north carolina state university carla d. savagegoddard/mini/2005/savage.pdf · carla d. savage...
TRANSCRIPT
![Page 1: North Carolina State University Carla D. Savagegoddard/MINI/2005/Savage.pdf · Carla D. Savage North Carolina State University. Matching \parentheses" in a bit string 1001100111000010](https://reader034.vdocuments.us/reader034/viewer/2022052005/601933d6518cf3194258269b/html5/thumbnails/1.jpg)
Symmetric Chains of Subsets and Necklaces
Carla D. Savage
North Carolina State University
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Matching “parentheses” in a bit string
1 0 0 1 1 0 0 1 1 1 0 0 0 0 1 0
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Matching “parentheses” in a bit string
1 0 0 1 1 0 0 1 1 1 0 0 0 0 1 0
unmatched 0’s and unmatched 1’s
1 * * * * * * * * 1 0 0 0 * * 0
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Matching “parentheses” in a bit string
1 0 0 1 1 0 0 1 1 1 0 0 0 0 1 0
unmatched 0’s and unmatched 1’s
1 * * * * * * * * 1 0 0 0 * * 0
Unmatched 0’s are always to the right of all unmatched 1’s.
Changing the first unmatched 0 to 1 or the last unmatched 1
to 0 does not change matching.
![Page 5: North Carolina State University Carla D. Savagegoddard/MINI/2005/Savage.pdf · Carla D. Savage North Carolina State University. Matching \parentheses" in a bit string 1001100111000010](https://reader034.vdocuments.us/reader034/viewer/2022052005/601933d6518cf3194258269b/html5/thumbnails/5.jpg)
Growing chains in the Boolean lattice Bn
1100 0110 0101 1010 0011 1001
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Growing chains in the Boolean lattice Bn
1110 0111 1011 1101
1100 0110 0101 1010 0011 1001
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Growing chains in the Boolean lattice Bn
1111
1110 0111 1011 1101
1100 0110 0101 1010 0011 1001
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Growing chains in the Boolean lattice Bn
1111
1110 0111 1011 1101
1100 0110 0101 1010 0011 1001
1000 0100 0010 0001
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Growing chains in the Boolean lattice Bn
1111
1110 0111 1011 1101
1100 0110 0101 1010 0011 1001
1000 0100 0010 0001
0000
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Growing chains in the Boolean lattice Bn
1111
1110 0111 1011 1101
1100 0110 0101 1010 0011 1001
1000 0100 0010 0001
0000
Gives a symmetric chain decomposition for every n
[Greene-Kleitman 1976].
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Pattern-avoiding permutations
Permutation π1π2 . . . πn avoids pattern “123” if there is no
i < j < k s.t. πi < πj < πk.
Amazing result [Simion & Schmidt 1985]: The number of
permutations that avoid 123 is independent of “123”.
Same holds true for permutations of a multiset 1a12a2 . . . nan
([SW 2005], extending [Albert et al 2001])
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Multiset permutations of 1a12a2 . . . nan avoiding 123
The number is the same for every permutation of a1, a2, . . . an
[Albert et al 2001]
Bijection - suffices to show for swap of adjacent ai:
1a1 . . . iai(i + 1)ai+1 . . . nan ↔ 1a1 . . . iai+1(i + 1)ai . . . nan
Use Greene Kleitman [SW2005].
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Example (assume ai > ai+1).
S = (12)(21)(31)(45)(52)(67)(71) → T = (12)(21)(31)(42)(55)(67)(71)
Start with a string x ∈ S
7 5 6 6 4 6 6 4 6 6 4 6 5 3 2 4 1 1 4
Replace i by ‘(’ and i + 1 by ‘)’
7 ) 6 6 ( 6 6 ( 6 6 ( 6 ) 3 2 ( 1 1 (
Match parentheses in the usual way. Change the leftmost
ai − ai+1 unmatched left parentheses to right parentheses.
7 ) 6 6 ) 6 6 ) 6 6 ( 6 ) 3 2 ) 1 1 (
Change ‘)’ back to i + 1 and change ‘(’ back to i.
7 5 6 6 5 6 6 5 6 6 4 6 5 3 2 5 1 1 4
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Venn Diagram for n sets
13
23
123 312
2
1
13
23123
1 2
23
13
12
1
2
3
12
23
2
3
123
12123
13
1
3
Dual is a graph whose vertices are the elements of Bn.
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Venn diagram for n sets: collection of n simple closed
curves in the plane, {Θ1, Θ2, . . . ,Θn}, such that for each
S ⊆ {1, 2, . . . , n} the region
⋂
i∈S
int(Θi) ∩⋂
i 6∈S
ext(Θi)
is nonempty and connected.
