ch3 - q-seriesqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt mon may 30 09:42:09 2005 1...
TRANSCRIPT
![Page 1: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/1.jpg)
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![Page 2: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/2.jpg)
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![Page 3: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/3.jpg)
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![Page 4: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/4.jpg)
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![Page 5: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/5.jpg)
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![Page 6: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/6.jpg)
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![Page 7: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/7.jpg)
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![Page 8: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/8.jpg)
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![Page 9: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/9.jpg)
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![Page 10: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/10.jpg)
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![Page 11: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/11.jpg)
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![Page 12: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/12.jpg)
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![Page 13: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/13.jpg)
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![Page 14: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/14.jpg)
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![Page 15: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/15.jpg)
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![Page 18: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/18.jpg)
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![Page 19: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/19.jpg)
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![Page 20: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/20.jpg)
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![Page 21: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/21.jpg)
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![Page 22: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/22.jpg)
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![Page 23: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/23.jpg)
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![Page 24: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/24.jpg)
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![Page 25: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/25.jpg)
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![Page 26: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/26.jpg)
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![Page 27: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/27.jpg)
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![Page 28: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/28.jpg)
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![Page 29: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/29.jpg)
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![Page 30: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/30.jpg)
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![Page 31: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/31.jpg)
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![Page 32: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/32.jpg)
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![Page 36: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/36.jpg)
36ch3_p36.txt Mon May 30 09:42:09 2005 1
EXAMPLE (6)===========Maple calculations for vector partitions of 4 and their crank mod 5
==============================================================================> read "FUNCS.txt":> V:=vecptns(4);> m:=matrix(nops(V),3):> for i from 1 to nops(V) do> m[i,1]:=V[i]:> m[i,2]:=vpw(V[i]):> m[i,3]:=vpcrank(V[i]):> od:> #Table of vector partitions of 4 with weight and crank> print(m); [[[], [], [1, 1, 1, 1]] 1 -4] [ ] [ [[], [], [1, 1, 2]] 1 -3] [ ] [ [[], [], [2, 2]] 1 -2] [ ] [ [[], [], [1, 3]] 1 -2] [ ] [ [[], [], [4]] 1 -1] [ ] [ [[], [1], [1, 1, 1]] 1 -2] [ ] [ [[], [1], [1, 2]] 1 -1] [ ] [ [[], [1], [3]] 1 0] [ ] [ [[], [1, 1], [1, 1]] 1 0] [ ] [ [[], [1, 1], [2]] 1 1] [ ] [ [[], [2], [1, 1]] 1 -1] [ ] [ [[], [2], [2]] 1 0] [ ] [ [[], [1, 1, 1], [1]] 1 2] [ ] [ [[], [1, 2], [1]] 1 1] [ ] [ [[], [3], [1]] 1 0] [ ] [[[], [1, 1, 1, 1], []] 1 4] [ ] [ [[], [1, 1, 2], []] 1 3] [ ] [ [[], [2, 2], []] 1 2] [ ] [ [[], [1, 3], []] 1 2] [ ] [ [[], [4], []] 1 1] [ ] [ [[1], [], [1, 1, 1]] -1 -3] [ ] [ [[1], [], [1, 2]] -1 -2] [ ] [ [[1], [], [3]] -1 -1] [ ] [ [[1], [1], [1, 1]] -1 -1] [ ] [ [[1], [1], [2]] -1 0] [ ]
![Page 37: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/37.jpg)
37ch3_p37.txt Mon May 30 09:42:09 2005 1
[ [[1], [1, 1], [1]] -1 1] [ ] [ [[1], [2], [1]] -1 0] [ ] [ [[1], [1, 1, 1], []] -1 3] [ ] [ [[1], [1, 2], []] -1 2] [ ] [ [[1], [3], []] -1 1] [ ] [ [[2], [], [1, 1]] -1 -2] [ ] [ [[2], [], [2]] -1 -1] [ ] [ [[2], [1], [1]] -1 0] [ ] [ [[2], [1, 1], []] -1 2] [ ] [ [[2], [2], []] -1 1] [ ] [ [[1, 2], [], [1]] 1 -1] [ ] [ [[3], [], [1]] -1 -1] [ ] [ [[1, 2], [1], []] 1 1] [ ] [ [[3], [1], []] -1 1] [ ] [ [[1, 3], [], []] 1 0] [ ] [ [[4], [], []] -1 0]
> > #Total weight of vector partitions of 4> > add(vpw(V[i]),i=1..nops(V)); 5> nops(V); 41
> #There are 41 vector partitions of 4 with total weight 5 #which is p(4).
