-
Notifications
You must be signed in to change notification settings - Fork 1
/
Copy pathIntroduction.nb
2084 lines (1989 loc) · 85.3 KB
/
Introduction.nb
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
(* Content-type: application/vnd.wolfram.mathematica *)
(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)
(* CreatedBy='Mathematica 12.1' *)
(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[ 158, 7]
NotebookDataLength[ 85105, 2076]
NotebookOptionsPosition[ 77272, 1932]
NotebookOutlinePosition[ 78801, 1978]
CellTagsIndexPosition[ 78599, 1968]
WindowFrame->Normal*)
(* Beginning of Notebook Content *)
Notebook[{
Cell["Polychrony as Chinampas. ", "Section",
CellChangeTimes->{{3.834685990474324*^9,
3.834685995998051*^9}},ExpressionUUID->"5bf4f851-17b1-4169-8a3e-\
43665bcd1f47"],
Cell[CellGroupData[{
Cell["\<\
Introduction\
\>", "Section",
CellChangeTimes->{{3.816171367318541*^9, 3.8161713704295454`*^9}, {
3.8161964466356173`*^9,
3.816196458357436*^9}},ExpressionUUID->"86bdc7cb-37f1-41c2-bbe8-\
bc026083df47"],
Cell["\<\
We start with a connected oriented graph with one minimum vertex and one \
maximum vertex. The direction is from left to right.\
\>", "Text",
CellChangeTimes->{
3.816195748445833*^9},ExpressionUUID->"6a773f62-6574-46b2-b072-\
790a9f153ca3"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"myline", "=",
GraphicsBox[
TagBox[RasterBox[CompressedData["
1:eJzt1cGKUzEYhuGjblx6C4IX4dal2xEvYAbr4KZCR1CvfqwLBZuDWJOT5Guf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"], {{0, 72.}, {175.2, 0}}, {0, 255},
ColorFunction->RGBColor,
ImageResolution->{120, 120}],
BoxForm`ImageTag["Byte", ColorSpace -> "RGB", Interleaving -> True],
Selectable->False],
DefaultBaseStyle->"ImageGraphics",
ImageSizeRaw->{175.2, 72.},
PlotRange->{{0, 175.2}, {0, 72.}}]}]], "Input",
CellChangeTimes->{{3.816171670767869*^9, 3.8161717627895727`*^9}, {
3.8161722247904224`*^9, 3.8161722269654217`*^9}, {3.816172299078423*^9,
3.816172309861556*^9}},
CellLabel->"In[4]:=",ExpressionUUID->"e3491dc0-0fda-46eb-8556-baf9780375d6"],
Cell[BoxData[
GraphicsBox[
TagBox[RasterBox[CompressedData["
1:eJzt1cGKUzEYhuGjblx6C4IX4dal2xEvYAbr4KZCR1CvfqwLBZuDWJOT5Guf
B0Z0Fi9/0vz15d3nm49Pl2V5eH784+b265vD4fb7uxfHf7zfP3y63+8+vN1/
2d3vDq/vnh1/+e348+rJsvz8+yMAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
AAAAAAAAAAAAAAAAAAAAAAAAIZZfRg8Cl2z5U/Nmk+BJtmGzuYghGaL5ri2F
+maZbd5sHpx2yJNswyZ/F7Froc05hyyzzZtNgpcn4nmENiOGnLb5eHH/+UZc
e2gzYsiUZn1wuMRrT2lGDJnSrA8O1/yWQj/KOZ9H6MHnvMzhmt9SGUxpVga3
aEYM2adZHxyuw7Vv8eRSmhFDTnjwslkfHG7rKwpqNn9yEUOmNCtrba2eF+Z0
7nvusEH/aMh1wSTsGvRh16CPnrtm3bhmnXftP5QDn4w94trgbIMW6AzlwJVj
r15Ch2b7D48oNQ+sj3LgmrF/Fxo2y2BQM2LIsrl6kMnV3Ekf5cA1Yy8b7FpK
swxWvoFuQwY1o83/hoOaq2YbMqJZBitrM5j/2oOaq2YbMroZre1x0j/Kyuaq
2YaMbkbb4jhtm1tc+0bPozThkNHNaFsc5zqbK5vWYs7mQ3Zr2rUT87/hoOb8
u5bStGuaZwXt2oTNgVKuKKJp1+ZvjtX8LCnXbteusHl5trif6/xOiBgyqMko
Wd8JrYJbNO0adBPxnQAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABA
hB9hkdrE
"], {{0, 72.}, {175.2, 0}}, {0, 255},
ColorFunction->RGBColor,
ImageResolution->{120, 120}],
BoxForm`ImageTag["Byte", ColorSpace -> "RGB", Interleaving -> True],
Selectable->False],
DefaultBaseStyle->"ImageGraphics",
ImageSizeRaw->{175.2, 72.},
PlotRange->{{0, 175.2}, {0, 72.}}]], "Output",
CellChangeTimes->{
3.8161722280484476`*^9, {3.8161723003874207`*^9, 3.816172310733557*^9}},
CellLabel->"Out[4]=",ExpressionUUID->"7934f6b4-7d44-4f54-b9bf-824c513dd10b"]
}, Open ]],
