{VERSION 6 0 "IBM INTEL NT" "6.0" }
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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 68 "Suppleme
ntal Proofs for \"Affine Partitions and Affine Grassmannians\"" }}
{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 257 41 "by Sara C. Billey and S
tephen A. Mitchell" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 
1 {PARA 3 "" 0 "" {TEXT -1 5 "Tools" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 
"truncatedpow:=(x,expon, deg)->if expon = 0 then 1; else add(x^(expon*
i), i=0..ceil(deg/expon));fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-tr
uncatedpowGf*6%%\"xG%&exponG%$degG6\"6$%)operatorG%&arrowGF*@%/9%\"\"!
\"\"\"-%$addG6$)9$*&F0F2%\"iGF2/F9;F1-%%ceilG6#*&9&F2F0!\"\"F*F*F*" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 71 "truncatedpoly:=(poly, var,deg)->add(coeff(poly,var,
i)*var^i, i=0..deg);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.truncatedpo
lyGf*6%%%polyG%$varG%$degG6\"6$%)operatorG%&arrowGF*-%$addG6$*&-%&coef
fG6%9$9%%\"iG\"\"\")F6F7F8/F7;\"\"!9&F*F*F*" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 57 "polyfy:=(alist)->add(alist[j]*t^(j-1), j=1..nops(a
list));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'polyfyGf*6#%&alistG6\"6$
%)operatorG%&arrowGF(-%$addG6$*&&9$6#%\"jG\"\"\")%\"tG,&F3F4F4!\"\"F4/
F3;F4-%%nopsG6#F1F(F(F(" }}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 8 "Type G_2" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 224 "## Cut Here and Paste into Maple to Simplify:\n\n
## Denominator corresponding to parts allowed with all: \n\ndeno[AFFIN
E-G,2]:= mul(1-t^i, i=[5,6]):\n\n## Repeaters: NIL\ngen[1]:=polyfy([1,
1,1,1,1,1])/mul(1-t^i, i=[]):\n\n## End Cut:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 72 "AffineParitionGF[G,2]:=simplify((add(gen[i], i=1
..1)/deno[AFFINE-G,2]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%1Affine
ParitionGFG6$%\"GG\"\"#*&\"\"\"F**&,&%\"tGF*F*!\"\"F*,&*$)F-\"\"&F*F*F
*F.F*F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "BottDenominator[
G,2]:=(mul(1-t^i, i = [1,5]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%0
BottDenominatorG6$%\"GG\"\"#*&,&\"\"\"F+%\"tG!\"\"F+,&F+F+*$)F,\"\"&F+
F-F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "simplify(BottDenomi
nator[G,2] * AffineParitionGF[G,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
-1 8 "Type F_4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 465 "## Cut He
re and Paste into Maple to Simplify:\n\n## Denominator corresponding t
o parts allowed with all: \n\ndeno[AFFINE-F,4]:= mul(1-t^i, i=[14,15,1
6]):\n\n## Repeaters: ((11 1))\ngen[1]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,
1,1,1,1,1,2,2,3,2,2,2,2,2,3,3,3,2,3,4,5,5,4,4,4,3,3,3,3,2,1,2,2,3,2,2,
2,2,1,1,1,1])/mul(1-t^i, i=[11]):\n\n## Repeaters: NIL\ngen[2]:=polyfy
([1,1,1,1,1,2,2,3,3,3,4,4,5,5,5,5,4,5,5,6,6,6,6,6,5,5,4,4,3,2,2,2,3,2,
2,2,2,1,1,1,1])/mul(1-t^i, i=[]):\n\n## End Cut:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 73 "AffineParitionGF[F,4]:=simplify((add(gen[i], \+
i=1..2))/ deno[AFFINE-F,4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%1Af
fineParitionGFG6$%\"FG\"\"%*&\"\"\"F***,&%\"tGF*F*!\"\"F*,&*$)F-\"\"&F
*F*F*F.F*,&*$)F-\"#6F*F*F*F.F*,&F*F.*$)F-\"\"(F*F*F*F." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "BottDenominator[F,4]:=(mul(1-t^i, i
 = [1,5,7,11]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%0BottDenominato
rG6$%\"FG\"\"%**,&\"\"\"F+%\"tG!\"\"F+,&F+F+*$)F,\"\"&F+F-F+,&F+F+*$)F
,\"\"(F+F-F+,&F+F+*$)F,\"#6F+F-F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "simplify(BottDenominator[F,4] * AffineParitionGF[F,4]
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 8 "Type E_6" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 8102 "\n\n## Cut Here and Paste into Maple to Simp
lify:\n\n\ndeno[AFFINE-E,6]:= mul(1-t^i, i=[20,21,22]);\n\n## Repeater
s: ((12 5) (18 2) (19 0))\ngen[1]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,0,0,0
,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
,1,1,1,1,2,2,2,2,3,2,2,2,2,2,3,5,6,7,8,10,11,13,13,14,13,13,11,10,9,9,
8,8,8,8,8,9,11,11,11,11,11,9,8,7,6,4,3,2,1])/mul(1-t^i, i=[12,18,19]):
\n\n## Repeaters: ((18 2) (19 0))\ngen[2]:=polyfy([0,0,0,0,0,0,0,0,0,0
,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,4
,6,7,8,9,10,11,12,14,15,16,16,18,19,21,21,22,23,23,21,20,19,18,16,15,1
4,12,11,11,12,11,11,11,11,9,8,7,6,4,3,2,1])/mul(1-t^i, i=[18,19]):\n\n
## Repeaters: ((12 1) (16 1) (19 0))\ngen[3]:=polyfy([0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,1,1,1,1,2,2,2,2,3,2,2,2,2,2,3,4,4,5,7,8,8,10,11,11,10,11,10,8,
8,8,7,7,9,10,10,12,15,15,16,18,18,16,15,15,12,10,10,9,7,7,8,8,8,10,11,
