5F1F1F9F8F8/-%$wabGF',$*&,&F1F1*,F(F6F*F4F,F1F+F1F)F1
F1F1F9F8F8/-%'wabacaGF',$*&,.F7F8F5F1F1F1FBF8*(F(F1F+F1F)F1F1F2F1F1F9F
8F8/-%%wabaGF',$*&,.F1F1**F(F=F,F1F*F1F+F1F8*,F(F=F,F1F*F1F+F1F)F1F1F7
F8F5F1F2F1F1F9F8F8/-%&wabacGF',$*&,*F5F1F1F1*(F(F4F+F1F*F1F8**F(F4F*F1
F+F1F)F1F1F1F9F8F8/-%&MarkbGF'F+/-%&MarkcGF'F,/-%&MarkdGF'F-/-%&MarkaG
F'F*/-%%MarkGF'F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "GF:=su
bs(\",w(z,u,a,b,c,d));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#GFG,$*&,&
\"\"\"F(**%\"zG\"\"%%\"cGF(%\"bGF(%\"aG\"\"#F(F(,:*&F*F(F.F(F(*,F*\"\"
'F,F(F.\"\"$F-F/%\"uGF(F(*,F*\"\"&F,F(%\"dGF(F-F(F.F/F(**F*F7F,F/F-F(F
.F/F(**F*F7F,F(F-F/F.F/F(F)!\"\"**F*F7F,F(F.F4F-F(F(**F*F3F,F(F.F4F-F/
F;F;F(*&F*F(F-F(F(*&F*F(F,F(F(*&F*F(F8F(F(F;F;" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 19 "The coefficient of " }{XPPEDIT 18 0 "a^k*b^l*c^m*d^p
*u^q*z^n" "*.)%\"aG%\"kG\"\"\")%\"bG%\"lGF&)%\"cG%\"mGF&)%\"dG%\"pGF&)
%\"uG%\"qGF&)%\"zG%\"nGF&" }{TEXT -1 34 " in the Taylor expansion of G
F in " }{XPPEDIT 18 0 "z" "I\"zG6\"" }{TEXT -1 34 " is the number of w
ords of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 6 " with " }
{XPPEDIT 18 0 "k " "I\"kG6\"" }{TEXT -1 30 " occurrences of the letter
 a, " }{XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 37 " occurrences of the \+
letter b ... and " }{XPPEDIT 18 0 "q" "I\"qG6\"" }{TEXT -1 93 " occurr
ences of the pattern. The probability generating function is obtained \+
by substituting " }{XPPEDIT 18 0 "a,b,c,d" "6&%\"aG%\"bG%\"cG%\"dG" }
{TEXT -1 148 " by the corresponding probability. Thus GF now takes int
o account both the specific pattern and the biased probabilities of th
e letters in the text." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "We can \+
again compute the probability that the pattern occurs twice in words o
f length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 41 " and compare th
is with the uniform model:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
34 "normal(coeff(series(GF,u,3),u,2));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#*.,&\"\"\"F%**%\"zG\"\"%%\"cGF%%\"bGF%%\"aG\"\"#F%F%F'\"#7F)F,F+
\"\"'F*F(,8*&F'F%F+F%!\"\"F&F%*,F'\"\"&F)F%%\"dGF%F*F%F+F,F1**F'F3F)F,
F*F%F+F,F1**F'F3F)F%F*F,F+F,F1*&F'F%F*F%F1**F'F3F)F%F+\"\"$F*F%F1**F'F
.F)F%F+F9F*F,F%F%F%*&F'F%F)F%F1*&F'F%F4F%F1!\"$" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 140 "Again we compute a linear recurrence satisfied by t
he Taylor coefficients from which we produce an efficient procedure fo
r their evaluation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "diff
eqtorec(y(z)-\",y(z),u(n)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
71 "nuprob:=rectoproc(subs([a=0.25,b=0.18,c=0.24,d=0.33],\"),u(n),reme
mber);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'nuprobG:6#%\"nG6\"6#%)rem
emberGE\\s3\"\"\"\"\"!\"#7$\"2+++++DiZ\"!#C\"\"&F-\"#9$\"6+++++++]t&))
!#G\"\"$F-\"\"(F-\"#6F-\"#:$\"9++++++++]Aw9!#I\"#;$\";+++++++]Q.a8A!#K
\"#<$\"=++++++++]:5o(4$!#M\"\"*F-\"#8$\"4++++++v'GW!#E\"\"#F-\"\")F-\"
\"'F-\"#5F-\"\"%F-F-F-,F-9!6#,&9$F,!#=F,$!:+++++++]P8Oz\"!#O-FR6#,&FUF
,!#<F,$\":++++++++DUd>\"FE-FR6#,&FUF,!#;F,$!9+++++++D#znx#FA-FR6#,&FUF
,!#:F,$\"8++++++++T(*\\#F=-FR6#,&FUF,!#9F,$!6++++++]?1<'F6-FR6#,&FUF,!
