Polyominoes, Gray Codes, and Venn Diagrams
Transcript of Polyominoes, Gray Codes, and Venn Diagrams
![Page 1: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/1.jpg)
Polyominoes, Gray Codes, and Venn Diagrams
Stirling Chow1 Frank Ruskey1
1Department of Computer ScienceUniversity of Victoria, CANADA
Northwest Theory Day, April 2005
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Definition - polyomino
I Made from unit squaresjoined along edges.
I No holes allowed.
I Must be connected.
I Translations allowed butnot flips or rotations.
(a)
(b)
(c)
(d)
(e)
![Page 3: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/3.jpg)
Definition - polyomino
I Made from unit squaresjoined along edges.
I No holes allowed.
I Must be connected.
I Translations allowed butnot flips or rotations.
(a)
(b)
(c)
(d)
(e)
![Page 4: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/4.jpg)
Definition - polyomino
I Made from unit squaresjoined along edges.
I No holes allowed.
I Must be connected.
I Translations allowed butnot flips or rotations.
(a)
(b)
(c)
(d)
(e)
![Page 5: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/5.jpg)
Definition - polyomino
I Made from unit squaresjoined along edges.
I No holes allowed.
I Must be connected.
I Translations allowed butnot flips or rotations.
(a)
(b)
(c)
(d)
(e)
![Page 6: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/6.jpg)
Definition - convexity conditions
I Column convex if intersections with vertical lines areconnected.
I Satisfy the recurrence: an = 5an−1 − 7an−2 + 4an−3
([Polya],nice proof:[Hickerson]).
I Convex if both row and column convex.
I Gray codes: column convex only.
I Further restriction: number of cells in each column is fixed.
(a) column−convexnot row−convex
(b) row−convexnot column−convex
(c) convex
![Page 7: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/7.jpg)
Definition - convexity conditions
I Column convex if intersections with vertical lines areconnected.
I Satisfy the recurrence: an = 5an−1 − 7an−2 + 4an−3
([Polya],nice proof:[Hickerson]).
I Convex if both row and column convex.
I Gray codes: column convex only.
I Further restriction: number of cells in each column is fixed.
(a) column−convexnot row−convex
(b) row−convexnot column−convex
(c) convex
![Page 8: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/8.jpg)
Definition - convexity conditions
I Column convex if intersections with vertical lines areconnected.
I Satisfy the recurrence: an = 5an−1 − 7an−2 + 4an−3
([Polya],nice proof:[Hickerson]).
I Convex if both row and column convex.
I Gray codes: column convex only.
I Further restriction: number of cells in each column is fixed.
(a) column−convexnot row−convex
(b) row−convexnot column−convex
(c) convex
![Page 9: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/9.jpg)
Definition - convexity conditions
I Column convex if intersections with vertical lines areconnected.
I Satisfy the recurrence: an = 5an−1 − 7an−2 + 4an−3
([Polya],nice proof:[Hickerson]).
I Convex if both row and column convex.
I Gray codes: column convex only.
I Further restriction: number of cells in each column is fixed.
(a) column−convexnot row−convex
(b) row−convexnot column−convex
(c) convex
![Page 10: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/10.jpg)
Definition - convexity conditions
I Column convex if intersections with vertical lines areconnected.
I Satisfy the recurrence: an = 5an−1 − 7an−2 + 4an−3
([Polya],nice proof:[Hickerson]).
I Convex if both row and column convex.
I Gray codes: column convex only.
I Further restriction: number of cells in each column is fixed.
(a) column−convexnot row−convex
(b) row−convexnot column−convex
(c) convex
![Page 11: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/11.jpg)
I Column counts: [a1, a2, . . . , ak ]; below is the set [1, 2, 2, 1].
I Shift limits: 〈A1,A2, . . . ,Ak−1〉, where Ai = ai + ai+1 − 1;thus [1, 2, 2, 1] → 〈2, 3, 2〉.
I Not unique [1, 3, 1, 3] → 〈3, 3, 3〉 and [2, 2, 2, 2] → 〈3, 3, 3〉.I No polyomino for 〈1, 2, 1〉.I Encode individual polyominoes as
(p1, p2, . . . , pk−1) ∈ A1 × A2 × · · · × Ak−1
(1,2,1)(1,2,0)(1,1,1)(1,1,0)(1,0,1)(1,0,0)
(0,0,0) (0,0,1) (0,1,0) (0,1,1) (0,2,0) (0,2,1)
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I Column counts: [a1, a2, . . . , ak ]; below is the set [1, 2, 2, 1].
I Shift limits: 〈A1,A2, . . . ,Ak−1〉, where Ai = ai + ai+1 − 1;thus [1, 2, 2, 1] → 〈2, 3, 2〉.