10,11,11,10,8,8,7,5,4,4,3,2,2,2,2,2,3,2,2,2,2,1,1,1,1])/mul(1-t^i, i=[
12,16,19]):\n\n## Repeaters: ((12 5) (16 1) (19 0))\ngen[4]:=polyfy([0
,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,2,2,2,2,2,3,4,4,5,7,8,8,10,11
,11,10,11,10,8,8,8,7,7,9,10,10,12,15,15,16,18,18,16,15,15,12,10,10,9,7
,7,8,8,8,10,11,10,11,11,10,8,8,7,5,4,4,3,2,2,2,2,2,3,2,2,2,2,1,1,1,1])
/mul(1-t^i, i=[12,16,19]):\n\n## Repeaters: ((16 1) (19 0))\ngen[5]:=p
olyfy([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
,0,0,0,1,1,1,1,2,3,3,4,6,7,8,9,10,11,12,13,13,14,16,17,18,20,23,24,25,
26,27,27,29,29,29,29,31,30,30,32,36,36,38,41,43,41,40,38,35,32,31,28,2
5,24,23,21,21,24,25,24,25,26,24,22,20,18,14,12,10,8,6,6,4,4,4,6,4,4,4,
4,2,2,2,2])/mul(1-t^i, i=[16,19]):\n\n## Repeaters: ((12 1) (19 0))\ng
en[6]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
,0,0,0,1,1,1,1,2,2,2,2,3,2,2,2,2,2,3,5,6,8,10,12,13,16,17,18,17,18,16,
14,13,13,12,13,15,17,18,21,25,27,29,31,31,29,27,25,21,18,16,14,11,10,1
0,10,10,12,13,13,13,13,12,10,9,8,6,5,4,3,2,2,2,2,2,3,2,2,2,2,1,1,1,1])
/mul(1-t^i, i=[12,19]):\n\n## Repeaters: ((12 5) (19 0))\ngen[7]:=poly
fy([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,
1,1,2,2,2,2,3,2,2,2,2,2,3,5,6,8,10,12,13,16,17,18,17,18,16,14,13,13,12
,13,15,17,18,21,25,27,29,31,31,29,27,25,21,18,16,14,11,10,10,10,10,12,
13,13,13,13,12,10,9,8,6,5,4,3,2,2,2,2,2,3,2,2,2,2,1,1,1,1])/mul(1-t^i,
 i=[12,19]):\n\n## Repeaters: ((19 0))\ngen[8]:=polyfy([0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,4,6,7,8,10,11,13,15,19,21,24,26,
30,33,39,43,47,50,54,54,54,54,55,54,55,56,56,55,57,61,64,67,71,72,71,6
8,64,57,52,47,42,35,32,29,27,25,28,29,30,29,30,28,26,22,20,16,14,10,8,
6,6,4,4,4,6,4,4,4,4,2,2,2,2])/mul(1-t^i, i=[19]):\n\n## Repeaters: ((1
2 1) (18 1) (19 1))\ngen[9]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1
,1,2,2,2,2,3,2,2,2,2,2,3,5,6,7,8,10,11,13,13,14,13,13,11,10,9,9,8,8,8,
8,8,9,11,11,11,11,11,9,8,7,6,4,3,2,1])/mul(1-t^i, i=[12,18,19]):\n\n##
 Repeaters: ((18 1) (19 1))\ngen[10]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,4,6,7,
8,9,10,11,12,14,15,16,16,18,19,21,21,22,23,23,21,20,19,18,16,15,14,12,
11,11,12,11,11,11,11,9,8,7,6,4,3,2,1])/mul(1-t^i, i=[18,19]):\n\n## Re
peaters: ((12 1) (16 1) (19 1))\ngen[11]:=polyfy([0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,1,1,1,1,2,2,2,2,3,2,2,2,2,2,3,4,4,5,7,8,8,10,11,11,10,11,10,8,8,8,
7,7,9,10,10,12,15,15,16,18,18,16,15,15,12,10,10,9,7,7,8,8,8,10,11,10,1
1,11,10,8,8,7,5,4,4,3,2,2,2,2,2,3,2,2,2,2,1,1,1,1])/mul(1-t^i, i=[12,1
6,19]):\n\n## Repeaters: ((12 5) (16 1) (19 1))\ngen[12]:=polyfy([0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,2,2,2,2,2,3,4,4,5,7,8,8,10,11,11
,10,11,10,8,8,8,7,7,9,10,10,12,15,15,16,18,18,16,15,15,12,10,10,9,7,7,
8,8,8,10,11,10,11,11,10,8,8,7,5,4,4,3,2,2,2,2,2,3,2,2,2,2,1,1,1,1])/mu
l(1-t^i, i=[12,16,19]):\n\n## Repeaters: ((16 1) (19 1))\ngen[13]:=pol
yfy([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
,0,0,1,1,1,1,2,3,3,4,6,7,8,9,10,11,12,13,13,14,16,17,18,20,23,24,25,26
,27,27,29,29,29,29,31,30,30,32,36,36,38,41,43,41,40,38,35,32,31,28,25,
24,23,21,21,24,25,24,25,26,24,22,20,18,14,12,10,8,6,6,4,4,4,6,4,4,4,4,
2,2,2,2])/mul(1-t^i, i=[16,19]):\n\n## Repeaters: ((12 1) (19 1))\ngen
[14]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,1,1,1,1,2,2,2,2,3,2,2,2,2,2,3,5,6,8,10,12,13,16,17,18,17,18,16,1
4,13,13,12,13,15,17,18,21,25,27,29,31,31,29,27,25,21,18,16,14,11,10,10
,10,10,12,13,13,13,13,12,10,9,8,6,5,4,3,2,2,2,2,2,3,2,2,2,2,1,1,1,1])/
mul(1-t^i, i=[12,19]):\n\n## Repeaters: ((12 5) (19 1))\ngen[15]:=poly
fy([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,
1,1,2,2,2,2,3,2,2,2,2,2,3,5,6,8,10,12,13,16,17,18,17,18,16,14,13,13,12
,13,15,17,18,21,25,27,29,31,31,29,27,25,21,18,16,14,11,10,10,10,10,12,
13,13,13,13,12,10,9,8,6,5,4,3,2,2,2,2,2,3,2,2,2,2,1,1,1,1])/mul(1-t^i,
 i=[12,19]):\n\n## Repeaters: ((19 1))\ngen[16]:=polyfy([0,0,0,0,0,0,0
,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,4,6,7,8,10,11,13,15,19,21,24,26
,30,33,39,43,47,50,54,54,54,54,55,54,55,56,56,55,57,61,64,67,71,72,71,
68,64,57,52,47,42,35,32,29,27,25,28,29,30,29,30,28,26,22,20,16,14,10,8
,6,6,4,4,4,6,4,4,4,4,2,2,2,2])/mul(1-t^i, i=[19]):\n\n## Repeaters: ((
12 1) (18 1))\ngen[17]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,2,2,2,2,2,3,5,6,7,8,10,11,13,1
3,14,13,13,11,10,9,9,8,8,8,8,8,9,11,11,11,11,11,9,8,7,6,4,3,2,1])/mul(
1-t^i, i=[12,18]):\n\n## Repeaters: ((18 1))\ngen[18]:=polyfy([0,0,0,0
,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,4,6,7,8,9,10,11,12,14,15,16