#8F,$\"5++++++vNL5FJ-FR6#,&FUF,!#7F,$!4+++++vpA.#F1-FR6#,&FUF,!#6F,$\"
3++++++m!e#!#A-FR6#,&FUF,!#5F,$!1+++++$yv'!#?-FR6#,&FUF,!\"*F,$\"/++++
+hlFV-FR6#,&FUF,!\")F,$!.++++P'QF]o-FR6#,&FUF,!\"(F,$\"-++++H))Fio-FR6
#,&FUF,!\"'F,$!,++]kY#Fep-FR6#,&FUF,!\"&F,$\"*+++V#Fbq-FR6#,&FUF,!\"%F
,$!'++\")F_r-FR6#,&FUF,!\"$F,$\"(+++\"F[s-FR6#,&FUF,!\"#F,$!&++$Fgs-FR
6#,&FUF,!\"\"F,$\"$+$FctF(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 207 "
The probabilities for this non-uniform model are rather different from
 those obtained in the uniform model: the pattern abacab is now less p
robable, since b and c are less probable than in the uniform model." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "for i from 0 by 500 to 100
00 do i,nuprob(i) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\"!F#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$\"$+&$\"S+\"[*))R%R^DT'*efa)Qa))oCVq*G
6m\"!#_" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+5$\"R@r=I`R]'G&p/loOnM*
zKz)R0$)Q'!#^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+:$\"SN&oyL$z1h-5-
X#\\]1\"H#3[k=(Qi8!#^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+?$\"Svj\"
GW)>&H:u>'4#eivswgJfOuvG#!#^" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+D$
\"STHpgHM!*z/s$34<O_JJ\"oJ\"[n6P$!#^" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6$\"%+I$\"S\")y)z7![@pW`(z\">`+FP27m$eH`d%!#^" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"%+N$\"Ru%R$Ra'*pP!zcD0**oRlAjni#oq'e!#]" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6$\"%+S$\"RHn*3/Y2m48;t(\\YrA4)eO^7u<s!#]" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+X$\"RuY&*y0+M^t$3+my01OHLI*o1Eg)!#
]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+]$\"Sb+MQm^N_>5wx!>9^3z#f'4VT
++\"!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+b$\"S9X8\\;$pd'oGA5')[C
e&>UB6T.$R6!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+g$\"Sb5u?GjkQ%*R
wF&[+z7&Q;.N>^w7!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+l$\"S3)*)))
Hp*=4O+$RmNb1lD'R)R4!Q59!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+q$
\"S6^]$p&f*eM:z(Q-v$[de;hw0T)R:!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$
\"%+v$\"SV=IG+`YDJqeY!Qg!z@1dF8I-k;!#]" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$\"%+!)$\"SEB+a.Z\\AARQO&Hh+p/bIY]EAy\"!#]" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$\"%+&)$\"S,];'QAq'p(He.)RF%yr!)3HH'R!R*=!#]" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$\"%+!*$\"S\">B:E0Cf\\D(oz1@xDP@e4Q7k)*>!#]" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"%+&*$\"SDqt/[)y\"p\"pa0<-'p(yMew\\
&49'4#!#]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"&++\"$\"Ss!Hg*)RJ\\^.w5