I Not unique [1, 3, 1, 3] → 〈3, 3, 3〉 and [2, 2, 2, 2] → 〈3, 3, 3〉.I No polyomino for 〈1, 2, 1〉.I Encode individual polyominoes as
(p1, p2, . . . , pk−1) ∈ A1 × A2 × · · · × Ak−1
(1,2,1)(1,2,0)(1,1,1)(1,1,0)(1,0,1)(1,0,0)
(0,0,0) (0,0,1) (0,1,0) (0,1,1) (0,2,0) (0,2,1)
![Page 13: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/13.jpg)
I Column counts: [a1, a2, . . . , ak ]; below is the set [1, 2, 2, 1].
I Shift limits: 〈A1,A2, . . . ,Ak−1〉, where Ai = ai + ai+1 − 1;thus [1, 2, 2, 1] → 〈2, 3, 2〉.
I Not unique [1, 3, 1, 3] → 〈3, 3, 3〉 and [2, 2, 2, 2] → 〈3, 3, 3〉.I No polyomino for 〈1, 2, 1〉.I Encode individual polyominoes as
(p1, p2, . . . , pk−1) ∈ A1 × A2 × · · · × Ak−1
(1,2,1)(1,2,0)(1,1,1)(1,1,0)(1,0,1)(1,0,0)
(0,0,0) (0,0,1) (0,1,0) (0,1,1) (0,2,0) (0,2,1)
![Page 14: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/14.jpg)
I Column counts: [a1, a2, . . . , ak ]; below is the set [1, 2, 2, 1].
I Shift limits: 〈A1,A2, . . . ,Ak−1〉, where Ai = ai + ai+1 − 1;thus [1, 2, 2, 1] → 〈2, 3, 2〉.
I Not unique [1, 3, 1, 3] → 〈3, 3, 3〉 and [2, 2, 2, 2] → 〈3, 3, 3〉.I No polyomino for 〈1, 2, 1〉.I Encode individual polyominoes as
(p1, p2, . . . , pk−1) ∈ A1 × A2 × · · · × Ak−1
(1,2,1)(1,2,0)(1,1,1)(1,1,0)(1,0,1)(1,0,0)
(0,0,0) (0,0,1) (0,1,0) (0,1,1) (0,2,0) (0,2,1)
![Page 15: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/15.jpg)
I Column counts: [a1, a2, . . . , ak ]; below is the set [1, 2, 2, 1].
I Shift limits: 〈A1,A2, . . . ,Ak−1〉, where Ai = ai + ai+1 − 1;thus [1, 2, 2, 1] → 〈2, 3, 2〉.
I Not unique [1, 3, 1, 3] → 〈3, 3, 3〉 and [2, 2, 2, 2] → 〈3, 3, 3〉.I No polyomino for 〈1, 2, 1〉.I Encode individual polyominoes as
(p1, p2, . . . , pk−1) ∈ A1 × A2 × · · · × Ak−1
(1,2,1)(1,2,0)(1,1,1)(1,1,0)(1,0,1)(1,0,0)
(0,0,0) (0,0,1) (0,1,0) (0,1,1) (0,2,0) (0,2,1)
![Page 16: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/16.jpg)
I The change pi := pi ± 1 corresponds to a small earthquake.fault line earthquake begins two unit shift
(1,1,3,2,3,1) (1,1,3,2,3,1) (1,1,5,2,3,1)
I Can generate these using Gray codes for mixed-radix numbers(H-path in k − 1 dimensional grid graph).
I A more interesting move is a single cell move within a column.
I Then pi := pi ± 1 and pi+1 := pi+1 ∓ 1 (except atextremities).
![Page 17: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/17.jpg)
I The change pi := pi ± 1 corresponds to a small earthquake.fault line earthquake begins two unit shift
(1,1,3,2,3,1) (1,1,3,2,3,1) (1,1,5,2,3,1)
I Can generate these using Gray codes for mixed-radix numbers(H-path in k − 1 dimensional grid graph).
I A more interesting move is a single cell move within a column.
I Then pi := pi ± 1 and pi+1 := pi+1 ∓ 1 (except atextremities).
![Page 18: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/18.jpg)
I The change pi := pi ± 1 corresponds to a small earthquake.fault line earthquake begins two unit shift
(1,1,3,2,3,1) (1,1,3,2,3,1) (1,1,5,2,3,1)
I Can generate these using Gray codes for mixed-radix numbers(H-path in k − 1 dimensional grid graph).
I A more interesting move is a single cell move within a column.
I Then pi := pi ± 1 and pi+1 := pi+1 ∓ 1 (except atextremities).
![Page 19: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/19.jpg)
I The change pi := pi ± 1 corresponds to a small earthquake.fault line earthquake begins two unit shift
(1,1,3,2,3,1) (1,1,3,2,3,1) (1,1,5,2,3,1)
I Can generate these using Gray codes for mixed-radix numbers(H-path in k − 1 dimensional grid graph).
I A more interesting move is a single cell move within a column.
I Then pi := pi ± 1 and pi+1 := pi+1 ∓ 1 (except atextremities).