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Back to Bn
0101
0010
1011
00111010
0001
1001
1101
0000
0111
0110
0100
1111
1110
1100
1000
Start with Greene-Kleitman chains.
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Back to Bn
0101
0010
1011
00111010
0001
1001
1101
0000
0111
0110
0100
1111
1110
1100
1000
Associate a graph.
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Back to Bn
0101
0010
1011
00111010
0001
1001
1101
0000
0111
0110
0100
1111
1110
1100
1000
Add “chain cover edges”.
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Back to Bn
{3}
{1,3}
{2,3,4}
{1,2,4}
{1,3,4}
{2,4}
{2}{1}
{1,2,3}
{1,4}
{4}
{1,2,3,4}
{1,2}
{}
{3,4}
{2,3}
Convert from bit string to set notation.
![Page 20: North Carolina State University Carla D. Savagegoddard/MINI/2005/Savage.pdf · Carla D. Savage North Carolina State University. Matching \parentheses" in a bit string 1001100111000010](https://reader034.vdocuments.us/reader034/viewer/2022052005/601933d6518cf3194258269b/html5/thumbnails/20.jpg)
Back to Bn
{3}
{1,3}
{2,3,4}
{1,2,4}
{1,3,4}
{2,4}
{2}{1}
{1,2,3}
{1,4}
{4}
{1,2,3,4}
{1,2}
{}
{3,4}
{2,3}
Take dual.
![Page 21: North Carolina State University Carla D. Savagegoddard/MINI/2005/Savage.pdf · Carla D. Savage North Carolina State University. Matching \parentheses" in a bit string 1001100111000010](https://reader034.vdocuments.us/reader034/viewer/2022052005/601933d6518cf3194258269b/html5/thumbnails/21.jpg)
Back to Bn
{3}
{1,3}
{2,3,4}
{1,2,4}
{1,3,4}
{2,4}
{2}{1}
{1,2,3}
{1,4}
{4}
{1,2,3,4}
{1,2}
{}
{3,4}
{2,3}
Take dual. First cross “1’ edges.
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Back to Bn
{3}
{1,3}
{2,3,4}
{1,2,4}
{1,3,4}
{2,4}
{2}{1}
{1,2,3}
{1,4}
{4}
{1,2,3,4}
{1,2}
{}
{3,4}
{2,3}
Take dual. First cross “1’ edges. Then cross “2 edges”.
![Page 23: North Carolina State University Carla D. Savagegoddard/MINI/2005/Savage.pdf · Carla D. Savage North Carolina State University. Matching \parentheses" in a bit string 1001100111000010](https://reader034.vdocuments.us/reader034/viewer/2022052005/601933d6518cf3194258269b/html5/thumbnails/23.jpg)
Back to Bn
{3}
{1,3}
{2,3,4}
{1,2,4}
{1,3,4}
{2,4}
{2}{1}
{1,2,3}
{1,4}
{4}
{1,2,3,4}
{1,2}
{}
{3,4}
{2,3}
Take dual. First cross “1’ edges. Then cross “2 edges”. Etc.
![Page 24: North Carolina State University Carla D. Savagegoddard/MINI/2005/Savage.pdf · Carla D. Savage North Carolina State University. Matching \parentheses" in a bit string 1001100111000010](https://reader034.vdocuments.us/reader034/viewer/2022052005/601933d6518cf3194258269b/html5/thumbnails/24.jpg)
Back to Bn
{3}
{1,3}
{2,3,4}
{1,2,4}
{1,3,4}
{2,4}
{2}{1}
{1,2,3}
{1,4}
{4}
{1,2,3,4}
{1,2}
{}
{3,4}
{2,3}
Take dual. First cross “1’ edges. Then cross “2 edges”. Etc.
![Page 25: North Carolina State University Carla D. Savagegoddard/MINI/2005/Savage.pdf · Carla D. Savage North Carolina State University. Matching \parentheses" in a bit string 1001100111000010](https://reader034.vdocuments.us/reader034/viewer/2022052005/601933d6518cf3194258269b/html5/thumbnails/25.jpg)
Back to Bn
{3}
{1,3}
{2,3,4}
{1,2,4}
{1,3,4}
{2,4}
{2}{1}
{1,2,3}
{1,4}
{4}
{1,2,3,4}
{1,2}
{}
{3,4}
{2,3}
Result: Venn diagram [GKS2004] (Can always find chain
cover edges and planar embedding of the chain cover graph.)