L?GVYej!=Lv?'=#!#]" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Here are th
e curves corresponding to the non-uniform and to the uniform model:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "plots[display](\{plot([se
q([100*i,p(100*i)],i=0..100)]),plot([seq([100*i,nuprob(100*i)],i=0..10
0)])\});" }}{PARA 13 "" 1 "" {INLPLOT "6$-%'CURVESG6$7aq7$\"\"!F(7$$\"
$+\"F($\"SYX.5hLk[*z&)eKd_4G4(z.;D)yK#!#`7$$\"$+#F($\"SsRs4l03(>v%ziq$
H*\\7[:rJ<G=5!#_7$$\"$+$F($\"S,kuRj+_2T6,**RHCUDfG(oi:#>BF47$$\"$+%F($
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 7896 0 0 0 0 0 0 }}}
{PARA 5 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 15 "
Markovian model" }}{PARA 0 "" 0 "" {TEXT -1 444 "By slightly modifying
 the automaton, it is possible to consider a Markovian model, where in
stead of giving the probabilities of occurrence of each letter, one gi
ves the probabilities of transition from one letter to the next one. T
he new automaton is almost the same as the previous one, except that t
he transitions are marked and three more states are added at the begin
ing. Here is the procedure generating the new automaton from the patte
rn:" }}{EXCHG {PARA 4 "> " 0 "" {MPLTEXT 1 0 858 "gengram2 := proc (pa
ttern::list(\{identical(a), identical(b),identical(c), identical(d)\})
)local i, eq, letter, state, j,alph;alph:=[a,b,c,d];for i to nops(patt
ern) do for letter in alph do for j from 0 to i-1 do if [op(1 .. i-j,p
attern)] =[op(j+1 .. i-1,pattern), letter] then state[letter] :=cat(w,
op(1 .. i-j,pattern));break fi od;if j=i then state[letter] := cat(w,l
etter) fi od;eq[i] := cat(w,op(1 .. i-1,pattern)) =Union(Epsilon,seq(P
rod(letter,`if`(i>1,cat(Mark,pattern[i-1],letter),cat(Markini,letter))
,state[letter]),    letter = alph))od;subs(cat(w,op(pattern)) =Prod(Ma
rk,w),\{Mark = Epsilon,seq(seq(cat(Mark,i,j)=Epsilon,j=alph),i=alph),s
eq(eq[i],i = 1 .. nops(pattern)),seq(letter = Atom,letter=alph),      \+
seq(cat(w,i)=Union(Epsilon,seq(Prod(j,cat(Mark,i,j),cat(w,j)),j=alph))
,i=subs(pattern[1]=NULL,alph)),seq(cat(Markini,i)=Epsilon,i=alph)\})en
d;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)gengram2G:6#'%(patternG-%%lis
tG6#<&-%*identicalG6#%\"aG-F.6#%\"bG-F.6#%\"cG-F.6#%\"dG6(%\"iG%#eqG%'
letterG%&stateG%\"jG%%alphG6\"FAC%>8)7&F0F3F6F9?(8$\"\"\"FH-%%nopsG6#9
$%%trueGC$?&8&FDFMC$?(8(\"\"!FH,&FGFH!\"\"FHFM@$/7#-%#opG6$;FH,&FGFHFS
FVFL7$-Fen6$;,&FSFHFHFHFUFLFPC$>&8'6#FP-%$catG6$%\"wGFZ%&breakG@$/FSFG
>F`o-Fdo6$FfoFP>&8%6#FG/-Fdo6$Ffo-Fen6$;FHFUFL-%&UnionG6$%(EpsilonG-%$
seqG6$-%%ProdG6%FP-%#ifG6%2FHFG-Fdo6%%%MarkG&FL6#FUFP-Fdo6$%(MarkiniGF