,16,18,19,21,21,22,23,23,21,20,19,18,16,15,14,12,11,11,12,11,11,11,11,
9,8,7,6,4,3,2,1])/mul(1-t^i, i=[18]):\n\n## Repeaters: ((12 5) (18 2))
\ngen[19]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
,0,0,0,0,1,1,1,1,2,2,2,2,3,2,2,2,2,2,3,5,6,7,8,10,11,13,13,14,13,13,11
,10,9,9,8,8,8,8,8,9,11,11,11,11,11,9,8,7,6,4,3,2,1])/mul(1-t^i, i=[12,
18]):\n\n## Repeaters: ((18 2))\ngen[20]:=polyfy([0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,4,6,7,8,9,10,11,12,14,15,16,16,18,19,21,
21,22,23,23,21,20,19,18,16,15,14,12,11,11,12,11,11,11,11,9,8,7,6,4,3,2
,1])/mul(1-t^i, i=[18]):\n\n## Repeaters: ((12 1) (16 1))\ngen[21]:=po
lyfy([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,
2,2,2,2,3,2,2,2,2,2,3,4,4,5,7,8,8,10,11,11,10,11,10,8,8,8,7,7,9,10,10,
12,15,15,16,18,18,16,15,15,12,10,10,9,7,7,8,8,8,10,11,10,11,11,10,8,8,
7,5,4,4,3,2,2,2,2,2,3,2,2,2,2,1,1,1,1])/mul(1-t^i, i=[12,16]):\n\n## R
epeaters: ((12 5) (16 1))\ngen[22]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,2,2,2,2,2,3,4,4,5,7,8,
8,10,11,11,10,11,10,8,8,8,7,7,9,10,10,12,15,15,16,18,18,16,15,15,12,10
,10,9,7,7,8,8,8,10,11,10,11,11,10,8,8,7,5,4,4,3,2,2,2,2,2,3,2,2,2,2,1,
1,1,1])/mul(1-t^i, i=[12,16]):\n\n## Repeaters: ((16 1))\ngen[23]:=pol
yfy([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,4,6,7,8,9,10,11,12,
13,13,14,16,17,18,20,23,24,25,26,27,27,29,29,29,29,31,30,30,32,36,36,3
8,41,43,41,40,38,35,32,31,28,25,24,23,21,21,24,25,24,25,26,24,22,20,18
,14,12,10,8,6,6,4,4,4,6,4,4,4,4,2,2,2,2])/mul(1-t^i, i=[16]):\n\n## Re
peaters: ((12 1))\ngen[24]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,
2,2,2,3,2,2,2,2,2,3,5,6,8,10,12,13,16,17,18,17,18,16,14,13,13,12,13,15
,17,18,21,25,27,29,31,31,29,27,25,21,18,16,14,11,10,10,10,10,12,13,13,
13,13,12,10,9,8,6,5,4,3,2,2,2,2,2,3,2,2,2,2,1,1,1,1])/mul(1-t^i, i=[12
]):\n\n## Repeaters: ((12 5))\ngen[25]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,
0,1,1,1,1,2,2,2,2,3,2,2,2,2,2,3,5,6,8,10,12,13,16,17,18,17,18,16,14,13
,13,12,13,15,17,18,21,25,27,29,31,31,29,27,25,21,18,16,14,11,10,10,10,
10,12,13,13,13,13,12,10,9,8,6,5,4,3,2,2,2,2,2,3,2,2,2,2,1,1,1,1])/mul(
1-t^i, i=[12]):\n\n## Repeaters: NIL\ngen[26]:=polyfy([1,1,1,1,2,3,3,4
,6,7,8,10,11,13,15,19,21,24,26,30,33,39,43,47,50,54,54,54,54,55,54,55,
56,56,55,57,61,64,67,71,72,71,68,64,57,52,47,42,35,32,29,27,25,28,29,3
0,29,30,28,26,22,20,16,14,10,8,6,6,4,4,4,6,4,4,4,4,2,2,2,2])/mul(1-t^i
, i=[]):\n\n## End Cut:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%denoG6$
,&%'AFFINEG\"\"\"%\"EG!\"\"\"\"'*(,&F)F)*$)%\"tG\"#?F)F+F),&F)F)*$)F1
\"#@F)F+F),&F)F)*$)F1\"#AF)F+F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 74 "AffineParitionGF[E,6]:=simplify((add(gen[i], i=1..26))/ deno[A
FFINE-E,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%1AffineParitionGFG6
$%\"EG\"\"'*&\"\"\"F**,,**$)%\"tG\"#7F*F**$)F/\"#6F*F*F/!\"\"F*F4F*,&F
*F4*$)F/\"\"(F*F*F*,2*$)F/\"\")F*F*F6F**$)F/F(F*F**$)F/\"\"&F*F**$)F/
\"\"$F*F4*$)F/\"\"#F*F4F/F4F*F4F*,2F6F*F=F4F?F**$)F/\"\"%F*F4FBF*FEF4F
/F*F*F4F*),&F/F*F*F4FGF*F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
55 "BottDenominator[E,6]:=(mul(1-t^i, i = [1,4,5,7,8,11]));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>&%0BottDenominatorG6$%\"EG\"\"'*.,&\"\"\"F+
%\"tG!\"\"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+,&F+F+*$)F,\"#6F+F-F+" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 55 "simplify(BottDenominator[E,6] * AffinePar
itionGF[E,6]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 8 "Type E_7" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12098 "## Cut Here and Paste int
o Maple to Simplify:\n\n## Parts allowed to add to add other affine pa
rtitions: ((31 0) (32 0)(33 0) (34 0))\n\ndeno[AFFINE-E,7]:= mul(1-t^i
, i=[31,32,33,34]):\n\n## Repeaters: ((28 0) (29 0) (30 0))\ngen[1]:=p
olyfy([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,5,6,7,8,1
0,11,13,14,16,17,19,21,22,24,25,27,28,30,31,32,33,34,35,35,36,36,36,35
,35,34,33,32,31,30,28,27,25,24,22,21,19,17,16,14,13,11,10,8,7,6,5,4,3,
3,2,2,1,1,1,1,1])/mul(1-t^i, i=[28,29,30]):\n\n## Repeaters: ((18 6) (
29 0) (30 0))\ngen[2]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3
,3,3,3,3,3,3,3,4,5,7,8,11,14,17,20,23,26,29,33,36,39,41,43,44,45,44,43
,42,41,39,38,38,38,40,42,45,49,55,60,67,73,80,87,94,100,105,110,112,11
3,112,110,105,100,94,88,81,74,67,62,57,51,47,44,43,42,42,42,43,45,47,4
8,49,49,49,48,46,43,40,37,33,30,26,22,18,15,12,9,6,4,3,2,1])/mul(1-t^i
, i=[18,29,30]):\n\n## Repeaters: ((29 0) (30 0))\ngen[3]:=polyfy([0,0
,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,5,6,7
,8,10,11,13,15,17,19,22,25,29,32,36,41,45,50,54,59,64,71,78,84,91,98,1