Cell["Since this graph has 9 vertex, I want you to think of it as", "Text",
CellChangeTimes->{
3.8161957657847505`*^9},ExpressionUUID->"bcc44699-35ee-4e6a-9a88-\
d227319f5517"],
Cell[BoxData[
RowBox[{"\[IndentingNewLine]", " ",
RowBox[{
RowBox[{
RowBox[{"x", "^", "9"}], "/",
RowBox[{
RowBox[{"(",
RowBox[{"1", " ", "-", " ", "x"}], ")"}], "^", "10"}]}], " ", "=", " ",
RowBox[{"sum", " ",
RowBox[{"(",
RowBox[{
RowBox[{"x", "^", "n"}], " ",
GraphicsBox[
TagBox[RasterBox[CompressedData["
1:eJzt3UFuG0cQBdBJssnSV8gBss/Wy2xt5AA2rBjeKIAcIMhhclZGtgGBkRiK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"], {{0, 94.2}, {180.6, 0}}, {0, 255},
ColorFunction->RGBColor,
ImageResolution->{120, 120}],
BoxForm`ImageTag["Byte", ColorSpace -> "RGB", Interleaving -> True],
Selectable->False],
DefaultBaseStyle->"ImageGraphics",
ImageSizeRaw->{180.6, 94.2},
PlotRange->{{0, 180.6}, {0, 94.2}}]}], ")"}]}]}]}]], "Input",
CellChangeTimes->{{3.8161717741835327`*^9, 3.816171799902857*^9},
3.816172007676258*^9, {3.8161957699583344`*^9, 3.816195809634447*^9}, {
3.816195905564397*^9,
3.816195955373496*^9}},ExpressionUUID->"0d89a750-020e-48a0-ac51-\
128fa24e0500"],
Cell[BoxData[
RowBox[{
RowBox[{"f", "[", "x_", "]"}], " ", ":=", " ",
RowBox[{
RowBox[{"x", "^", "9"}], "/",
RowBox[{
RowBox[{"(",
RowBox[{"1", "-", "x"}], ")"}], "^", "10"}]}]}]], "Input",
InitializationCell->True,
CellChangeTimes->{{3.8161970551812916`*^9, 3.8161970687914977`*^9}, {
3.816736549253096*^9, 3.816736569894264*^9}},
CellLabel->"In[1]:=",ExpressionUUID->"49b44512-d9fd-4e92-890f-7d5a3920ea83"],
Cell[TextData[{
"Why? well this is the magic of the project.\nWe are going to ask \
Mathematica to count all possible ways to label our figure with numbers from \
1 to n preserving the order . For the line \n with 9 vertex: \
\[LineSeparator]\n",
Cell[BoxData[
GraphicsBox[
TagBox[RasterBox[CompressedData["
1:eJzt3E2O40QUAOAAG5ZcgQOwR9mxZDuIA8yIzohNI80gIW6RC3CNXMaHaZx0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"], {{0, 94.2}, {180.6, 0}}, {0, 255},
ColorFunction->RGBColor,
ImageResolution->{120, 120}],
BoxForm`ImageTag["Byte", ColorSpace -> "RGB", Interleaving -> True],
Selectable->False],
DefaultBaseStyle->"ImageGraphics",
ImageSizeRaw->{180.6, 94.2},
PlotRange->{{0, 180.6}, {0, 94.2}}]], "Input",
CellChangeTimes->{{3.816172491264717*^9, 3.816172492132763*^9},
3.8161725484993353`*^9},ExpressionUUID->
"4e234b7d-d76f-437e-abc2-202950b5b285"],
"\n"
}], "Text",
CellChangeTimes->{{3.816196085996751*^9, 3.8161961064257236`*^9}, {
3.8161961434247417`*^9, 3.8161961826913104`*^9},
3.8346857425519047`*^9},ExpressionUUID->"f089560d-07f7-4972-9edd-\
5799fd684c04"],
Cell["\<\
We choose a label for \[OpenCurlyQuote]a\[CloseCurlyQuote] between 1 and n-8, \
the reason is that \[OpenCurlyQuote]a<b<c<d<e<f<g<h<i\[CloseCurlyQuote] \
implies that \[OpenCurlyQuote]a\[CloseCurlyQuote] cannot be n, or n-1,... n-7.\
\>", "Text",
CellChangeTimes->{{3.8161739423949766`*^9, 3.816173995114994*^9}, {
3.816174088250981*^9,
3.816174122898978*^9}},ExpressionUUID->"b5e92875-e959-4edc-aaa0-\
0afab715572b"],
Cell["\<\
We choose a label for \[OpenCurlyQuote]b\[CloseCurlyQuote] between \
\[OpenCurlyQuote]a+1\[CloseCurlyQuote] and n-7, the reason is that \
\[OpenCurlyQuote]a<b<c<d<e<f<g<h<i\[CloseCurlyQuote] implies that what ever \
value \[OpenCurlyQuote]a\[CloseCurlyQuote] has, \[OpenCurlyQuote]b\
\[CloseCurlyQuote] can be at least \[OpenCurlyQuote]a+1\[CloseCurlyQuote] and \
cannot be n, or n-1,... n-6.