> > > # vector partitions of 4 with crank = 0 mod 5> > V0:=vecptnsC(4,0,5);V0 := [[[], [1], [3]], [[], [1, 1], [1, 1]], [[], [2], [2]], [[], [3], [1]],
[[1], [1], [2]], [[1], [2], [1]], [[2], [1], [1]], [[1, 3], [], []],
[[4], [], []]]
> nops(V0); 9
> add(vpw(V0[i]),i=1..nops(V0)); 1
> > # There are 9 vector partitions of 4 with crank = 0 mod 5> # and total weight = 1.> > # vector partitions of 4 with crank = 1 mod 5
![Page 38: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/38.jpg)
38ch3_p38.txt Mon May 30 09:42:09 2005 1
> > V1:=vecptnsC(4,1,5);V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[], [1, 2], [1]],
[[], [4], []], [[1], [1, 1], [1]], [[1], [3], []], [[2], [2], []],
[[1, 2], [1], []], [[3], [1], []]]
> nops(V1); 9
> add(vpw(V1[i]),i=1..nops(V1)); 1
> > # There are 9 vector partitions of 4 with crank = 1 mod 5> # and total weight = 1.> > # vector partitions of 4 with crank = 2 mod 5> > V2:=vecptnsC(4,2,5);V2 := [[[], [], [1, 1, 2]], [[], [1, 1, 1], [1]], [[], [2, 2], []],
[[], [1, 3], []], [[1], [], [1, 1, 1]], [[1], [1, 2], []],
[[2], [1, 1], []]]
> nops(V2); 7
> add(vpw(V2[i]),i=1..nops(V2)); 1
> > # There are 7 vector partitions of 4 with crank = 0 mod 5> # and total weight = 1.> > # vector partitions of 4 with crank = 3 mod 5> > V3:=vecptnsC(4,3,5);V3 := [[[], [], [2, 2]], [[], [], [1, 3]], [[], [1], [1, 1, 1]],
[[], [1, 1, 2], []], [[1], [], [1, 2]], [[1], [1, 1, 1], []],
[[2], [], [1, 1]]]
> nops(V3); 7
> add(vpw(V3[i]),i=1..nops(V3)); 1
> > # There are 7 vector partitions of 4 with crank = 3 mod 5> # and total weight = 1.> > # vector partitions of 4 with crank = 4 mod 5> > V4:=vecptnsC(4,4,5);V4 := [[[], [], [4]], [[], [1], [1, 2]], [[], [2], [1, 1]],
[[], [1, 1, 1, 1], []], [[1], [], [3]], [[1], [1], [1, 1]], [[2], [], [2]],
[[1, 2], [], [1]], [[3], [], [1]]]
![Page 39: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/39.jpg)
39ch3_p39.txt Mon May 30 09:42:09 2005 1
> nops(V4); 9
> add(vpw(V4[i]),i=1..nops(V4)); 1
> > # There are 9 vector partitions of 4 with crank = 4 mod 5> # and total weight = 1.>
==============================================================================
We see that
NV(0,5,4)=NV(1,5,4)=NV(2,5,4)=NV(3,5,4)=NV(4,5,4)=1=p(4)/5
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![Page 44: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/44.jpg)
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![Page 45: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/45.jpg)
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![Page 46: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/46.jpg)
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![Page 47: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/47.jpg)
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![Page 48: ch3 - q-SERIESqseries.org/fgarvan/qs/notes/ch3/ch3.pdf · 38 ch3_p38.txt Mon May 30 09:42:09 2005 1 > > V1:=vecptnsC(4,1,5); V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[],](https://reader033.vdocument.in/reader033/viewer/2022052011/60277182696512472032ec67/html5/thumbnails/48.jpg)
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