05,111,117,122,127,131,134,137,140,143,144,146,147,149,151,151,153,154
,156,157,159,161,163,165,165,164,162,160,155,150,144,137,129,121,113,1
05,97,88,80,74,69,64,60,57,55,54,53,52,52,51,50,48,46,43,40,37,33,30,2
6,22,18,15,12,9,6,4,3,2,1])/mul(1-t^i, i=[29,30]):\n\n## Repeaters: ((
28 0) (30 0))\ngen[4]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
,0,0,0,1,1,1,1,1,2,2,3,3,4,5,6,7,8,10,11,13,14,16,17,19,21,22,24,25,27
,28,30,31,32,33,34,35,35,36,36,36,35,35,34,33,32,31,30,28,27,25,24,22,
21,19,17,16,14,13,11,10,8,7,6,5,4,3,3,2,2,1,1,1,1,1])/mul(1-t^i, i=[28
,30]):\n\n## Repeaters: ((18 6) (26 3) (30 0))\ngen[5]:=polyfy([0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,7,7,9,11,12,15,17,20,22,
24,26,28,30,30,31,32,32,31,30,29,28,27,27,26,27,28,29,34,37,42,47,53,5
9,65,72,78,84,89,93,96,98,97,96,94,92,87,83,78,73,71,67,66,65,66,68,72
,77,83,90,98,106,113,121,126,132,136,138,138,136,132,126,121,113,106,9
8,90,83,77,72,68,66,65,66,67,71,73,78,83,87,92,94,96,97,98,96,93,89,84
,78,72,65,59,53,47,42,37,34,29,28,27,26,27,27,28,29,30,31,32,32,31,30,
30,28,26,24,22,20,17,15,12,11,9,7,7,5,4,3,3,3,3,3,3,3,3,3,3,3,2,2,2,2,
1,1,1,1,1])/mul(1-t^i, i=[18,26,30]):\n\n## Repeaters: ((26 3) (30 0))
\ngen[6]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,
2,2,3,3,4,5,6,7,8,10,11,13,14,16,17,20,21,24,24,26,28,29,32,34,38,39,4
2,44,47,50,53,55,58,61,64,66,69,72,75,79,81,84,88,91,97,100,104,107,11
2,116,121,128,134,140,145,151,155,161,163,166,167,170,169,169,168,166,
167,164,164,161,162,161,164,165,167,169,172,176,179,186,189,195,198,20
2,203,204,203,200,197,192,187,180,173,166,159,152,145,139,133,130,126,
126,123,123,123,123,124,124,125,124,125,123,121,117,114,108,103,96,90,
83,77,71,64,60,53,50,46,43,41,40,39,38,38,37,37,36,35,33,33,31,29,27,2
5,23,20,18,14,13,11,9,8,6,5,4,4,3,3,3,3,3,3,3,3,3,2,2,2,2,1,1,1,1,1])/
mul(1-t^i, i=[26,30]):\n\n## Repeaters: ((18 6) (30 0))\ngen[7]:=polyf
y([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4
,5,7,8,11,14,17,21,25,29,32,36,40,44,47,49,51,53,53,52,51,50,48,47,47,
47,49,52,57,65,74,84,97,110,124,138,153,167,180,192,202,210,214,215,21
3,209,202,193,183,173,162,154,147,142,139,138,142,148,157,167,180,195,
210,224,238,250,260,267,270,270,265,257,245,233,217,201,184,167,152,13
7,125,114,107,101,99,98,100,103,108,114,119,125,128,131,132,132,129,12
5,119,112,104,95,86,77,69,60,53,46,41,36,33,31,29,30,30,31,32,33,34,35
,35,34,33,32,30,28,26,23,21,18,16,13,11,9,7,7,5,4,3,3,3,3,3,3,3,3,3,3,
3,2,2,2,2,1,1,1,1,1])/mul(1-t^i, i=[18,30]):\n\n## Repeaters: ((30 0))
\ngen[8]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,1,1,1,1,1,2,2,3,3,4,5,6,7,8,10,11,13,15,17,19,22,25,29,32,36,4
1,45,51,56,62,67,74,82,89,98,106,115,123,132,140,148,156,163,171,178,1
87,193,202,209,220,229,238,248,258,269,279,292,303,317,329,343,353,364
,372,378,382,384,384,381,378,371,367,360,354,346,340,336,333,332,330,3
32,334,338,341,348,353,359,363,365,366,363,359,350,342,329,316,300,284
,268,251,236,220,208,194,186,177,173,168,166,165,164,165,164,165,163,1
63,159,156,150,145,137,129,120,111,102,92,84,75,69,61,56,51,47,45,43,4
2,41,41,40,40,39,38,36,35,33,31,29,26,24,21,19,15,13,11,9,8,6,5,4,4,3,
3,3,3,3,3,3,3,3,2,2,2,2,1,1,1,1,1])/mul(1-t^i, i=[30]):\n\n## Repeater
s: ((18 6) (26 3) (30 1))\ngen[9]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,0,0,0
,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3
,3,3,3,3,3,3,3,3,3,3,4,5,7,7,9,11,12,15,17,20,22,24,26,28,30,30,31,32,
32,31,30,29,28,27,27,26,27,28,29,34,37,42,47,53,59,65,72,78,84,89,93,9
6,98,97,96,94,92,87,83,78,73,71,67,66,65,66,68,72,77,83,90,98,106,113,
121,126,132,136,138,138,136,132,126,121,113,106,98,90,83,77,72,68,66,6
5,66,67,71,73,78,83,87,92,94,96,97,98,96,93,89,84,78,72,65,59,53,47,42
,37,34,29,28,27,26,27,27,28,29,30,31,32,32,31,30,30,28,26,24,22,20,17,
15,12,11,9,7,7,5,4,3,3,3,3,3,3,3,3,3,3,3,2,2,2,2,1,1,1,1,1])/mul(1-t^i
, i=[18,26,30]):\n\n## Repeaters: ((26 3) (30 1))\ngen[10]:=polyfy([0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,5,6,7,8,10
,11,13,14,16,17,20,21,24,24,26,28,29,32,34,38,39,42,44,47,50,53,55,58,
61,64,66,69,72,75,79,81,84,88,91,97,100,104,107,112,116,121,128,134,14
0,145,151,155,161,163,166,167,170,169,169,168,166,167,164,164,161,162,
161,164,165,167,169,172,176,179,186,189,195,198,202,203,204,203,200,19
7,192,187,180,173,166,159,152,145,139,133,130,126,126,123,123,123,123,
124,124,125,124,125,123,121,117,114,108,103,96,90,83,77,71,64,60,53,50
,46,43,41,40,39,38,38,37,37,36,35,33,33,31,29,27,25,23,20,18,14,13,11,
9,8,6,5,4,4,3,3,3,3,3,3,3,3,3,2,2,2,2,1,1,1,1,1])/mul(1-t^i, i=[26,30]
):\n\n## Repeaters: ((18 6) (30 1))\ngen[11]:=polyfy([0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,7,8,11,14,17,21,