![Page 20: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/20.jpg)
I Define two operations (and their inverses) on A-sequences:τi : pi := pi + 1 (only at the endpoints)σi : pi := pi − 1; pi+1 := pi+1 + 1;
I The “problem” of Ai = 1.I Implies that ai , ai+1 = 1. (Since Ai = ai + ai+1 − 1)I I.e., cells i and i + 1 are frozen in place.
I If two or more Ai = 1 then underlying graph is disconnected;otherwise it is connected.
I The graph is denoted G ([a1, . . . , ak ]) or G (〈A1, . . . ,Ak−1〉).I An interesting case occurs when Ak−1 = 1 and Ai > 1,
for i = 1, 2, . . . , k − 2: the right-frozen case.
I Free = neither side frozen.
![Page 21: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/21.jpg)
I Define two operations (and their inverses) on A-sequences:τi : pi := pi + 1 (only at the endpoints)σi : pi := pi − 1; pi+1 := pi+1 + 1;
I The “problem” of Ai = 1.I Implies that ai , ai+1 = 1. (Since Ai = ai + ai+1 − 1)I I.e., cells i and i + 1 are frozen in place.
I If two or more Ai = 1 then underlying graph is disconnected;otherwise it is connected.
I The graph is denoted G ([a1, . . . , ak ]) or G (〈A1, . . . ,Ak−1〉).I An interesting case occurs when Ak−1 = 1 and Ai > 1,
for i = 1, 2, . . . , k − 2: the right-frozen case.
I Free = neither side frozen.
![Page 22: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/22.jpg)
I Define two operations (and their inverses) on A-sequences:τi : pi := pi + 1 (only at the endpoints)σi : pi := pi − 1; pi+1 := pi+1 + 1;
I The “problem” of Ai = 1.I Implies that ai , ai+1 = 1. (Since Ai = ai + ai+1 − 1)I I.e., cells i and i + 1 are frozen in place.
I If two or more Ai = 1 then underlying graph is disconnected;otherwise it is connected.
I The graph is denoted G ([a1, . . . , ak ]) or G (〈A1, . . . ,Ak−1〉).I An interesting case occurs when Ak−1 = 1 and Ai > 1,
for i = 1, 2, . . . , k − 2: the right-frozen case.
I Free = neither side frozen.
![Page 23: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/23.jpg)
I Define two operations (and their inverses) on A-sequences:τi : pi := pi + 1 (only at the endpoints)σi : pi := pi − 1; pi+1 := pi+1 + 1;
I The “problem” of Ai = 1.I Implies that ai , ai+1 = 1. (Since Ai = ai + ai+1 − 1)I I.e., cells i and i + 1 are frozen in place.
I If two or more Ai = 1 then underlying graph is disconnected;otherwise it is connected.
I The graph is denoted G ([a1, . . . , ak ]) or G (〈A1, . . . ,Ak−1〉).I An interesting case occurs when Ak−1 = 1 and Ai > 1,
for i = 1, 2, . . . , k − 2: the right-frozen case.
I Free = neither side frozen.
![Page 24: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/24.jpg)
I Define two operations (and their inverses) on A-sequences:τi : pi := pi + 1 (only at the endpoints)σi : pi := pi − 1; pi+1 := pi+1 + 1;
I The “problem” of Ai = 1.I Implies that ai , ai+1 = 1. (Since Ai = ai + ai+1 − 1)I I.e., cells i and i + 1 are frozen in place.
I If two or more Ai = 1 then underlying graph is disconnected;otherwise it is connected.
I The graph is denoted G ([a1, . . . , ak ]) or G (〈A1, . . . ,Ak−1〉).I An interesting case occurs when Ak−1 = 1 and Ai > 1,
for i = 1, 2, . . . , k − 2: the right-frozen case.
I Free = neither side frozen.
![Page 25: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/25.jpg)
I Define two operations (and their inverses) on A-sequences:τi : pi := pi + 1 (only at the endpoints)σi : pi := pi − 1; pi+1 := pi+1 + 1;
I The “problem” of Ai = 1.I Implies that ai , ai+1 = 1. (Since Ai = ai + ai+1 − 1)I I.e., cells i and i + 1 are frozen in place.
I If two or more Ai = 1 then underlying graph is disconnected;otherwise it is connected.
I The graph is denoted G ([a1, . . . , ak ]) or G (〈A1, . . . ,Ak−1〉).I An interesting case occurs when Ak−1 = 1 and Ai > 1,
for i = 1, 2, . . . , k − 2: the right-frozen case.
I Free = neither side frozen.
![Page 26: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/26.jpg)
I Define two operations (and their inverses) on A-sequences:τi : pi := pi + 1 (only at the endpoints)σi : pi := pi − 1; pi+1 := pi+1 + 1;
I The “problem” of Ai = 1.I Implies that ai , ai+1 = 1. (Since Ai = ai + ai+1 − 1)I I.e., cells i and i + 1 are frozen in place.
I If two or more Ai = 1 then underlying graph is disconnected;otherwise it is connected.
I The graph is denoted G ([a1, . . . , ak ]) or G (〈A1, . . . ,Ak−1〉).I An interesting case occurs when Ak−1 = 1 and Ai > 1,
for i = 1, 2, . . . , k − 2: the right-frozen case.