We are computing the sum below:\
\>", "Text",
CellChangeTimes->{{3.8161740242440195`*^9, 3.816174074618984*^9}, {
3.816174114242979*^9, 3.8161741426679764`*^9},
3.816196277460713*^9},ExpressionUUID->"fb0a42a2-fc36-4f71-90e4-\
b72482d89a46"],
Cell[BoxData[""], "Input",
CellChangeTimes->{{3.816174006376975*^9,
3.816174026498979*^9}},ExpressionUUID->"d2f5d4bb-59fb-4b65-beeb-\
c04b7990804f"],
Cell[BoxData[""], "Input",
CellChangeTimes->{{3.8161725524203415`*^9, 3.816172552951336*^9}, {
3.8161739314510765`*^9,
3.8161739333149757`*^9}},ExpressionUUID->"de726d66-861c-4a03-8ed2-\
6b2c8f4f0522"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"ToExpression", "[",
RowBox[{
"\"\<\\\\ \
\\sum_{a=1}^{n-8}\\sum_{b=a+1}^{n-7}\\sum_{c=b+1}^{n-6}\\sum_{d=c+1}^{n-5}\\\
sum_{e=d+1}^{n-4}\\sum_{f=e+1}^{n-3}\\sum_{g=e+1}^{n-2}\\sum_{h=g+1}^{n-1}\\\
sum_{i=h+1}^{n} 1\>\"", ",", "TeXForm"}], "]"}]], "Input",
CellChangeTimes->{{3.8161736925976334`*^9, 3.8161737125236435`*^9}, {
3.816173742820294*^9, 3.816173846740079*^9}},
EmphasizeSyntaxErrors->True,
CellLabel->"In[2]:=",ExpressionUUID->"cafcc6fe-fb23-4c9f-9938-1018fb8f2f39"],
Cell[BoxData[
TemplateBox[{
"Sum", "vloc",
"\"The variable \\!\\(\\*RowBox[{\\\"b\\\", \\\"\[Equal]\\\", \
RowBox[{\\\"a\\\", \\\"+\\\", \\\"1\\\"}]}]\\) cannot be localized so that it \
can be assigned to numerical values.\"", 2, 2, 19, 28434973565249803309,
"Local"},
"MessageTemplate"]], "Message", "MSG",
CellChangeTimes->{3.816173713883668*^9, 3.8161738490000725`*^9,
3.834685763868513*^9},
CellLabel->
"During evaluation of \
In[2]:=",ExpressionUUID->"ea586b4f-9507-42e5-b785-9721c4236e4d"],
Cell[BoxData[
RowBox[{
UnderoverscriptBox["\[Sum]",
RowBox[{"a", "=", "1"}],
RowBox[{"n", "-", "8"}]],
RowBox[{
UnderoverscriptBox["\[Sum]",
RowBox[{"b", "\[Equal]",
RowBox[{"a", "+", "1"}]}],
RowBox[{"n", "-", "7"}]],
RowBox[{
UnderoverscriptBox["\[Sum]",
RowBox[{"c", "\[Equal]",
RowBox[{"b", "+", "1"}]}],
RowBox[{"n", "-", "6"}]],
RowBox[{
UnderoverscriptBox["\[Sum]",
RowBox[{"d", "\[Equal]",
RowBox[{"c", "+", "1"}]}],
RowBox[{"n", "-", "5"}]],
RowBox[{
UnderoverscriptBox["\[Sum]",
RowBox[{"e", "\[Equal]",
RowBox[{"d", "+", "1"}]}],
RowBox[{"n", "-", "4"}]],
RowBox[{
UnderoverscriptBox["\[Sum]",
RowBox[{"f", "\[Equal]",
RowBox[{"e", "+", "1"}]}],
RowBox[{"n", "-", "3"}]],
RowBox[{
UnderoverscriptBox["\[Sum]",
RowBox[{"g", "\[Equal]",
RowBox[{"e", "+", "1"}]}],
RowBox[{"n", "-", "2"}]],
RowBox[{
UnderoverscriptBox["\[Sum]",
RowBox[{"h", "\[Equal]",
RowBox[{"g", "+", "1"}]}],
RowBox[{"n", "-", "1"}]],
RowBox[{
UnderoverscriptBox["\[Sum]",
RowBox[{"i", "\[Equal]",
RowBox[{"h", "+", "1"}]}], "n"], "1"}]}]}]}]}]}]}]}]}]], "Output",
CellChangeTimes->{{3.8161736943646708`*^9, 3.81617371390466*^9},
3.816173849014044*^9, 3.8346857639377527`*^9},
CellLabel->"Out[2]=",ExpressionUUID->"43a0a2eb-7f5e-4f59-8330-9114f0f284cc"]
}, Open ]],
Cell["\<\
each term assign a value to the letters \[OpenCurlyQuote]a,b,c,d,e,f,g,h,i\
\[CloseCurlyQuote]. In Mathematica this is equivalent to:\
\>", "Text",
CellChangeTimes->{{3.816174160058978*^9, 3.8161742003149786`*^9}, {
3.8161742470029783`*^9, 3.81617426281898*^9},
3.834685770694645*^9},ExpressionUUID->"ef9d0eca-ef10-40f1-8460-\
82318c4f6c3a"],
Cell["", "Input",
CellChangeTimes->{{3.816172055364253*^9, 3.8161721027880898`*^9}, {
3.8161722376204453`*^9, 3.8161722379724207`*^9}, 3.8161722797894225`*^9, {
3.8161723164865303`*^9, 3.816172388051546*^9}, 3.816172436693529*^9},
CellLabel->"In[8]:=",ExpressionUUID->"c3939f60-82af-4455-8c1f-ad3a0f6348e6"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"Sum", "[",
RowBox[{"1", ",",
RowBox[{"{",
RowBox[{"a", ",", "1", ",",
RowBox[{"n", "-", "8"}]}], "}"}], ",", " ",
RowBox[{"{",
RowBox[{"b", ",",