25,29,32,36,40,44,47,49,51,53,53,52,51,50,48,47,47,47,49,52,57,65,74,8
4,97,110,124,138,153,167,180,192,202,210,214,215,213,209,202,193,183,1
73,162,154,147,142,139,138,142,148,157,167,180,195,210,224,238,250,260
,267,270,270,265,257,245,233,217,201,184,167,152,137,125,114,107,101,9
9,98,100,103,108,114,119,125,128,131,132,132,129,125,119,112,104,95,86
,77,69,60,53,46,41,36,33,31,29,30,30,31,32,33,34,35,35,34,33,32,30,28,
26,23,21,18,16,13,11,9,7,7,5,4,3,3,3,3,3,3,3,3,3,3,3,2,2,2,2,1,1,1,1,1
])/mul(1-t^i, i=[18,30]):\n\n## Repeaters: ((30 1))\ngen[12]:=polyfy([
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,
2,2,3,3,4,5,6,7,8,10,11,13,15,17,19,22,25,29,32,36,41,45,51,56,62,67,7
4,82,89,98,106,115,123,132,140,148,156,163,171,178,187,193,202,209,220
,229,238,248,258,269,279,292,303,317,329,343,353,364,372,378,382,384,3
84,381,378,371,367,360,354,346,340,336,333,332,330,332,334,338,341,348
,353,359,363,365,366,363,359,350,342,329,316,300,284,268,251,236,220,2
08,194,186,177,173,168,166,165,164,165,164,165,163,163,159,156,150,145
,137,129,120,111,102,92,84,75,69,61,56,51,47,45,43,42,41,41,40,40,39,3
8,36,35,33,31,29,26,24,21,19,15,13,11,9,8,6,5,4,4,3,3,3,3,3,3,3,3,3,2,
2,2,2,1,1,1,1,1])/mul(1-t^i, i=[30]):\n\n## Repeaters: ((28 0) (29 0))
\ngen[13]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1
,1,2,2,3,3,4,5,6,7,8,10,11,13,14,16,17,19,21,22,24,25,27,28,30,31,32,3
3,34,35,35,36,36,36,35,35,34,33,32,31,30,28,27,25,24,22,21,19,17,16,14
,13,11,10,8,7,6,5,4,3,3,2,2,1,1,1,1,1])/mul(1-t^i, i=[28,29]):\n\n## R
epeaters: ((18 6) (29 0))\ngen[14]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,
1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,7,8,11,14,17,20,23,26,29,33,
36,39,41,43,44,45,44,43,42,41,39,38,38,38,40,42,45,49,55,60,67,73,80,8
7,94,100,105,110,112,113,112,110,105,100,94,88,81,74,67,62,57,51,47,44
,43,42,42,42,43,45,47,48,49,49,49,48,46,43,40,37,33,30,26,22,18,15,12,
9,6,4,3,2,1])/mul(1-t^i, i=[18,29]):\n\n## Repeaters: ((29 0))\ngen[15
]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1
,1,1,1,1,2,2,3,3,4,5,6,7,8,10,11,13,15,17,19,22,25,29,32,36,41,45,50,5
4,59,64,71,78,84,91,98,105,111,117,122,127,131,134,137,140,143,144,146
,147,149,151,151,153,154,156,157,159,161,163,165,165,164,162,160,155,1
50,144,137,129,121,113,105,97,88,80,74,69,64,60,57,55,54,53,52,52,51,5
0,48,46,43,40,37,33,30,26,22,18,15,12,9,6,4,3,2,1])/mul(1-t^i, i=[29])
:\n\n## Repeaters: ((28 0))\ngen[16]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,5,6,7,8,10,11,13,1
4,16,17,19,21,22,24,25,27,28,30,31,32,33,34,35,35,36,36,36,35,35,34,33
,32,31,30,28,27,25,24,22,21,19,17,16,14,13,11,10,8,7,6,5,4,3,3,2,2,1,1
,1,1,1])/mul(1-t^i, i=[28]):\n\n## Repeaters: ((18 6) (26 3))\ngen[17]
:=polyfy([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,
5,7,7,9,11,12,15,17,20,22,24,26,28,30,30,31,32,32,31,30,29,28,27,27,26
,27,28,29,34,37,42,47,53,59,65,72,78,84,89,93,96,98,97,96,94,92,87,83,
78,73,71,67,66,65,66,68,72,77,83,90,98,106,113,121,126,132,136,138,138
,136,132,126,121,113,106,98,90,83,77,72,68,66,65,66,67,71,73,78,83,87,
92,94,96,97,98,96,93,89,84,78,72,65,59,53,47,42,37,34,29,28,27,26,27,2
7,28,29,30,31,32,32,31,30,30,28,26,24,22,20,17,15,12,11,9,7,7,5,4,3,3,
3,3,3,3,3,3,3,3,3,2,2,2,2,1,1,1,1,1])/mul(1-t^i, i=[18,26]):\n\n## Rep
eaters: ((26 3))\ngen[18]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,4,5,6,7,8,10,11,13,14,16,17,20,21,2
4,24,26,28,29,32,34,38,39,42,44,47,50,53,55,58,61,64,66,69,72,75,79,81
,84,88,91,97,100,104,107,112,116,121,128,134,140,145,151,155,161,163,1
66,167,170,169,169,168,166,167,164,164,161,162,161,164,165,167,169,172
,176,179,186,189,195,198,202,203,204,203,200,197,192,187,180,173,166,1
59,152,145,139,133,130,126,126,123,123,123,123,124,124,125,124,125,123
,121,117,114,108,103,96,90,83,77,71,64,60,53,50,46,43,41,40,39,38,38,3
7,37,36,35,33,33,31,29,27,25,23,20,18,14,13,11,9,8,6,5,4,4,3,3,3,3,3,3
,3,3,3,2,2,2,2,1,1,1,1,1])/mul(1-t^i, i=[26]):\n\n## Repeaters: ((18 6
))\ngen[19]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2
,2,2,3,3,3,3,3,3,3,3,3,3,3,4,5,7,8,11,14,17,21,25,29,32,36,40,44,47,49
,51,53,53,52,51,50,48,47,47,47,49,52,57,65,74,84,97,110,124,138,153,16
7,180,192,202,210,214,215,213,209,202,193,183,173,162,154,147,142,139,
138,142,148,157,167,180,195,210,224,238,250,260,267,270,270,265,257,24
5,233,217,201,184,167,152,137,125,114,107,101,99,98,100,103,108,114,11
9,125,128,131,132,132,129,125,119,112,104,95,86,77,69,60,53,46,41,36,3
3,31,29,30,30,31,32,33,34,35,35,34,33,32,30,28,26,23,21,18,16,13,11,9,
7,7,5,4,3,3,3,3,3,3,3,3,3,3,3,2,2,2,2,1,1,1,1,1])/mul(1-t^i, i=[18]):