I Free = neither side frozen.
![Page 27: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/27.jpg)
I Define two operations (and their inverses) on A-sequences:τi : pi := pi + 1 (only at the endpoints)σi : pi := pi − 1; pi+1 := pi+1 + 1;
I The “problem” of Ai = 1.I Implies that ai , ai+1 = 1. (Since Ai = ai + ai+1 − 1)I I.e., cells i and i + 1 are frozen in place.
I If two or more Ai = 1 then underlying graph is disconnected;otherwise it is connected.
I The graph is denoted G ([a1, . . . , ak ]) or G (〈A1, . . . ,Ak−1〉).I An interesting case occurs when Ak−1 = 1 and Ai > 1,
for i = 1, 2, . . . , k − 2: the right-frozen case.
I Free = neither side frozen.
![Page 28: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/28.jpg)
1,1,0
2,0,0 0,1,0
2,1,0
1,0,0
0,0,0
LemmaIf k is even, then G ([a1, . . . , ak ]) isbipartite.
Proof.Define partite sets according to theparity of∑
j odd
pj (=∑
j
jpj mod 2).
Example
G ([2, 2, 1, 1]) is bipartite.
![Page 29: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/29.jpg)
2,1
0,0
1,1
2.0 0,1
1,0
LemmaIf k > 3 is odd, then G ([a1, . . . , ak ])is bipartite iff A1 = 1 or Ak−1 = 1.
Proof.Odd cycle: τ1, σ1, . . . , σk−2, τ
−1k−1:
Example
G ([2, 2, 1]) is not bipartite.
![Page 30: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/30.jpg)
2,1
0,0
1,1
2.0 0,1
1,0
LemmaIf k > 3 is odd, then G ([a1, . . . , ak ])is bipartite iff A1 = 1 or Ak−1 = 1.
Proof.Odd cycle: τ1, σ1, . . . , σk−2, τ
−1k−1:
0000,
Example
G ([2, 2, 1]) is not bipartite.
![Page 31: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/31.jpg)
2,1
0,0
1,1
2.0 0,1
1,0
LemmaIf k > 3 is odd, then G ([a1, . . . , ak ])is bipartite iff A1 = 1 or Ak−1 = 1.
Proof.Odd cycle: τ1, σ1, . . . , σk−2, τ
−1k−1:
0000,1000,
Example
G ([2, 2, 1]) is not bipartite.
![Page 32: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/32.jpg)
2,1
0,0
1,1
2.0 0,1
1,0
LemmaIf k > 3 is odd, then G ([a1, . . . , ak ])is bipartite iff A1 = 1 or Ak−1 = 1.
Proof.Odd cycle: τ1, σ1, . . . , σk−2, τ
−1k−1:
0000,1000,0100,
Example
G ([2, 2, 1]) is not bipartite.
![Page 33: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/33.jpg)
2,1
0,0
1,1
2.0 0,1
1,0
LemmaIf k > 3 is odd, then G ([a1, . . . , ak ])is bipartite iff A1 = 1 or Ak−1 = 1.
Proof.Odd cycle: τ1, σ1, . . . , σk−2, τ
−1k−1:
0000,1000,0100,0010,
Example
G ([2, 2, 1]) is not bipartite.
![Page 34: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/34.jpg)
2,1
0,0
1,1
2.0 0,1
1,0
LemmaIf k > 3 is odd, then G ([a1, . . . , ak ])is bipartite iff A1 = 1 or Ak−1 = 1.
Proof.Odd cycle: τ1, σ1, . . . , σk−2, τ
−1k−1:
0000,1000,0100,0010,0001,
Example
G ([2, 2, 1]) is not bipartite.
![Page 35: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/35.jpg)
2,1
0,0
1,1
2.0 0,1
1,0
LemmaIf k > 3 is odd, then G ([a1, . . . , ak ])is bipartite iff A1 = 1 or Ak−1 = 1.
Proof.Odd cycle: τ1, σ1, . . . , σk−2, τ
−1k−1:
0000, 0000.1000,0100,0010,0001,
Example
G ([2, 2, 1]) is not bipartite.
![Page 36: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/36.jpg)
LemmaIf k > 2 is even, then G (a) has no Hamilton path if A2i+1 is oddfor all i , unless a2 = a3 = · · · = ak−1 = 1.
Proof.Define a sign-reversing involution... If all A2i+1 are odd, thenparity difference is
d(A) =∏
j even
Aj .
Example
G (〈3, 2, 3, 2, 3〉) has no H-path.G (〈3, 1, 1, 1, 3〉) is a 3 by 3 grid and has a H-path.Conjecture: There is a H-path if k even and some A2i+1 is even.
![Page 37: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/37.jpg)
TheoremIf G (〈A1,A2, . . . ,Ak−1〉) has a Hamilton path H and BC is evenand BC 6= 6, then G (〈B,A1,A2, . . . ,Ak−1,C 〉) has a Hamiltonpath H ′.