PF`o/FPFD-%%subsG6$/-Fdo6$Ffo-FenFK-F_q6$FgqFfo<(/FgqFjp-F\\q6$/-Fdo6$
F\\rFGFjp/FGFD-F\\q6$/-Fdo6$FfoFG-Fhp6$Fjp-F\\q6$-F_q6%FS-Fdo6%FgqFGFS
-Fdo6$FfoFS/FSFD/FG-F_r6$/&FL6#FH%%NULLGFD-F\\q6$/FP%%AtomGF]r-F\\q6$-
F\\q6$/FjsFjpF^tF^s-F\\q6$F^p/FG;FHFIFAFA" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 102 "For instance, the same pattern abacab as above yields th
e automaton described by the following grammar" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 27 "G:=gengram2([a,b,a,c,a,b]);" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#>%\"GG<D/%%MarkG%(EpsilonG/%\"aG%%AtomG/%\"bGF+/%\"cG
F+/%\"dGF+/%#waG-%&UnionG6'F(-%%ProdG6%F*%'MarkaaGF3-F86%F-%'MarkabG%$
wabG-F86%F/%'MarkacG%#wcG-F86%F1%'MarkadG%#wdG/%\"wG-F56'F(-F86%F*%)Ma
rkiniaGF3-F86%F-%)MarkinibG%#wbG-F86%F/%)MarkinicGFB-F86%F1%)MarkinidG
FF/%%wabaG-F56'F(F7F;-F86%F/FA%&wabacGFC/F>-F56'F(-F86%F*%'MarkbaGFY-F
86%F-%'MarkbbGFQ-F86%F/%'MarkbcGFB-F86%F1%'MarkbdGFF/FF-F56'F(-F86%F*%
'MarkdaGF3-F86%F-%'MarkdbGFQ-F86%F/%'MarkdcGFB-F86%F1%'MarkddGFF/FB-F5
6'F(-F86%F*%'MarkcaGF3-F86%F-%'MarkcbGFQ-F86%F/%'MarkccGFB-F86%F1%'Mar
kcdGFF/FTF(/FWF(/FPF(/FMF(/Fhn-F56'F(-F86%F*F\\q%'wabacaGF]qF`qFcq/F:F
(/FAF(/F=F(/F_r-F56'F(F7-F86%F-F=-F86$F'FHF?FC/FfpF(/FeqF(/F]pF(/F`pF(
/FcpF(/FgoF(/F\\qF(/F_qF(/FbqF(/FEF(/FaoF(/FdoF(/F^oF(/FQ-F56'F(-F86%F
*F^oF3F_oFboFeo" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "From there, g
enerating functions follow. For instance, we recover the generating fu
nction obtained before when all transitions are equiprobable:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "subs(gfsolve(G,unlabelled,z
,[[u,Mark],seq([1/4,cat(Markini,i)],i=[a,b,c,d]),seq(seq([1/4,cat(Mark
,i,j)],j=[a,b,c,d]),i=[a,b,c,d])]),w(z,u));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*&,&\"$c#\"\"\"*$%\"zG\"\"%F'F',.!%'4%F'F)\"%'4%*$F)
\"\"'!\"\"*$F)\"\"&\"#;*&F)F/%\"uGF'F'F(!#;F0F6" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 39 "The rest of the treatment is as before." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 17 "Multiple
 patterns" }}{PARA 0 "" 0 "" {TEXT -1 284 "It is also possible to writ
e an automaton which will recognize not only a fixed pattern, but a se
t of possible patterns. The procedure gengram3 below takes as input a \+
list of patterns, and produces the minimal grammar recognizing all wor
ds on 4 letters, the patterns being ``marked''." }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 913 "gengram3:=proc (patterns::list(list(\{identic
al(a),identical(b),identical(c),identical(d)\}))) local alpha, states,
 state, eq, n, trans, letter, nbst, st, indst, i; alpha:=[a,b,c,d]; nb