\n\n## Repeaters: NIL\ngen[20]:=polyfy([1,1,1,1,1,2,2,3,3,4,5,6,7,8,10
,11,13,15,17,19,22,25,29,32,36,41,45,51,56,62,67,74,82,89,98,106,115,1
23,132,140,148,156,163,171,178,187,193,202,209,220,229,238,248,258,269
,279,292,303,317,329,343,353,364,372,378,382,384,384,381,378,371,367,3
60,354,346,340,336,333,332,330,332,334,338,341,348,353,359,363,365,366
,363,359,350,342,329,316,300,284,268,251,236,220,208,194,186,177,173,1
68,166,165,164,165,164,165,163,163,159,156,150,145,137,129,120,111,102
,92,84,75,69,61,56,51,47,45,43,42,41,41,40,40,39,38,36,35,33,31,29,26,
24,21,19,15,13,11,9,8,6,5,4,4,3,3,3,3,3,3,3,3,3,2,2,2,2,1,1,1,1,1])/mu
l(1-t^i, i=[]):\n\n## End Cut:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 74 "AffineParitionGF[E,7]:=simplify((add(gen[i], i=1..20))/ deno[A
FFINE-E,7]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%1AffineParitionGFG6
$%\"EG\"\"(,$*&\"\"\"F+*0,&*$)%\"tG\"#<F+F+F+!\"\"F+,.*$)F0\"#8F+F+*$)
F0\"#7F+F+*$)F0\"#6F+F+*$)F0\"\"#F+F2F0F2F+F2F+),&F0F+F+F2\"\"$F+,&F+F
2*$)F0F(F+F+F+,&F4F+F+F2F+,,*$)F0\"\"%F+F+*$)F0FBF+F+F=F+F0F+F+F+F+,(*
$)F0\"\"'F+F+FKF+F+F+F+F2F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
55 "BottDenominator[E,7]:=mul(1-t^j, j=[1,5,7,9,11,13,17]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>&%0BottDenominatorG6$%\"EG\"\"(*0,&\"\"\"F+
%\"tG!\"\"F+,&F+F+*$)F,\"\"&F+F-F+,&F+F+*$)F,F(F+F-F+,&F+F+*$)F,\"\"*F
+F-F+,&F+F+*$)F,\"#6F+F-F+,&F+F+*$)F,\"#8F+F-F+,&F+F+*$)F,\"#<F+F-F+" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "simplify(BottDenominator[
E,7] * AffineParitionGF[E,7]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"
\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 8 "Type
 E_8" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11604 "\n## Cut Here and
 Paste into Maple to Simplify:\n\n## Denominator corresponding to part
s allowed with all: \n\ndeno[AFFINE-E,8]:= mul(1-t^i, i=[53,54,55,56,5
7,58]):\n\n## Repeaters: ((51 0) (52 0))\ngen[1]:=polyfy([0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,
2,2,2,3,3,4,5,5,5,6,7,8,9,10,11,12,14,15,17,18,20,22,25,27,29,32,35,38
,42,46,49,53,58,63,68,74,79,85,91,98,104,111,118,125,133,142,150,159,1
69,179,189,200,211,221,233,245,257,268,281,293,306,319,332,344,357,370
,383,397,411,424,437,451,463,476,489,502,513,526,538,549,560,572,582,5
91,601,609,617,625,633,639,646,652,657,662,667,670,672,675,676,676,676
,675,672,670,667,662,657,652,646,639,633,625,617,609,601,591,582,572,5
60,549,538,526,513,502,489,476,463,451,437,424,411,397,383,370,357,344
,332,319,306,293,281,268,257,245,233,221,211,200,189,179,169,159,150,1
42,133,125,118,111,104,98,91,85,79,74,68,63,58,53,49,46,42,38,35,32,29
,27,25,22,20,18,17,15,14,12,11,10,9,8,7,6,5,5,5,4,3,3,2,2,2,2,1,1,1,1,
1,1,1])/mul(1-t^i, i=[51,52]):\n\n## Repeaters: ((46 1) (52 0))\ngen[2
]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1
,1,1,1,1,1,2,2,2,2,3,3,4,5,5,5,6,7,8,9,10,11,12,14,15,17,18,20,21,24,2
5,27,29,32,33,36,38,40,41,44,46,48,50,52,54,55,59,61,65,67,71,72,75,76
,79,81,84,86,89,91,95,98,101,104,107,110,113,118,122,127,131,137,142,1
48,154,161,167,174,180,188,196,204,212,222,231,238,247,255,263,270,280
,288,295,302,311,319,328,339,349,358,366,374,381,389,395,403,409,417,4
23,431,437,444,449,455,461,466,472,478,485,491,499,505,514,520,530,536
,546,552,563,569,582,592,605,615,628,637,647,657,668,678,687,698,705,7
15,723,737,746,761,770,783,790,801,807,817,823,831,837,845,850,856,861
,865,868,870,873,873,875,875,877,876,878,877,878,877,877,875,874,871,8
70,870,871,873,875,877,876,877,876,877,876,877,875,873,871,870,870,871
,874,875,877,877,878,877,878,876,877,875,875,873,873,870,868,865,861,8
56,850,845,837,831,823,817,807,801,790,783,770,761,746,737,723,715,705
,698,687,678,668,657,647,637,628,615,605,592,582,569,563,552,546,536,5
30,520,514,505,499,491,485,478,472,466,461,455,449,444,437,431,423,417
,409,403,395,389,381,374,366,358,349,339,328,319,311,302,295,288,280,2
70,263,255,247,238,231,222,212,204,196,188,180,174,167,161,154,148,142
,137,131,127,122,118,113,110,107,104,101,98,95,91,89,86,84,81,79,76,75
,72,71,67,65,61,59,55,54,52,50,48,46,44,41,40,38,36,33,32,29,27,25,24,
21,20,18,17,15,14,12,11,10,9,8,7,6,5,5,5,4,3,3,2,2,2,2,1,1,1,1,1,1,1])
/mul(1-t^i, i=[46,52]):\n\n## Repeaters: ((52 0))\ngen[3]:=polyfy([0,0
,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5,5,5,6,7,8
,9,10,11,12,14,15,17,18,20,22,25,27,29,32,35,38,42,46,49,53,58,63,68,7
4,79,85,91,99,106,114,122,130,138,147,156,166,177,188,200,212,225,238,
253,267,282,296,312,328,346,364,382,400,419,439,459,481,502,525,547,57
2,595,621,645,672,698,727,754,783,811,841,871,902,933,962,993,1023,105