Proof.Convert each edge of H into a path of BC vertices in H ′. Thestructure of B,C is a B by C grid graph. Need to be careful aboutthe τ moves in H.... Need to show the existence of particular typesof H-cycles in grid graphs....
TheoremIf A has the form (N>3)
0|1(O>3N>1)∗(N)0|1E or it’s reverse, then
there is a 1-move Gray code for A
Proof.Add pairs of columns on the left...
![Page 38: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/38.jpg)
TheoremIf G (〈A1,A2, . . . ,Ak−1〉) has a Hamilton path H and BC is evenand BC 6= 6, then G (〈B,A1,A2, . . . ,Ak−1,C 〉) has a Hamiltonpath H ′.
Example
Converting G (2, 2) into G (2, 2, 2, 2).0 0 1 10 1 0 1
![Page 39: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/39.jpg)
TheoremIf G (〈A1,A2, . . . ,Ak−1〉) has a Hamilton path H and BC is evenand BC 6= 6, then G (〈B,A1,A2, . . . ,Ak−1,C 〉) has a Hamiltonpath H ′.
Example
Converting G (2, 2) into G (2, 2, 2, 2).0 0 1 10 1 0 1
0 0 0 00 0 0 0
![Page 40: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/40.jpg)
TheoremIf G (〈A1,A2, . . . ,Ak−1〉) has a Hamilton path H and BC is evenand BC 6= 6, then G (〈B,A1,A2, . . . ,Ak−1,C 〉) has a Hamiltonpath H ′.
Example
Converting G (2, 2) into G (2, 2, 2, 2).0 0 1 10 1 0 1
? ? ? ?0 0 0 0 00 0 0 0 1? ? ? ?
![Page 41: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/41.jpg)
TheoremIf G (〈A1,A2, . . . ,Ak−1〉) has a Hamilton path H and BC is evenand BC 6= 6, then G (〈B,A1,A2, . . . ,Ak−1,C 〉) has a Hamiltonpath H ′.
Example
Converting G (2, 2) into G (2, 2, 2, 2).0 0 1 10 1 0 1
? ? ? ?0 0 0 0 00 0 0 0 1? ? ? 1 0
![Page 42: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/42.jpg)
TheoremIf G (〈A1,A2, . . . ,Ak−1〉) has a Hamilton path H and BC is evenand BC 6= 6, then G (〈B,A1,A2, . . . ,Ak−1,C 〉) has a Hamiltonpath H ′.
Example
Converting G (2, 2) into G (2, 2, 2, 2).0 0 1 10 1 0 1
0 1 1 0 00 0 0 0 00 0 0 0 10 0 1 1 0
![Page 43: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/43.jpg)
TheoremIf G (〈A1,A2, . . . ,Ak−1〉) has a Hamilton path H and BC is evenand BC 6= 6, then G (〈B,A1,A2, . . . ,Ak−1,C 〉) has a Hamiltonpath H ′.
Example
Converting G (2, 2) into G (2, 2, 2, 2).0 0 1 10 1 0 1
0 1 1 0 0 ? ? ?0 0 0 0 0 0 0 0 10 0 0 0 1 1 1 1 00 0 1 1 0 ? ? ?
![Page 44: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/44.jpg)
TheoremIf G (〈A1,A2, . . . ,Ak−1〉) has a Hamilton path H and BC is evenand BC 6= 6, then G (〈B,A1,A2, . . . ,Ak−1,C 〉) has a Hamiltonpath H ′.
Example
Converting G (2, 2) into G (2, 2, 2, 2).0 0 1 10 1 0 1
0 1 1 0 0 0 1 1 10 0 0 0 0 0 0 0 10 0 0 0 1 1 1 1 00 0 1 1 0 1 1 0 0
![Page 45: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/45.jpg)
TheoremIf G (〈A1,A2, . . . ,Ak−1〉) has a Hamilton path H and BC is evenand BC 6= 6, then G (〈B,A1,A2, . . . ,Ak−1,C 〉) has a Hamiltonpath H ′.
Example
Converting G (2, 2) into G (2, 2, 2, 2).0 0 1 10 1 0 1
0 1 1 0 0 0 1 1 1 ? ? ?0 0 0 0 0 0 0 0 1 1 1 1 10 0 0 0 1 1 1 1 0 0 0 0 10 0 1 1 0 1 1 0 0 ? ? ?
![Page 46: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/46.jpg)
TheoremIf G (〈A1,A2, . . . ,Ak−1〉) has a Hamilton path H and BC is evenand BC 6= 6, then G (〈B,A1,A2, . . . ,Ak−1,C 〉) has a Hamiltonpath H ′.