RowBox[{"a", "+", "1"}], ",",
RowBox[{"n", "-", "7"}]}], "}"}], ",", " ",
RowBox[{"{",
RowBox[{"c", ",",
RowBox[{"b", "+", "1"}], ",",
RowBox[{"n", "-", "6"}]}], "}"}], ",", " ",
RowBox[{"{",
RowBox[{"d", ",",
RowBox[{"c", "+", "1"}], ",",
RowBox[{"n", "-", "5"}]}], "}"}], ",", " ",
RowBox[{"{",
RowBox[{"e", ",",
RowBox[{"d", "+", "1"}], ",",
RowBox[{"n", "-", "4"}]}], "}"}], ",", " ",
RowBox[{"{",
RowBox[{"f", ",",
RowBox[{"e", "+", "1"}], ",",
RowBox[{"n", "-", "3"}]}], "}"}], ",",
RowBox[{"{",
RowBox[{"g", ",",
RowBox[{"f", "+", "1"}], ",",
RowBox[{"n", "-", "2"}]}], "}"}], ",", " ",
RowBox[{"{",
RowBox[{"h", ",",
RowBox[{"g", "+", "1"}], " ", ",",
RowBox[{"n", "-", "1"}]}], "}"}], ",",
RowBox[{"{",
RowBox[{"i", ",",
RowBox[{"h", "+", "1"}], ",", "n"}], "}"}]}], "]"}]], "Input",
CellChangeTimes->{{3.81617211670113*^9, 3.816172205141425*^9}, {
3.8161725692043705`*^9, 3.816172575084376*^9}, {3.8161726094230113`*^9,
3.8161726980480123`*^9}, {3.8161734108309584`*^9, 3.8161735474683*^9}},
CellLabel->"In[14]:=",ExpressionUUID->"2ddebded-677c-4fd6-a4ee-a9421cc54f88"],
Cell[BoxData[
FractionBox[
RowBox[{
RowBox[{"40320", " ", "n"}], "-",
RowBox[{"109584", " ",
SuperscriptBox["n", "2"]}], "+",
RowBox[{"118124", " ",
SuperscriptBox["n", "3"]}], "-",
RowBox[{"67284", " ",
SuperscriptBox["n", "4"]}], "+",
RowBox[{"22449", " ",
SuperscriptBox["n", "5"]}], "-",
RowBox[{"4536", " ",
SuperscriptBox["n", "6"]}], "+",
RowBox[{"546", " ",
SuperscriptBox["n", "7"]}], "-",
RowBox[{"36", " ",
SuperscriptBox["n", "8"]}], "+",
SuperscriptBox["n", "9"]}], "362880"]], "Output",
CellChangeTimes->{3.8161725767453637`*^9, 3.8161726990410104`*^9,
3.816173526739922*^9, 3.8161735830192914`*^9},
CellLabel->"Out[14]=",ExpressionUUID->"d12f5226-82a6-432e-9719-78f60680887e"]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"FullSimplify", "[",
FractionBox[
RowBox[{
RowBox[{"40320", " ", "n"}], "-",
RowBox[{"109584", " ",
SuperscriptBox["n", "2"]}], "+",
RowBox[{"118124", " ",
SuperscriptBox["n", "3"]}], "-",
RowBox[{"67284", " ",
SuperscriptBox["n", "4"]}], "+",
RowBox[{"22449", " ",
SuperscriptBox["n", "5"]}], "-",
RowBox[{"4536", " ",
SuperscriptBox["n", "6"]}], "+",
RowBox[{"546", " ",
SuperscriptBox["n", "7"]}], "-",
RowBox[{"36", " ",
SuperscriptBox["n", "8"]}], "+",
SuperscriptBox["n", "9"]}], "362880"], "]"}]], "Input",
NumberMarks->False,
CellLabel->"In[7]:=",ExpressionUUID->"b16e632f-89f6-4640-a5b4-ecf56937d8bf"],
Cell[BoxData[
FractionBox[
RowBox[{
RowBox[{"(",
RowBox[{
RowBox[{"-", "8"}], "+", "n"}], ")"}], " ",
RowBox[{"(",
RowBox[{
RowBox[{"-", "7"}], "+", "n"}], ")"}], " ",
RowBox[{"(",
RowBox[{
RowBox[{"-", "6"}], "+", "n"}], ")"}], " ",
RowBox[{"(",
RowBox[{
RowBox[{"-", "5"}], "+", "n"}], ")"}], " ",
RowBox[{"(",
RowBox[{
RowBox[{"-", "4"}], "+", "n"}], ")"}], " ",
RowBox[{"(",
RowBox[{
RowBox[{"-", "3"}], "+", "n"}], ")"}], " ",
RowBox[{"(",
RowBox[{
RowBox[{"-", "2"}], "+", "n"}], ")"}], " ",
RowBox[{"(",
RowBox[{
RowBox[{"-", "1"}], "+", "n"}], ")"}], " ", "n"}], "362880"]], "Output",
CellChangeTimes->{3.8167379971547155`*^9, 3.8167868310959854`*^9},
CellLabel->"Out[7]=",ExpressionUUID->"14e849a5-6252-4995-a61b-49625a46bcc7"]
}, Open ]],
Cell[BoxData[
RowBox[{
FractionBox[
RowBox[{
RowBox[{"(",
RowBox[{
RowBox[{"-", "8"}], "+", "n"}], ")"}], " ",
RowBox[{"(",
RowBox[{
RowBox[{"-", "7"}], "+", "n"}], ")"}], " ",
RowBox[{"(",
RowBox[{
RowBox[{"-", "6"}], "+", "n"}], ")"}], " ",
RowBox[{"(",
RowBox[{
RowBox[{"-", "5"}], "+", "n"}], ")"}], " ",
RowBox[{"(",
RowBox[{
RowBox[{"-", "4"}], "+", "n"}], ")"}], " ",
RowBox[{"(",
RowBox[{
RowBox[{"-", "3"}], "+", "n"}], ")"}], " ",
RowBox[{"(",
RowBox[{
RowBox[{"-", "2"}], "+", "n"}], ")"}], " ",
RowBox[{"(",
RowBox[{
RowBox[{"-", "1"}], "+", "n"}], ")"}], " ", "n"}], "362880"], "=",