st:=1; states[1]:=[]; for indst while indst<=nbst do state:=states[ind
st]; n:=nops(state); for letter in alpha do if member([op(state),lette
r],patterns) then trans[letter]:=Prod(Mark,w) elif member([op(state),l
etter],map( proc(x,n) if nops(x)>n then [op(1..n+1,x)] fi end, pattern
s,nops(state))) then trans[letter]:=cat(w,op(state),letter); nbst:=nbs
t+1; states[nbst]:=[op(state),letter] else for i from indst by -1 to 2
 do st:=states[i]; if st=[op(n-nops(st)+2..n,state),letter] then trans
[letter]:=cat(w,op(st)); break fi od; if i=1 then trans[letter]:=w fi \+
fi od; eq[indst]:=cat(w,op(state))= Union(Epsilon,seq(Prod(letter,tran
s[letter]),letter=alpha)) od; \{seq(eq[i],i=1..nbst),Mark=Epsilon,seq(
letter=Atom,letter=alpha)\} end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%
)gengram3G:6#'%)patternsG-%%listG6#-F*6#<&-%*identicalG6#%\"aG-F06#%\"
bG-F06#%\"cG-F06#%\"dG6-%&alphaG%'statesG%&stateG%#eqG%\"nG%&transG%'l
etterG%%nbstG%#stG%&indstG%\"iG6\"FHC'>8$7&F2F5F8F;>8+\"\"\">&8%6#FO7
\"?(8-FOFOFH1FVFNC&>8&&FR6#FV>8(-%%nopsG6#FZ?&8*FK%%trueG@'-%'memberG6
$7$-%#opGF[oF]o9$>&8)6#F]o-%%ProdG6$%%MarkG%\"wG-Fao6$Fco-%$mapG6%:6$%
\"xGFAFHFHFH@$29%-Fjn6#Ffo7#-Feo6$;FO,&FjpFOFOFOFfoFHFHFfoFinC%>Fho-%$
catG6%F_pFdoF]o>FN,&FNFOFOFO>&FR6#FNFcoC$?(8.FV!\"\"\"\"#F^oC$>8,&FR6#
F^r@$/Fcr7$-Feo6$;,(FhnFO-Fjn6#FcrF_rF`rFOFhnFZF]oC$>Fho-Feq6$F_p-FeoF
^s%&breakG@$/F^rFO>FhoF_p>&8'Ffn/-Feq6$F_pFdo-%&UnionG6$%(EpsilonG-%$s
eqG6$-F\\p6$F]oFho/F]oFK<%/F^pFat-Fct6$&FjsFer/F^r;FOFN-Fct6$/F]o%%Ato
mGFgtFHFH" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "For instance, here i
s the grammar recognizing the words abab and abacab:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 39 "G:=gengram3([[a,b,a,b],[a,b,a,c,a,b]]);" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"GG<-/%\"wG-%&UnionG6'%(EpsilonG-
%%ProdG6$%\"aG%#waG-F-6$%\"bGF'-F-6$%\"cGF'-F-6$%\"dGF'/F0-F)6'F+F,-F-
6$F3%$wabGF4F7/F?-F)6'F+-F-6$F/%%wabaGF1F4F7/%%MarkGF+/F/%%AtomG/F3FI/
F6FI/F9FI/%&wabacG-F)6'F+-F-6$F/%'wabacaGF1F4F7/FS-F)6'F+F,-F-6$F3-F-6
$FGF'F4F7/FE-F)6'F+F,FW-F-6$F6FNF7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 35 "Again, generating functions follow:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 48 "subs(gfsolve(G,unlabelled,z,[[u,Mark]]),w(z,u));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,(\"\"\"F&*$%\"zG\"\"#F&*$F(\"\"%
F&F&,4!\"\"F&F(F+F'F-*$F(\"\"$F+F*!\"#*&F(F+%\"uGF&F&*$F(\"\"&F+*$F(\"
\"'F-*&F(F6F2F&F&F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "The coef
ficient of " }{XPPEDIT 18 0 "u^k*z^n" "*&)%\"uG%\"kG\"\"\")%\"zG%\"nGF
&" }{TEXT -1 84 " in the Taylor expansion of this rational function is