Example
Converting G (2, 2) into G (2, 2, 2, 2).0 0 1 10 1 0 1
0 1 1 0 0 0 1 1 1 0 0 1 10 0 0 0 0 0 0 0 1 1 1 1 10 0 0 0 1 1 1 1 0 0 0 0 10 0 1 1 0 1 1 0 0 0 1 1 0
![Page 47: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/47.jpg)
TheoremIf G (〈A1,A2, . . . ,Ak−1〉) has a Hamilton path H and BC is evenand BC 6= 6, then G (〈B,A1,A2, . . . ,Ak−1,C 〉) has a Hamiltonpath H ′.
Example
Converting G (2, 2) into G (2, 2, 2, 2).0 0 1 10 1 0 1
0 1 1 0 0 0 1 1 1 0 0 1 1 ? ? ?0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 10 0 0 0 1 1 1 1 0 0 0 0 1 1 1 10 0 1 1 0 1 1 0 0 0 1 1 0 ? ? ?
![Page 48: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/48.jpg)
TheoremIf G (〈A1,A2, . . . ,Ak−1〉) has a Hamilton path H and BC is evenand BC 6= 6, then G (〈B,A1,A2, . . . ,Ak−1,C 〉) has a Hamiltonpath H ′.
Example
Converting G (2, 2) into G (2, 2, 2, 2).0 0 1 10 1 0 1
0 1 1 0 0 0 1 1 1 0 0 1 1 0 0 10 0 0 0 0 0 0 0 1 1 1 1 1 1 1 10 0 0 0 1 1 1 1 0 0 0 0 1 1 1 10 0 1 1 0 1 1 0 0 0 1 1 0 0 1 1
![Page 49: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/49.jpg)
TheoremIf A has the form (N>3)
0|1(O>3N>1)∗(N)0|1E or it’s reverse, then
there is a 1-move Gray code for A
![Page 50: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/50.jpg)
Right-frozen case – a poset
I Operations are τ1 and σ1, σ2, . . ..
I Underlying poset G (〈A1,A2, . . . ,Ak−1, 1〉) orders p-sequences:(p1, p2, . . . , pk−1) ≤ (q1, q2, . . . , qk−1) iff
k−1∑i=j
pi ≤k−1∑i=j
qi
I The operations are the cover relations.
I G (〈2, 2, . . . , 2︸ ︷︷ ︸n
, 1〉) is M(n) (e.g., [Lindstrom][Stanley])
I Is a distributive lattice in general.
I Join irreducibles: A′1A
′2 · · ·A′
i0 · · · 0xA′j+1 · · ·A′
k−1, where0 ≤ x < A′
j := Aj − 1. There are∑
j jA′j of them.
I Is self-dual under the mappingp1 · · · pk−1 7→ (A′
1 − p1) · · · (A′k−1 − pk−1).
![Page 51: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/51.jpg)
Right-frozen case – a poset
I Operations are τ1 and σ1, σ2, . . ..
I Underlying poset G (〈A1,A2, . . . ,Ak−1, 1〉) orders p-sequences:(p1, p2, . . . , pk−1) ≤ (q1, q2, . . . , qk−1) iff
k−1∑i=j
pi ≤k−1∑i=j
qi
I The operations are the cover relations.
I G (〈2, 2, . . . , 2︸ ︷︷ ︸n
, 1〉) is M(n) (e.g., [Lindstrom][Stanley])
I Is a distributive lattice in general.
I Join irreducibles: A′1A
′2 · · ·A′
i0 · · · 0xA′j+1 · · ·A′
k−1, where0 ≤ x < A′
j := Aj − 1. There are∑
j jA′j of them.
I Is self-dual under the mappingp1 · · · pk−1 7→ (A′
1 − p1) · · · (A′k−1 − pk−1).
![Page 52: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/52.jpg)
Right-frozen case – a poset
I Operations are τ1 and σ1, σ2, . . ..
I Underlying poset G (〈A1,A2, . . . ,Ak−1, 1〉) orders p-sequences:(p1, p2, . . . , pk−1) ≤ (q1, q2, . . . , qk−1) iff
k−1∑i=j
pi ≤k−1∑i=j
qi
I The operations are the cover relations.
I G (〈2, 2, . . . , 2︸ ︷︷ ︸n
, 1〉) is M(n) (e.g., [Lindstrom][Stanley])
I Is a distributive lattice in general.
I Join irreducibles: A′1A
′2 · · ·A′
i0 · · · 0xA′j+1 · · ·A′
k−1, where0 ≤ x < A′
j := Aj − 1. There are∑
j jA′j of them.
I Is self-dual under the mappingp1 · · · pk−1 7→ (A′
1 − p1) · · · (A′k−1 − pk−1).
![Page 53: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/53.jpg)
Right-frozen case – a poset
I Operations are τ1 and σ1, σ2, . . ..
I Underlying poset G (〈A1,A2, . . . ,Ak−1, 1〉) orders p-sequences:(p1, p2, . . . , pk−1) ≤ (q1, q2, . . . , qk−1) iff
k−1∑i=j
pi ≤k−1∑i=j
qi
I The operations are the cover relations.
I G (〈2, 2, . . . , 2︸ ︷︷ ︸n
, 1〉) is M(n) (e.g., [Lindstrom][Stanley])
I Is a distributive lattice in general.