GraphicsBox[
TagBox[RasterBox[CompressedData["
1:eJzt3UFuG0cQBdBJssnSV8gBss/Wy2xt5AA2rBjeKIAcIMhhclZGtgGBkRiK
7O7pqmq9B9iQDbCnpvG/h+RQ8k/v/3jz+/fbtn3+8f63N+/+en139+7vt6/u
//Db7edPH29vPvx6++fNx5u7X97/cP+X/9z/+vm7bfvy9QEAAAAAAAAAAADI
zef9IJwaQjg1hHBqCG22r4as078IvEyj6qOG0GZgd9QQGuggxBpbHDWEaw1v
TXgNR73RBHOs18EHeSaB8/LUcDsSPgxMs0dKG9Z8Wj23L3kJ9nv1dPmy5699
7mCytl2Tecnil/wj4Kkpa4uq4bUv/QZesjWRVPYO5Mn1ewqliawnqoZjFwxf
CppNyOHxIYbcgPAJHxYz81KY9pWdJhJo5qUw8114NSTEtM9YHtdwp5UTrgaX
mJY6NYSTZkZu+IdCj1dOuxo8a34Nd1o5+YLwf9bo4MP6aVeDM1aq4cGtfAqa
nLRaNRy+Gpw0v4a1DqGGTLDYM9KDGlLNepfC4UdRQ3a13qvChwMlXxAeLHkp
3ONAashO5kdLDeGRVZ+RHtL8KDl41sI13EPp4ckp5Blp6SSXHp6cFn5huJPq
85OQGjZY4BRIRQ0bLHAKpKKGDRY4BVJRwwYLnAJ5hMRpgQwvcArkIU5t7Buj
yFIzW8costTM1jGKLDWzdQwhSD3sHkMIUg+7xxCC1MPuMYQg9bB79JOiTjaQ
flLUyQbST4o62UD6SVEnG0g/KepkA+kkQv3sIZ1EqJ89pIf8DGEb6ZEzP9sp
0UOdk3w8kkuYn5MjJW9i5tlILlV4nr3qZW5i2sHIL1V4LhkmbRNzTkUJScJz
VbmSzPxIzqkoIUN4rr3A5bwgJhyJKsLD0zZA+NhPJRyJKmLD03xdy5b5bPNQ
S3gNJz9wJ9nmoZDwDqohhNcw5LHDpRqGctRwiFTDUI4aDpFqGGoJD88aNcwz
CRWF50cNITw/agjh+VFDCM/PAjVMMgZ1ZYhQwwzfHvL0gSGnk2EPKS1JhBq+
veJwNHzgz8dIsoGUlidFDd9sGNi+R5NAj2wp2q4UPW+6DaScuhFK0sFD5T0k
iboRSlLDDDNQXd0UqSHLqJuiDJNnmIEF1A1S+OThA7CMulkKnzx8AJZRN0vu
FbKMunFSQ5ZRNE7hb5MW3TdyKhonNWQlReMUW8Oim0ZaRROlhqykaKJ0kJUU
DVXU2EW3i+SK5koNWUnRXIWMHf72LKsqmquoGs4/KC9B0WhNHjvPd/qzqorp
qjgznFEx0hVnhjMqRrrizHBeuVSXGxieJdWQgSZCODWEcGoI4dQQMtBECOfz
WpCBGkI4NYQMNBHCnamhF48wzdOuKSDM99//11oBAQAAAAAAAAAAAAAAAAAA
AAAAAAAAAAAAAADgRfgXZ+K3eQ==
"], {{0, 94.2}, {180.6, 0}}, {0, 255},
ColorFunction->RGBColor,
ImageResolution->{120, 120}],
BoxForm`ImageTag["Byte", ColorSpace -> "RGB", Interleaving -> True],
Selectable->False],
DefaultBaseStyle->"ImageGraphics",
ImageSizeRaw->{180.6, 94.2},
PlotRange->{{0, 180.6}, {0, 94.2}}]}]], "Input",
CellChangeTimes->{{3.8161963083549776`*^9,
3.81619632356875*^9}},ExpressionUUID->"fd87331c-faa0-4b6e-bf52-\
be65ca1f638e"],
Cell[TextData[{
"So if we consider a series $sum x^n a_n$ where $a_n$ stands for the number \
of labelings of our figure we obtain ",
Cell[BoxData[
RowBox[{" ",
RowBox[{"sum", " ",
RowBox[{"(",
RowBox[{
RowBox[{"x", "^", "n"}], " ",
GraphicsBox[
TagBox[RasterBox[CompressedData["
1:eJzt3UFuG0cQBdBJssnSV8gBss/Wy2xt5AA2rBjeKIAcIMhhclZGtgGBkRiK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"], {{0, 94.2}, {180.6, 0}}, {0, 255},
ColorFunction->RGBColor,
ImageResolution->{120, 120}],
BoxForm`ImageTag["Byte", ColorSpace -> "RGB", Interleaving -> True],
Selectable->False],
DefaultBaseStyle->"ImageGraphics",
ImageSizeRaw->{180.6, 94.2},
PlotRange->{{0, 180.6}, {0, 94.2}}]}], ")"}]}]}]],ExpressionUUID->
"958c3b44-e254-4f1f-877c-89ce96cd4846"],
"=",
Cell[BoxData[
RowBox[{
RowBox[{"x", "^", "9"}], "/",
RowBox[{
RowBox[{"(",
RowBox[{"1", " ", "-", " ", "x"}], ")"}], "^", "10", " "}]}]],
CellChangeTimes->{{3.8161717741835327`*^9, 3.816171799902857*^9},
3.816172007676258*^9, {3.8161957699583344`*^9, 3.816195809634447*^9}, {
3.816195905564397*^9, 3.816195955373496*^9}},ExpressionUUID->
"a876db57-e8e6-4bdd-bd95-518a980a902e"],
"\n\nThe Mathematica expression Sum[1,{a,1,n-8}, {b,a+1,n-7}, {c,b+1,n-6}, \
{d,c+1,n-5}, {e,d+1,n-4}, {f,e+1,n-3},{g,f+1,n-2}, {h,g+1 ,n-1},{i,h+1,n}] is \
just a lot of for loops, a lot of iterated sums, each one for each {x,x_i, \
x_f }. For example:\n",
Cell[BoxData[
RowBox[{"Sum", "[",
RowBox[{"1", ",", " ",
RowBox[{"{",
RowBox[{"i", ",", " ", "3", ",", " ", "8"}], "}"}]}], "]"}]], "Input",
ExpressionUUID->"cbd3b2a9-0a90-451f-85ea-f885d67413a9"],
"\n",
Cell[BoxData["6"], "Output",
CellChangeTimes->{3.816172887799859*^9},ExpressionUUID->
"52105230-cc85-4127-87d5-9e4be92aa2f7"]
}], "Text",
CellChangeTimes->{
3.8161724143285418`*^9, 3.816196080643053*^9, {3.816196359401285*^9,