 the number of words of length " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT 
-1 6 " with " }{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 75 " occurrences
 of the patterns abab and abacab. Here are the first few terms:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "map(expand,series(\",z,10));
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#+9%\"zG\"\"\"\"\"!\"\"%\"\"\"\"#;
\"\"#\"#k\"\"$,&\"$b#F%%\"uGF%\"\"%,&\"%;5F%F/\"\")\"\"&,&\"%[SF%F/\"#
[\"\"',&\"&Gh\"F%F/\"$c#\"\"(,(\"&eU'F%F/\"%x7*$F/\"\"#F%\"\"),(\"'?gD
F%F/\"%7hF@\"#7\"\"*-%\"OG6#F%\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 22 "For instance the term " }{XPPEDIT 18 0 "48*u*z^6" "*(\"#[\"\"\"
%\"uGF$%\"zG\"\"'" }{TEXT -1 364 " corresponds to the 47 words of leng
th 6 containing abab (ababab is counted only once), plus the word abac
ab itself. Of course, it would not be difficult to modify the grammar \+
so as to take into accounts overlapping words differently. From this g
enerating function, the treatment proceeds as before. Again, non-unifo
rm and Markovian extensions could be considered." }}}}{SECT 0 {PARA 4 
"" 0 "" {TEXT -1 20 "Patterns with errors" }}{PARA 0 "" 0 "" {TEXT -1 
612 "It is also possible to accomodate patterns whose occurrence is ex
act except at one unspecified position. A direct way would be to apply
 the previous technique for multiple patterns after having generated a
ll possible patterns obtained by introducing one error in the pattern \+
under study. However, the number of patterns obtained this way may be \+
much too large for this technique to be practical. A better way is to \+
produce directly the automaton corresponding to occurrences of the pat
tern with at most one error, and this turns out not to be too difficul
t. The following procedure generates all words of length " }{XPPEDIT 
18 0 "n" "I\"nG6\"" }{TEXT -1 148 ", exact occurrences of the pattern \+
being tagged with a mark as before (Mark0err), while occurrences with \+
one mismatch are tagged by a Mark1err mark." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 1479 "gengram4:=proc (pattern::list(\{identical(a),iden
tical(b),identical(c),identical(d)\})) local alpha, state, eq, n, lett
er, st, i, eq2, staterr, j; alpha:=[a,b,c,d]; for i to nops(pattern) d
o staterr[i]:=\{\}; for letter in alpha do for j from 0 to i do if j=i
 or [op(1 .. i - j, pattern)] = [op(j + 1 .. i - 1, pattern), letter] \+
then if j=0 then state[letter] := cat(w, op(1 .. i , pattern)) else st
aterr[i]:=staterr[i] union \{[[op(1..i,pattern)], [op(1..i-j,pattern)]
]\}; state[letter]:=cat(w,op(1..i,pattern),`|`, op(1..i-j,pattern)) fi