I Join irreducibles: A′1A
′2 · · ·A′
i0 · · · 0xA′j+1 · · ·A′
k−1, where0 ≤ x < A′
j := Aj − 1. There are∑
j jA′j of them.
I Is self-dual under the mappingp1 · · · pk−1 7→ (A′
1 − p1) · · · (A′k−1 − pk−1).
![Page 54: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/54.jpg)
Right-frozen case – a poset
I Operations are τ1 and σ1, σ2, . . ..
I Underlying poset G (〈A1,A2, . . . ,Ak−1, 1〉) orders p-sequences:(p1, p2, . . . , pk−1) ≤ (q1, q2, . . . , qk−1) iff
k−1∑i=j
pi ≤k−1∑i=j
qi
I The operations are the cover relations.
I G (〈2, 2, . . . , 2︸ ︷︷ ︸n
, 1〉) is M(n) (e.g., [Lindstrom][Stanley])
I Is a distributive lattice in general.
I Join irreducibles: A′1A
′2 · · ·A′
i0 · · · 0xA′j+1 · · ·A′
k−1, where0 ≤ x < A′
j := Aj − 1. There are∑
j jA′j of them.
I Is self-dual under the mappingp1 · · · pk−1 7→ (A′
1 − p1) · · · (A′k−1 − pk−1).
![Page 55: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/55.jpg)
Right-frozen case – a poset
I Operations are τ1 and σ1, σ2, . . ..
I Underlying poset G (〈A1,A2, . . . ,Ak−1, 1〉) orders p-sequences:(p1, p2, . . . , pk−1) ≤ (q1, q2, . . . , qk−1) iff
k−1∑i=j
pi ≤k−1∑i=j
qi
I The operations are the cover relations.
I G (〈2, 2, . . . , 2︸ ︷︷ ︸n
, 1〉) is M(n) (e.g., [Lindstrom][Stanley])
I Is a distributive lattice in general.
I Join irreducibles: A′1A
′2 · · ·A′
i0 · · · 0xA′j+1 · · ·A′
k−1, where0 ≤ x < A′
j := Aj − 1. There are∑
j jA′j of them.
I Is self-dual under the mappingp1 · · · pk−1 7→ (A′
1 − p1) · · · (A′k−1 − pk−1).
![Page 56: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/56.jpg)
Right-frozen case – a poset
I Operations are τ1 and σ1, σ2, . . ..
I Underlying poset G (〈A1,A2, . . . ,Ak−1, 1〉) orders p-sequences:(p1, p2, . . . , pk−1) ≤ (q1, q2, . . . , qk−1) iff
k−1∑i=j
pi ≤k−1∑i=j
qi
I The operations are the cover relations.
I G (〈2, 2, . . . , 2︸ ︷︷ ︸n
, 1〉) is M(n) (e.g., [Lindstrom][Stanley])
I Is a distributive lattice in general.
I Join irreducibles: A′1A
′2 · · ·A′
i0 · · · 0xA′j+1 · · ·A′
k−1, where0 ≤ x < A′
j := Aj − 1. There are∑
j jA′j of them.
I Is self-dual under the mappingp1 · · · pk−1 7→ (A′
1 − p1) · · · (A′k−1 − pk−1).
![Page 57: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/57.jpg)
Further properties
I The rank of a p-sequence is
r(p) =k−1∑j=1
jpj .
I Rank generating function is
k−1∏j=1
1− z jAj
1− z j
I Since G is the cover graph of a distributive lattice, its prism isHamiltonian ([Pruesse & R]). Thus G 2 is Hamiltonian. Notthe same as moving two cells!
![Page 58: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/58.jpg)
Further properties
I The rank of a p-sequence is
r(p) =k−1∑j=1
jpj .
I Rank generating function is
k−1∏j=1
1− z jAj
1− z j
I Since G is the cover graph of a distributive lattice, its prism isHamiltonian ([Pruesse & R]). Thus G 2 is Hamiltonian. Notthe same as moving two cells!
![Page 59: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/59.jpg)
Further properties
I The rank of a p-sequence is
r(p) =k−1∑j=1
jpj .
I Rank generating function is
k−1∏j=1
1− z jAj
1− z j
I Since G is the cover graph of a distributive lattice, its prism isHamiltonian ([Pruesse & R]). Thus G 2 is Hamiltonian. Notthe same as moving two cells!
![Page 60: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/60.jpg)
LemmaThe graph G (〈2, . . . , 2, 1〉) has a Hamilton path if and only if(n+1
2
)is even and n 6= 5.
Proof.This follows from the results of Savage, Shields, and West.
TheoremFor all n > 0 the graph G (〈2, 2, . . . , 2︸ ︷︷ ︸
n
〉) is Hamiltonian.
Proof.Induction on n with a suitably strengthened hypothesis...
![Page 61: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/61.jpg)
LemmaThe graph G (〈2, . . . , 2, 1〉) has a Hamilton path if and only if(n+1
2
)is even and n 6= 5.