6,1087,1122,1153,1187,1219,1254,1286,1321,1354,1389,1421,1456,1489,152
2,1554,1586,1617,1646,1676,1704,1734,1761,1789,1814,1841,1864,1890,191
3,1938,1957,1979,1998,2018,2034,2053,2070,2086,2102,2118,2133,2145,215
9,2170,2183,2191,2202,2208,2217,2222,2231,2237,2245,2249,2255,2260,226
4,2267,2270,2273,2273,2275,2274,2275,2272,2271,2266,2263,2257,2252,224
5,2239,2230,2220,2210,2199,2187,2173,2161,2144,2129,2110,2094,2073,205
6,2035,2019,1999,1982,1961,1944,1923,1904,1885,1866,1845,1822,1803,178
0,1761,1739,1723,1701,1685,1665,1650,1630,1615,1595,1581,1563,1548,153
1,1517,1501,1485,1471,1455,1440,1422,1408,1389,1373,1353,1337,1316,130
0,1278,1261,1238,1218,1193,1173,1149,1127,1105,1085,1064,1042,1023,100
1,981,959,941,918,898,874,854,831,813,792,775,756,739,721,706,690,674,
659,645,632,618,607,595,584,572,563,552,542,530,520,509,499,488,478,46
8,457,447,436,426,412,400,387,377,364,355,344,334,322,313,303,293,282,
272,262,250,240,229,220,209,201,192,185,175,168,160,154,146,141,134,12
9,123,119,115,111,107,103,100,96,93,89,87,83,81,78,77,73,72,68,66,62,6
0,56,54,52,50,48,46,44,41,40,38,36,33,32,29,27,25,24,21,20,18,17,15,14
,12,11,10,9,8,7,6,5,5,5,4,3,3,2,2,2,2,1,1,1,1,1,1,1])/mul(1-t^i, i=[52
]):\n\n## Repeaters: ((46 1) (52 1))\ngen[4]:=polyfy([0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5,
5,5,6,7,8,9,10,11,12,14,15,17,18,20,21,24,25,27,29,32,33,36,38,40,41,4
4,46,48,50,52,54,55,59,61,65,67,71,72,75,76,79,81,84,86,89,91,95,98,10
1,104,107,110,113,118,122,127,131,137,142,148,154,161,167,174,180,188,
196,204,212,222,231,238,247,255,263,270,280,288,295,302,311,319,328,33
9,349,358,366,374,381,389,395,403,409,417,423,431,437,444,449,455,461,
466,472,478,485,491,499,505,514,520,530,536,546,552,563,569,582,592,60
5,615,628,637,647,657,668,678,687,698,705,715,723,737,746,761,770,783,
790,801,807,817,823,831,837,845,850,856,861,865,868,870,873,873,875,87
5,877,876,878,877,878,877,877,875,874,871,870,870,871,873,875,877,876,
877,876,877,876,877,875,873,871,870,870,871,874,875,877,877,878,877,87
8,876,877,875,875,873,873,870,868,865,861,856,850,845,837,831,823,817,
807,801,790,783,770,761,746,737,723,715,705,698,687,678,668,657,647,63
7,628,615,605,592,582,569,563,552,546,536,530,520,514,505,499,491,485,
478,472,466,461,455,449,444,437,431,423,417,409,403,395,389,381,374,36
6,358,349,339,328,319,311,302,295,288,280,270,263,255,247,238,231,222,
212,204,196,188,180,174,167,161,154,148,142,137,131,127,122,118,113,11
0,107,104,101,98,95,91,89,86,84,81,79,76,75,72,71,67,65,61,59,55,54,52
,50,48,46,44,41,40,38,36,33,32,29,27,25,24,21,20,18,17,15,14,12,11,10,
9,8,7,6,5,5,5,4,3,3,2,2,2,2,1,1,1,1,1,1,1])/mul(1-t^i, i=[46,52]):\n\n
## Repeaters: ((52 1))\ngen[5]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5,5,5,6,7,8,9,10,11,12,14,15,17,18,20,
22,25,27,29,32,35,38,42,46,49,53,58,63,68,74,79,85,91,99,106,114,122,1
30,138,147,156,166,177,188,200,212,225,238,253,267,282,296,312,328,346
,364,382,400,419,439,459,481,502,525,547,572,595,621,645,672,698,727,7
54,783,811,841,871,902,933,962,993,1023,1056,1087,1122,1153,1187,1219,
1254,1286,1321,1354,1389,1421,1456,1489,1522,1554,1586,1617,1646,1676,
1704,1734,1761,1789,1814,1841,1864,1890,1913,1938,1957,1979,1998,2018,
2034,2053,2070,2086,2102,2118,2133,2145,2159,2170,2183,2191,2202,2208,
2217,2222,2231,2237,2245,2249,2255,2260,2264,2267,2270,2273,2273,2275,
2274,2275,2272,2271,2266,2263,2257,2252,2245,2239,2230,2220,2210,2199,
2187,2173,2161,2144,2129,2110,2094,2073,2056,2035,2019,1999,1982,1961,
1944,1923,1904,1885,1866,1845,1822,1803,1780,1761,1739,1723,1701,1685,
1665,1650,1630,1615,1595,1581,1563,1548,1531,1517,1501,1485,1471,1455,
1440,1422,1408,1389,1373,1353,1337,1316,1300,1278,1261,1238,1218,1193,
1173,1149,1127,1105,1085,1064,1042,1023,1001,981,959,941,918,898,874,8
54,831,813,792,775,756,739,721,706,690,674,659,645,632,618,607,595,584
,572,563,552,542,530,520,509,499,488,478,468,457,447,436,426,412,400,3
87,377,364,355,344,334,322,313,303,293,282,272,262,250,240,229,220,209
,201,192,185,175,168,160,154,146,141,134,129,123,119,115,111,107,103,1
00,96,93,89,87,83,81,78,77,73,72,68,66,62,60,56,54,52,50,48,46,44,41,4
0,38,36,33,32,29,27,25,24,21,20,18,17,15,14,12,11,10,9,8,7,6,5,5,5,4,3
,3,2,2,2,2,1,1,1,1,1,1,1])/mul(1-t^i, i=[52]):\n\n## Repeaters: ((51 0
))\ngen[6]:=polyfy([0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,
2,2,3,3,4,5,5,5,6,7,8,9,10,11,12,14,15,17,18,20,22,25,27,29,32,35,38,4
2,46,49,53,58,63,68,74,79,85,91,98,104,111,118,125,133,142,150,159,169
,179,189,200,211,221,233,245,257,268,281,293,306,319,332,344,357,370,3
83,397,411,424,437,451,463,476,489,502,513,526,538,549,560,572,582,591