3.816196435503292*^9}, {3.8161964769070654`*^9, 3.8161964956955395`*^9}, {
3.8161965610158796`*^9, 3.8161965669259014`*^9},
3.834685786016102*^9},ExpressionUUID->"024bd625-eb2c-4677-93b1-\
92b745cb8f6d"],
Cell["", "Text",
CellChangeTimes->{{3.816171927986204*^9, 3.816171947238204*^9}, {
3.816196018388873*^9, 3.8161960229071894`*^9},
3.816196510921241*^9},ExpressionUUID->"ee6448a2-f247-4b49-a8af-\
2001afb3a3c2"]
}, Open ]],
Cell[CellGroupData[{
Cell["Main steps.", "Chapter",
CellChangeTimes->{{3.8161964889235635`*^9, 3.816196508072571*^9}, {
3.816197863113875*^9, 3.816197865003409*^9}},
CellTags->"main",ExpressionUUID->"b6ab9871-6816-48b4-a2bd-0b0978c1ed01"],
Cell["\<\
Now for any new image that we draw over the line with 9 vertex, we are going \
to create a new power series using the following steps. Start with adding \
some handles to the line with 9 vertex, remember that our graphs have a \
direction from left to right.\
\>", "Text",
CellChangeTimes->{{3.816196506456235*^9,
3.8161965138656273`*^9}},ExpressionUUID->"279799f9-aea1-49f2-b324-\
181fe65dc2ec"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"myotherline", "=",
GraphicsBox[
TagBox[RasterBox[CompressedData["
1:eJzt3E2O41QQwPEAG5ZcgQOwR9mxZDuIA8yIzohNI80gIW6RC3CNXMaHaZJ0
Pp7jZ8d2Vb2qZ/9/EmiY7q7Uq1dlJ7abHz/99WH37Waz+fr98V8fPv7zy5cv
H//97Yfjf/z++vXPz68vf/z6+vfL55cvP3/67viX/x3/+embzeb05zcAAAAA
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADA0ubKOxFgpTZt3ukAa8QYojCa
rYsxREn0WxZlQUn0WxZlQUn0WxZlQUn0WxZlQUn0WxZlQUn0W9emh3deWCw6
LdU3gJQIpuixmzEzSJVggQZ7N34GV14oWKDB3qbP4GoLBQv02Nvoz4MrrxIs
cLR/N3Lo+r7ZNXfUbcwMrqHNRs5gWoe1lQh2xo/hgttsYLHDFVhJfWBq0gwu
uNMGljlpDJdaH9gZM25rmMThBT5d+7KLA2tjxjD7bY45q3u6uqljuLD6wNTw
DG4GzwhL6jT5GI78HqCr2zmTunEZzTZmUTPGcBnFgbVs20wdwwU024wRk4QC
Un09s6pj/sjlzBvD2osDa5KGWUyzjV/I+PUuozIoQ9gty5hExhCO5EO0gDGc
tITZ31ljZVCGSp/U3m+Tkrf7ZqyT4vjU229Ti8AYQpdikyhOdGFT05aMYUVl
QTG6HVJjv80YE+vvx6qoH6hrPPLPS3jS91dXE5Rk0R51TWKZbOuqCUoy6o26
Wq5YqhXVBCXZNUYtk1gyzyoKgvJMG6OKriuZZBUFQWHWJ4L4J8TyGQYvCMor
0BLBu658esELgvLKj2GoxnPJLWw14KJYE4ZtPJfEwlYDLor1Q9gToldWMasB
FyWbIWDjOR4cAlYDLgp3QsATomM+0UoBL+U7IdQk+iYTpw7wxRj6ZuKeANx5
9UCQSYyQhnsCcMcYkgN8+c6C+yS6J5BNwyUHOHJvgFBjWPjV+9JgEtfGffd9
O9B9+QEzQWFBDsJeOQRZfjYZx0xQWJCt9xqHIMsPmAxKirP15TMJdSrs5uOb
DKylGx1n68sPRZy13wRMCRa63S7fdK3OUU9s0msJ41hkpRITAamPodE4WzSk
9drVM5QH7EZWjInZIo/hQHqSmMORI4+h1tRYx1QJuCrqfa7bOerpPY0siV8g
yfXEfFvTKVu9z9V3RDe94bC6a7dIcrUx5QEjs25FeQF10xsOq7t2iyRXG1Me
MDL1Jq90ly3WbpHkamPKA0Zm3YoqNSwWU33t6kkKA9YbUx4wsgKtaNGNtcSs
IsmAC+/GlAeMrMYdqaUbhdHKJFlLTGG0YrKlAGKa2s8FJkjOpZJAEN7zd+Fd
BsCT9/xdeJcB8OQ9f3felQDceA+fSHctDyvyqCgwmdMA6eiuRbiibH0KxNTf
V1RF0mDuumuRrOgWQTFmN2BFMatIshszu5DgJDVx112LZEUbgzGsJWY3oLA9
iiVZUcylit/eFcXMipZkFTG7AYXRgou/IxXFzIqWZNUxl0p3pbXvsjBmVrQk
q465VBYr1Y1psSNGndMVMMmqYy6VxUrXGTMzhBp5qidZLCZjOF789q4oZvwx
rCUmY0hMrYCMYcCYMdVSvSpiMobxY4alvsxadoQxXGHMVbEo3ToPF1UkWVFM
BFTX4UIroEVMxhCIoIrDBQAAAAAAAAAAAEw1TXM47He77Xaz3Tfe2YzxnvE+
esbNMcPzjbPISb7dy3nKdnfQD78/V2G7i10FP5cC3RlsgrLDrp1x1A5vDrtt
BWV9LKd+qmmPRd2sEJKtiNsvLUmLh8z4vfV2h3PTHQfy8qeojmdtu0FhDEdK
ChWyqTPuR454GV9zi5dZr3sHWAwKb0pHYQwV3YsZLbMBtmOIceI2dZ+4Gd8y
q6mhGcMI4jZ1n7gZM4brcboMl3zwPb7rFl0DeGjqTnSdCwzNYb9LL8ye057z
s8cf1LtE05xDb1tJzV1u6+poSvVY0U5YY/87Y9ikOyWNntwPi3bMnO+4Cepb
nYzhXj/6W39/jjn29va2OLHrbF8PM43oTNa5+aNVvNyLbG9XYbuvOuPVkjHc
7bPlnrWECu+HjXRe2e301GrR+e8nkjsWyaGvaR0TZ5fwthfb+0H12j3PYt7X
t+v8rHRncxc0Fd6cWb4pvcXO5zz/dVsx0u7SmKCD4d0QL8c1tRejcZmz/5NW
uj+zwmc75xr5ybYMXHQUfzZMDg75as5tGcMx7Amt8Ip9TZT8vUbw5Yxhl8LV
iqEQsmPZQIscvzSc8NClf+mi+2LLD2p2Y9jXzgoN0DspGiO04DG8fujdqr4/
y4UQFVFyE21wIqSt13mGK3Gq6nb2gy+GZ8P8VmjcqGQMp0muGJ4+xR2aRmMb
xo2h7KP/5B0oNIbarWF6K+V+KeB2Wen6HNpW8nKM4RTZkZCP4aiGn1VC0WeL
oVHRe1Oq3BvmTyR1b/uIb1cwhpPkB850DG9fkl4JmbEJQ5eHxOec9F2pZnfY