; break fi od; od; eq[i] := cat(w, op(1 .. i - 1, pattern)) = Union(Ep
silon, seq(Prod(letter, state[letter]), letter = alpha)) od; for i to \+
nops(pattern)-1 do for st in staterr[i] do n:=nops(st[2]); for letter \+
in alpha do for j from 0 to n+1 do if j=n+1 or [op(1..n-j+1,pattern)]=
 [op(j+1..n,st[2]),letter] then if pattern[i+1]=letter then state[lett
er]:=cat(w,op(1..i+1,pattern),`|`, op(1..n-j+1,pattern)); staterr[i+1]
:=staterr[i+1] union \{[[op(1..i+1,pattern)],[op(1..n-j+1,pattern)]]\}
 else state[letter]:=cat(w,op(1..n-j+1,pattern)) fi; break fi od od; e
q2[st] := cat(w,op(st[1]),`|`,op(st[2])) = Union(Epsilon, seq(Prod(let
ter, state[letter]), letter = alpha)) od od; \{seq(eq[i],i=1..nops(pat
tern)), seq(seq(eq2[st],st=staterr[i]),i=1..nops(pattern)-1), cat(w,op
(pattern))=Prod(Mark0err,w), seq(cat(w,op(st[1]),`|`,op(st[2]))=Prod(M
ark1err,w), st=staterr[nops(pattern)]), Mark0err=Epsilon,Mark1err=Epsi
lon,seq(letter=Atom,letter=alpha)\} end;" }}{PARA 12 "" 1 "" {XPPMATH 
20 "6#>%)gengram4G:6#'%(patternG-%%listG6#<&-%*identicalG6#%\"aG-F.6#%
\"bG-F.6#%\"cG-F.6#%\"dG6,%&alphaG%&stateG%#eqG%\"nG%'letterG%#stG%\"i
G%$eq2G%(staterrG%\"jG6\"FEC&>8$7&F0F3F6F9?(8*\"\"\"FL-%%nopsG6#9$%%tr
ueGC%>&8,6#FK<\"?&8(FHFQ?(8-\"\"!FLFKFQ@$5/FenFK/7#-%#opG6$;FL,&FKFLFe
n!\"\"FP7$-F]o6$;,&FenFLFLFL,&FKFLFaoFLFPFYC$@%/FenFfn>&8%6#FY-%$catG6
$%\"wG-F]o6$;FLFKFPC$>FT-%&unionG6$FT<#7$7#FcpF[o>F\\p-F`p6&FbpFcp%\"|
grGF\\o%&breakG>&8&FV/-F`p6$Fbp-F]o6$;FLFgoFP-%&UnionG6$%(EpsilonG-%$s
eqG6$-%%ProdG6$FYF\\p/FYFH?(FKFLFL,&FMFLFaoFLFQ?&8)FTFQC%>8'-FN6#&Fjr6
#\"\"#?&FYFHFQ?(FenFfnFL,&F]sFLFLFLFQ@$5/FenFes/7#-F]o6$;FL,(F]sFLFenF
aoFLFLFP7$-F]o6$;FfoF]sF`sFYC$@%/&FP6#,&FKFLFLFLFYC$>F\\p-F`p6&Fbp-F]o
6$;FLFhtFPFaqF[t>&FUFgt-Fip6$Fau<#7$7#F]uFjs>F\\p-F`p6$FbpF[tFbq>&8+6#
Fjr/-F`p6&Fbp-F]o6#&Fjr6#FLFaq-F]oF_sF\\r<)/%)Mark0errGF_r/%)Mark1errG
F_r-Far6$/FY%%AtomGFfr-Far6$/F_v-Fdr6$FjvFbp/Fjr&FU6#FM/-F`p6$Fbp-F]oF
O-Fdr6$FhvFbp-Far6$-Far6$F[v/FjrFT/FK;FLFhr-Far6$Fdq/FK;FLFMFEFE" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "Here is a simple example. The gen
erated automaton is almost optimal (in general it has O(1) too many st
ates):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "gengram4([a,b,b,c
]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<5/%\"aG%%AtomG/%\"bGF&/%\"cGF&
/%\"dGF&/%#waG-%&UnionG6'%(EpsilonG-%%ProdG6$F%%&wab|graG-F46$F(%$wabG
-F46$F*%%wab|grG-F46$F,F</F9-F06'F2-F46$F%%'wabb|graG-F46$F(%%wabbG-F4
6$F*%&wabb|grG-F46$F,FJ/FG-F06'F2-F46$F%%(wabbc|graG-F46$F(%'wabbc|grG
-F46$F*%&wabbcG-F46$F,FU/%$wa|grG-F06'F2-F46$F%F.-F46$F(F<-F46$F*%\"wG
-F46$F,F_o/F6-F06'F2Fin-F46$F(%(wabb|grabGF]oF`o/F<-F06'F2Fin-F46$F(FJ