,601,609,617,625,633,639,646,652,657,662,667,670,672,675,676,676,676,6
75,672,670,667,662,657,652,646,639,633,625,617,609,601,591,582,572,560
,549,538,526,513,502,489,476,463,451,437,424,411,397,383,370,357,344,3
32,319,306,293,281,268,257,245,233,221,211,200,189,179,169,159,150,142
,133,125,118,111,104,98,91,85,79,74,68,63,58,53,49,46,42,38,35,32,29,2
7,25,22,20,18,17,15,14,12,11,10,9,8,7,6,5,5,5,4,3,3,2,2,2,2,1,1,1,1,1,
1,1])/mul(1-t^i, i=[51]):\n\n## Repeaters: ((46 1))\ngen[7]:=polyfy([0
,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5,5,5,6,7,8,9,10,11,1
2,14,15,17,18,20,21,24,25,27,29,32,33,36,38,40,41,44,46,48,50,52,54,55
,59,61,65,67,71,72,75,76,79,81,84,86,89,91,95,98,101,104,107,110,113,1
18,122,127,131,137,142,148,154,161,167,174,180,188,196,204,212,222,231
,238,247,255,263,270,280,288,295,302,311,319,328,339,349,358,366,374,3
81,389,395,403,409,417,423,431,437,444,449,455,461,466,472,478,485,491
,499,505,514,520,530,536,546,552,563,569,582,592,605,615,628,637,647,6
57,668,678,687,698,705,715,723,737,746,761,770,783,790,801,807,817,823
,831,837,845,850,856,861,865,868,870,873,873,875,875,877,876,878,877,8
78,877,877,875,874,871,870,870,871,873,875,877,876,877,876,877,876,877
,875,873,871,870,870,871,874,875,877,877,878,877,878,876,877,875,875,8
73,873,870,868,865,861,856,850,845,837,831,823,817,807,801,790,783,770
,761,746,737,723,715,705,698,687,678,668,657,647,637,628,615,605,592,5
82,569,563,552,546,536,530,520,514,505,499,491,485,478,472,466,461,455
,449,444,437,431,423,417,409,403,395,389,381,374,366,358,349,339,328,3
19,311,302,295,288,280,270,263,255,247,238,231,222,212,204,196,188,180
,174,167,161,154,148,142,137,131,127,122,118,113,110,107,104,101,98,95
,91,89,86,84,81,79,76,75,72,71,67,65,61,59,55,54,52,50,48,46,44,41,40,
38,36,33,32,29,27,25,24,21,20,18,17,15,14,12,11,10,9,8,7,6,5,5,5,4,3,3
,2,2,2,2,1,1,1,1,1,1,1])/mul(1-t^i, i=[46]):\n\n## Repeaters: NIL\ngen
[8]:=polyfy([1,1,1,1,1,1,1,2,2,2,2,3,3,4,5,5,5,6,7,8,9,10,11,12,14,15,
17,18,20,22,25,27,29,32,35,38,42,46,49,53,58,63,68,74,79,85,91,99,106,
114,122,130,138,147,156,166,177,188,200,212,225,238,253,267,282,296,31
2,328,346,364,382,400,419,439,459,481,502,525,547,572,595,621,645,672,
698,727,754,783,811,841,871,902,933,962,993,1023,1056,1087,1122,1153,1
187,1219,1254,1286,1321,1354,1389,1421,1456,1489,1522,1554,1586,1617,1
646,1676,1704,1734,1761,1789,1814,1841,1864,1890,1913,1938,1957,1979,1
998,2018,2034,2053,2070,2086,2102,2118,2133,2145,2159,2170,2183,2191,2
202,2208,2217,2222,2231,2237,2245,2249,2255,2260,2264,2267,2270,2273,2
273,2275,2274,2275,2272,2271,2266,2263,2257,2252,2245,2239,2230,2220,2
210,2199,2187,2173,2161,2144,2129,2110,2094,2073,2056,2035,2019,1999,1
982,1961,1944,1923,1904,1885,1866,1845,1822,1803,1780,1761,1739,1723,1
701,1685,1665,1650,1630,1615,1595,1581,1563,1548,1531,1517,1501,1485,1
471,1455,1440,1422,1408,1389,1373,1353,1337,1316,1300,1278,1261,1238,1
218,1193,1173,1149,1127,1105,1085,1064,1042,1023,1001,981,959,941,918,
898,874,854,831,813,792,775,756,739,721,706,690,674,659,645,632,618,60
7,595,584,572,563,552,542,530,520,509,499,488,478,468,457,447,436,426,
412,400,387,377,364,355,344,334,322,313,303,293,282,272,262,250,240,22
9,220,209,201,192,185,175,168,160,154,146,141,134,129,123,119,115,111,
107,103,100,96,93,89,87,83,81,78,77,73,72,68,66,62,60,56,54,52,50,48,4
6,44,41,40,38,36,33,32,29,27,25,24,21,20,18,17,15,14,12,11,10,9,8,7,6,
5,5,5,4,3,3,2,2,2,2,1,1,1,1,1,1,1])/mul(1-t^i, i=[]):\n\n## End Cut:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "AffineParitionGF[E,8]:=si
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{XPPMATH 20 "6#>&%1AffineParitionGFG6$%\"EG\"\")*&\"\"\"F**2,&*$)%\"tG
\"#HF*F*F*!\"\"F*,&*$)F/\"#>F*F*F*F1F*,&F*F1*$)F/\"\"(F*F*F*,&*$)F/\"#
6F*F*F*F1F*),&F/F*F*F1\"\"#F*,&F*F1*$)F/\"#BF*F*F*,&*$)F/\"#8F*F*F*F1F
*,D*$)F/\"#;F*F**$)F/\"#:F*F**$)F/\"#9F*F*FFF**$)F/\"#7F*F*F;F**$)F/\"
#5F*F**$)F/\"\"*F*F**$)F/F(F*F*F7F**$)F/\"\"'F*F**$)F/\"\"&F*F**$)F/\"
\"%F*F**$)F/\"\"$F*F**$)F/F@F*F*F/F*F*F*F*F1" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 60 "BottDenominator[E,8]:=mul(1-t^j, j=[1,7,11,13,17
,19,23,29]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%0BottDenominatorG6$
%\"EG\"\")*2,&\"\"\"F+%\"tG!\"\"F+,&F+F+*$)F,\"\"(F+F-F+,&F+F+*$)F,\"#
6F+F-F+,&F+F+*$)F,\"#8F+F-F+,&F+F+*$)F,\"#<F+F-F+,&F+F+*$)F,\"#>F+F-F+
,&F+F+*$)F,\"#BF+F-F+,&F+F+*$)F,\"#HF+F-F+" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 55 "simplify(BottDenominator[E,8] * AffineParitionGF[E,
8]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}}{MARK "5" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 
1 2 33 1 1 }