j2HrUrlmUMZwnMwhvHX1XmEM09sKzeUNj+yR3NansNaB+3Q3Yjjl9qxk79Lo
LPp8H+52R+T8+MLMNduO4TV6sTHUePNe4SPLw7oXFo7tI/8llYEbzxpP0Eju
a+du3Z8OCyqf7QaeCxDcNrz3nOnvDOXTnrdfyS86PWxJUmfBlQfDkvhIzgTX
M8vDbM6o1vvBv2k/xKb2DNsl79MDWLOev2tndUnqYdGCQ/XDk2HzP2n1zYji
GeDpGE6vRc8l40vSwsdgevJdxjkR6/XkyT66HLDWvE/hNnO+Tt5XMoaAlfvl
mfzXRXeWAIxxGbPeKXv2dQBit99xys7Z9f/3whACpu6/FZLc5bx9YlS5uQTg
ucc7P5en2cSPswEAAAAAAAAAAAAAAAAAAAAAAAAAhP4HIvYYMQ==
"], {{0, 94.2}, {180.6, 0}}, {0, 255},
ColorFunction->RGBColor,
ImageResolution->{120, 120}],
BoxForm`ImageTag["Byte", ColorSpace -> "RGB", Interleaving -> True],
Selectable->False],
DefaultBaseStyle->"ImageGraphics",
ImageSizeRaw->{180.6, 94.2},
PlotRange->{{0, 180.6}, {0, 94.2}}]}]], "Input",
CellChangeTimes->{{3.81617470222865*^9, 3.816174705525059*^9}},
CellLabel->"In[23]:=",ExpressionUUID->"a22d22dd-3e9f-4631-ad11-8992e52d65fd"],
Cell[BoxData[
GraphicsBox[
TagBox[RasterBox[CompressedData["
1:eJzt3E2O41QQwPEAG5ZcgQOwR9mxZDuIA8yIzohNI80gIW6RC3CNXMaHaZJ0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"], {{0, 94.2}, {180.6, 0}}, {0, 255},
ColorFunction->RGBColor,
ImageResolution->{120, 120}],
BoxForm`ImageTag["Byte", ColorSpace -> "RGB", Interleaving -> True],
Selectable->False],
DefaultBaseStyle->"ImageGraphics",
ImageSizeRaw->{180.6, 94.2},
PlotRange->{{0, 180.6}, {0, 94.2}}]], "Output",
CellChangeTimes->{3.8161747059970503`*^9},
CellLabel->"Out[23]=",ExpressionUUID->"3eed9159-48c6-4a10-8156-b1bbe157cb28"]
}, Open ]],
Cell["I labeled the vertex to easy our work.", "Text",
CellChangeTimes->{
3.8161960418354836`*^9, {3.8161965234035416`*^9,
3.816196523771219*^9}},ExpressionUUID->"9136a322-15c5-4dcb-a0f1-\
948480493309"],
Cell[BoxData[
RowBox[{"\[IndentingNewLine]",
StyleBox[
RowBox[{"Obtaining", " ", "the", " ", "differential", " ", "equation"}],
"Section"]}]], "Input",
CellChangeTimes->{{3.816171853906876*^9,
3.8161718657997217`*^9}},ExpressionUUID->"604ac2fc-adc8-4fbe-b591-\
53416b5ef028"],
Cell[CellGroupData[{
Cell["Adding handles.", "Subsection",
CellChangeTimes->{{3.816174329458706*^9,
3.8161743400266914`*^9}},ExpressionUUID->"cac03236-4563-4c3d-bed9-\
d39ed3c493b5"],
Cell["\<\
Lets now compute the expression for adding two handles to the line with 9 \
points
\
\>", "Text",
CellChangeTimes->{{3.816174347444687*^9, 3.8161743505556774`*^9}, {
3.8161746842746515`*^9, 3.8161746934426503`*^9},
3.816250667208026*^9},ExpressionUUID->"1284e97b-dd94-480e-b5e3-\
2782e80cd2e2"],
Cell[CellGroupData[{
Cell[BoxData["myotherline"], "Input",
CellChangeTimes->{{3.8161747157730875`*^9, 3.8161747171630735`*^9}},
CellLabel->"In[24]:=",ExpressionUUID->"24a43699-63d8-4ee0-8b4c-d21a2ccbfb2c"],
Cell[BoxData[
GraphicsBox[
TagBox[RasterBox[CompressedData["
1:eJzt3E2O41QQwPEAG5ZcgQOwR9mxZDuIA8yIzohNI80gIW6RC3CNXMaHaZJ0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"], {{0, 94.2}, {180.6, 0}}, {0, 255},
ColorFunction->RGBColor,
ImageResolution->{120, 120}],
BoxForm`ImageTag["Byte", ColorSpace -> "RGB", Interleaving -> True],
Selectable->False],
DefaultBaseStyle->"ImageGraphics",
ImageSizeRaw->{180.6, 94.2},
PlotRange->{{0, 180.6}, {0, 94.2}}]], "Output",
CellChangeTimes->{3.8161747177510533`*^9},
CellLabel->"Out[24]=",ExpressionUUID->"f05a7368-7387-411b-8f4b-3d140ab4280e"]
}, Open ]],
Cell[BoxData[""], "Input",
CellChangeTimes->{{3.816174731543052*^9,
3.816174732603092*^9}},ExpressionUUID->"0d788f65-0d8b-4bc3-bb8c-\
728bd4c5cb80"],
Cell["\<\
We now need to find a label for those points in the handle, but if \
\[OpenCurlyQuote]a\[CloseCurlyQuote] and \[OpenCurlyQuote]c\[CloseCurlyQuote] \
have a value then the first handle can have values: a+1, a+2,...,c-1, which \
are in total (c-a-1) values. The second handle can have (f-d-1) values so in \
total we are adding (c-a-1)(f-d-1) new labelings.