![Page 26: North Carolina State University Carla D. Savagegoddard/MINI/2005/Savage.pdf · Carla D. Savage North Carolina State University. Matching \parentheses" in a bit string 1001100111000010](https://reader034.vdocuments.us/reader034/viewer/2022052005/601933d6518cf3194258269b/html5/thumbnails/26.jpg)
Bonus
10010
1011001110
01100
11110
11100
11000
10011
0000100010
00101
00110
00100
01001
01010
01000
00000
00011��������
������
������
��������
��������
���
���
����
���
���
����
���
���
����
��������
����
��������
��������
������
������
��������
��������
������
������
����
����
����
����
���
���
����
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������
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Theorem [KRSW 2004] The face bounded by a chain and its
parent can always be “quadrangulated” by chords joining
vertices which differ in one bit, giving a Venn diagram with
(2n − 2)/2 vertices (half simple).
![Page 27: North Carolina State University Carla D. Savagegoddard/MINI/2005/Savage.pdf · Carla D. Savage North Carolina State University. Matching \parentheses" in a bit string 1001100111000010](https://reader034.vdocuments.us/reader034/viewer/2022052005/601933d6518cf3194258269b/html5/thumbnails/27.jpg)
Rotationally symmetric Venn diagrams?
No if n is composite [Henderson 1963].
For prime n?
Work in 1/n th of Bn (necklaces) to get SCD.
Embed in 1/n th pie slice of plane.
Rotate.
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Necklace - equivalence class of bitstrings under rotation
{11110, 01111, 10111, 11011, 11101},{10110, 01011, 10101, 11010, 01101}, etc.
When n is prime:
*if* SCD for then then
necklace reps. rotate once, three more times ...
11111
11110 01111
11100 10110 01110 01011
11000 10100 01100 01010
10000 01000
00000
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Getting symmetry from necklace SCD
1
1312
134123
1234
235
45
4
124
25
5
15
125 1235
12345
234
1245
14514
23 2
24
245
2345
345
34
1345
135
35
3
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Getting symmetry from necklace SCD
1312
134123
1234
1
25
5
15
1235
12345
234125
235
34
23 2
24
245
2345
345
1245
45
43
35
135
1345
14145
124
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Getting symmetry from necklace SCD
123515
234
235
2345
13
125
45 34
23
5
134
1245
4
3
2
12345
1234
345145
25
24
135
1345
245124
1
12
123
35
14
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Getting symmetry from necklace SCD
123515
234
235
2345
13
125
45 34
23
5
134
1245
4
3
2
12345
1234
345145
25
24
135
1345
245124
1
12
123
35
14
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Focus
How to choose necklace representatives so that the “necklace
subposet” has a SCD with the chain cover property?
Solution
Choose as reps: bit string with the lex min block code. Then
Greene-Kleitman works and you get:
Theorem [GKS 2004] Rotationally symmetric Venn diagrams
exist for all prime n.
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Block code of a binary string:
11000 1110 100000 10 110 (5, 4, 6, 2, 3)
01100011101000001011 (∞)
10110001110100000101 (∞)
110 11000 1110 100000 10 (3, 5, 4, 6, 2)
When n is prime:
- x ∈ Bn has n distinct rotations
- no 2 rotations have same finite block code.
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SCD in necklace poset for n = 7
1111110
1011110 1111100 1101110
1010110 1011100 1111000 1101100 1001110
1010100 1011000 1110000 1101000 1001100
1010000 1100000 1001000
1000000
(2,5) (2,5) (7) (3,4) (3,4)
G-K chains preserve paren matching and block code!
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Necklace poset when n is composite - Does it still have
a SCD?
[111110] (6)
[101110] (6) [111100] (6) [110110] (3)
[101100] (6) [111000] (6) [110100] (6) [101010] (2)
[101000] (6) [110000] (6) [100100] (3)
[100000] (6)
(2,4) (6) (3,3) (2,2,2)
(It shouldn’t, but it seems to?)
![Page 37: North Carolina State University Carla D. Savagegoddard/MINI/2005/Savage.pdf · Carla D. Savage North Carolina State University. Matching \parentheses" in a bit string 1001100111000010](https://reader034.vdocuments.us/reader034/viewer/2022052005/601933d6518cf3194258269b/html5/thumbnails/37.jpg)
For composite n,
Necklace poset is unimodal and symmetric [Stanley 1984].
[Jiang 2003]:
So are subposets induced by any block code.
Block code aperiodic implies subposet has SCD.
Thus, suffices to check periodic block codes.
Necklace poset has SCD for all n ≤ 16.
e.g. checking n = 12 ...
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Independent family of curves (regions need not be
connected)
A
A
B
B
C C
AB
BC AC
ABC
C
Not a Venn diagram
(independent family of curves)
Symmetric independent family of curves
(but not a Venn diagram)
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[Grunbaum 1999]:
The minimum number of regions in a rotationally symmetric
independent family of curves is
2 + n(Cn − 2),
where Cn is the number of n-bit necklaces.
Rotationally symmetric independent families of curves can be
constructed for any n.
But can this be done with the minimum number of regions?
Existence of a SCD with the chain cover property in the
necklace poset would solve this problem.