F]oF`o/FJ-F06'F2Fin-F46$F(F_o-F46$F*FUF`o/FD-F06'F2FinF7FbpF`o/FU-F46$
%)Mark1errGF_o/FX-F46$%)Mark0errGF_o/FRFhp/Fgo-F06'F2FinFEFbpF`o/F^qF2
/FjpF2/F_o-F06'F2Fin-F46$F(Ffn-F46$F*Ffn-F46$F,Ffn" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 452 "There are two kinds of states: states correspond
ing to words without error, which as before are represented by w.prefi
x, and states with one error which contain a `|', possibly followed by
 the last few letters which were read, when these form a prefix of the
 pattern. For instance, wabb|ab is the state reached by words which co
ntain the string abb with one error, their last two letters being ab.
\nFrom there, we derive trivariate generating functions:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "subs(gfsolve(\",unlabelled,z,[[u,Ma
rk0err],[v,Mark1err]]),w(z,u,v));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,
$*&,4!\"\"\"\"\"%\"zG!\"%*$F(\"\"#!\"'*$F(\"\"$F,*$F(\"\"%F)*$F(\"\"&!
\"$*$F(\"\")F.*$F(\"\"'!\"#*$F(\"\"*F.F',L*&F(F2%\"uGF'F)*&F(\"\"(F=F'
!\"&F'F'F1\"#@F-!#=*&F(F2%\"vGF'!#I*&F(F7F=F'F,*&F(F7FDF'!#@F4F3F9F3F6
\"#<*$F(F?F.*&F(F?FDF'F,*&F(F:FDF'F:*&F(\"#5FDF'F:*&F(FNF=F'F.*&F(F:F=
F'F.*&F(F0F=F'F&F*!#5*&F(F0FDF'!#7F/!\"(F&F&" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 19 "The coefficient of " }{XPPEDIT 18 0 "u^k*v^l*z^n" "*()%
\"uG%\"kG\"\"\")%\"vG%\"lGF&)%\"zG%\"nGF&" }{TEXT -1 54 " in the Taylo
r expansion of this rational function at " }{XPPEDIT 18 0 "z=0" "/%\"z
G\"\"!" }{TEXT -1 34 " is the number or words of length " }{XPPEDIT 
18 0 "n" "I\"nG6\"" }{TEXT -1 31 " where the pattern abbc occurs " }
{XPPEDIT 18 0 "k" "I\"kG6\"" }{TEXT -1 25 " times without error and " 
}{XPPEDIT 18 0 "l" "I\"lG6\"" }{TEXT -1 149 " times with exactly one e
rror. Again, the Markovian model could also be treated, and extensions
 to multiple errors are likely to be possible as well." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 10 "Conclusion" }}{PARA 0 "" 0 "" {TEXT -1 575 "Various proba
bilistic parameters related to the occurrence of specific patterns in \+
random words can be computed very easily using combstruct and gfun. He
re, combstruct is used to model the combinatorics of the problem and g
fun is very helpful to compute expansions to very large orders, thanks
 to the rational type of the corresponding generating functions. The m
odel itself can be modified in various directions, to take into accoun
t different probability models or different ways of counting occurrenc
es of the pattern when they overlap, or several patterns simultaneousl
y." }}}}{MARK "5 0 0" 46 }{VIEWOPTS 0 0 0 1 1 1803 }