Then the expression
\
\>", "Text",
CellChangeTimes->{{3.816174739275091*^9, 3.816174825611055*^9}, {
3.8161966024018707`*^9, 3.8161966044752235`*^9}},
CellTags->"analysis",ExpressionUUID->"d1452a5a-42c6-48e8-960a-889ccea0ca4e"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"ToExpression", "[",
RowBox[{
"\"\<\\\\ \
\\sum_{a=1}^{n-8}\\sum_{b=a+1}^{n-7}\\sum_{c=b+1}^{n-6}\\sum_{d=c+1}^{n-5}\\\
sum_{e=d+1}^{n-4}\\sum_{f=e+1}^{n-3}\\sum_{g=e+1}^{n-2}\\sum_{h=g+1}^{n-1}\\\
sum_{i=h+1}^{n} (c-a-1)(f-d-1)\>\"", ",", "TeXForm"}], "]"}]], "Input",
CellChangeTimes->{{3.8161748516430616`*^9, 3.816174851831052*^9}},
EmphasizeSyntaxErrors->True,
CellLabel->"In[25]:=",ExpressionUUID->"603324f7-cc08-4806-af46-ff27fa01d53a"],
Cell[BoxData[
TemplateBox[{
"Sum", "vloc",
"\"The variable \\!\\(\\*RowBox[{\\\"b\\\", \\\"\[Equal]\\\", \
RowBox[{\\\"a\\\", \\\"+\\\", \\\"1\\\"}]}]\\) cannot be localized so that it \
can be assigned to numerical values.\"", 2, 25, 52, 28313626184081491958,
"Local"},
"MessageTemplate"]], "Message", "MSG",
CellChangeTimes->{3.816174855368085*^9},
CellLabel->
"During evaluation of \
In[25]:=",ExpressionUUID->"4721170f-dec2-4985-aa9b-1730c18a9d0b"],
Cell[BoxData[
RowBox[{
UnderoverscriptBox["\[Sum]",
RowBox[{"a", "=", "1"}],
RowBox[{"n", "-", "8"}]],
RowBox[{
UnderoverscriptBox["\[Sum]",
RowBox[{"b", "\[Equal]",
RowBox[{"a", "+", "1"}]}],
RowBox[{"n", "-", "7"}]],
RowBox[{
UnderoverscriptBox["\[Sum]",
RowBox[{"c", "\[Equal]",
RowBox[{"b", "+", "1"}]}],
RowBox[{"n", "-", "6"}]],
RowBox[{
UnderoverscriptBox["\[Sum]",
RowBox[{"d", "\[Equal]",
RowBox[{"c", "+", "1"}]}],
RowBox[{"n", "-", "5"}]],
RowBox[{
UnderoverscriptBox["\[Sum]",
RowBox[{"e", "\[Equal]",
RowBox[{"d", "+", "1"}]}],
RowBox[{"n", "-", "4"}]],
RowBox[{
UnderoverscriptBox["\[Sum]",
RowBox[{"f", "\[Equal]",
RowBox[{"e", "+", "1"}]}],
RowBox[{"n", "-", "3"}]],
RowBox[{
UnderoverscriptBox["\[Sum]",
RowBox[{"g", "\[Equal]",
RowBox[{"e", "+", "1"}]}],
RowBox[{"n", "-", "2"}]],
RowBox[{
UnderoverscriptBox["\[Sum]",
RowBox[{"h", "\[Equal]",
RowBox[{"g", "+", "1"}]}],
RowBox[{"n", "-", "1"}]],
RowBox[{
UnderoverscriptBox["\[Sum]",
RowBox[{"i", "\[Equal]",
RowBox[{"h", "+", "1"}]}], "n"],
RowBox[{
RowBox[{"(",
RowBox[{"c", "-", "a", "-", "1"}], ")"}], " ",
RowBox[{"(",
RowBox[{"f", "-", "d", "-", "1"}],
")"}]}]}]}]}]}]}]}]}]}]}]], "Output",
CellChangeTimes->{3.8161748553820887`*^9},
CellLabel->"Out[25]=",ExpressionUUID->"e2f2aab5-d64d-43d4-94c7-67938bfdd8a1"]
}, Open ]],
Cell[BoxData[""], "Input",
CellChangeTimes->{{3.8161748588701663`*^9,
3.81617485945817*^9}},ExpressionUUID->"eebc0717-3e9d-4eaa-a300-\
0d2618b1e0ab"],
Cell["\<\
Counts all possible labelings of this new figure, in Mathematica this is \
equivalent to:
\
\>", "Text",
CellChangeTimes->{{3.816174864947152*^9, 3.8161749043221807`*^9},
3.8346858112244844`*^9},
CellTags->"codeone",ExpressionUUID->"3847261d-324c-427c-ac2d-a5bb53b2e8c4"],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"Sum", "[",
RowBox[{