Proof.This follows from the results of Savage, Shields, and West.
TheoremFor all n > 0 the graph G (〈2, 2, . . . , 2︸ ︷︷ ︸
n
〉) is Hamiltonian.
Proof.Induction on n with a suitably strengthened hypothesis...
![Page 62: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/62.jpg)
LemmaThe graph G (〈2, . . . , 2, 1〉) has a Hamilton path if and only if(n+1
2
)is even and n 6= 5.
Proof.This follows from the results of Savage, Shields, and West.
TheoremFor all n > 0 the graph G (〈2, 2, . . . , 2︸ ︷︷ ︸
n
〉) is Hamiltonian.
Proof.Induction on n with a suitably strengthened hypothesis...
![Page 63: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/63.jpg)
LemmaThe graph G (〈2, . . . , 2, 1〉) has a Hamilton path if and only if(n+1
2
)is even and n 6= 5.
Proof.This follows from the results of Savage, Shields, and West.
TheoremFor all n > 0 the graph G (〈2, 2, . . . , 2︸ ︷︷ ︸
n
〉) is Hamiltonian.
Proof.Induction on n with a suitably strengthened hypothesis...
![Page 64: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/64.jpg)
LemmaThere is no Hamilton path in G (〈A1,A2, . . . ,Ak , 1〉) if (1) has thesame parity as A1A2 · · ·Ak .
k∑j=1
j(Aj − 1) =k∑
j=1
jA′j . (1)
Proof.Note that 00 · · · 00 and A′
1A′2 · · ·A′
k0 are pendant vertices in aA1A2 · · ·Ak vertex bipartite graph G and that their distance fromeach other is precisely (1).
![Page 65: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/65.jpg)
LemmaThere is no Hamilton path in G (〈A1,A2, . . . ,Ak , 1〉) if (1) has thesame parity as A1A2 · · ·Ak .
k∑j=1
j(Aj − 1) =k∑
j=1
jA′j . (1)
Proof.Note that 00 · · · 00 and A′
1A′2 · · ·A′
k0 are pendant vertices in aA1A2 · · ·Ak vertex bipartite graph G and that their distance fromeach other is precisely (1).
![Page 66: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/66.jpg)
44
43
42
41
40
40
30
20
10
00
34
00
10
20
30
40
44
34
2443
33 14
34
24
43042342
42 04
14
1332
41 03032241
1231
21 02
11 00
01
0240
01
I The Poset G (〈5, 5, 1〉). Joinirreducibles in yellow.
I The subposet of join-irreducibles.
I Both are self-dual.
I Note: G (〈5, 5, 1〉) notrank-unimodal.
![Page 67: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/67.jpg)
44
43
42
41
40
40
30
20
10
00
34
00
10
20
30
40
44
34
2443
33 14
34
24
43042342
42 04
14
1332
41 03032241
1231
21 02
11 00
01
0240
01
I The Poset G (〈5, 5, 1〉). Joinirreducibles in yellow.
I The subposet of join-irreducibles.
I Both are self-dual.
I Note: G (〈5, 5, 1〉) notrank-unimodal.
![Page 68: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/68.jpg)
44
43
42
41
40
40
30
20
10
00
34
00
10
20
30
40
44
34
2443
33 14
34
24
43042342
42 04
14
1332
41 03032241
1231
21 02
11 00
01
0240
01
I The Poset G (〈5, 5, 1〉). Joinirreducibles in yellow.
I The subposet of join-irreducibles.
I Both are self-dual.
I Note: G (〈5, 5, 1〉) notrank-unimodal.
![Page 69: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/69.jpg)
44
43
42
41
40
40
30
20
10
00
34
00
10
20
30
40
44
34
2443
33 14
34
24
43042342
42 04
14
1332
41 03032241
1231
21 02
11 00
01
0240
01
I The Poset G (〈5, 5, 1〉). Joinirreducibles in yellow.
I The subposet of join-irreducibles.
I Both are self-dual.
I Note: G (〈5, 5, 1〉) notrank-unimodal.
![Page 70: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/70.jpg)
Posets - Join Irreducibles
200 001
122
000
220
221
212 022
202
201
012
022
Expanding
I G (〈3, 3, 3, 1〉).I Ends with 2 or
not.
I G (〈3, 3, 3, 3, 1〉).
![Page 71: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/71.jpg)
Posets - Join Irreducibles
0222
0122
0222
1222
2012
2022
2122
2212
2202
2002
0002
0012
2221
2220 2201
2200 2001
00012000
0000
Expanding
I G (〈3, 3, 3, 1〉).I Ends with 2 or
not.
I G (〈3, 3, 3, 3, 1〉).
![Page 72: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/72.jpg)
Posets - Join Irreducibles
0222
0122
0222
1222
2012
2022
2122
2212
2202
2002
0002
0012
2221
2220 2201
2200 2001
00012000
0000
Expanding
I G (〈3, 3, 3, 1〉).I Ends with 2 or
not.
I G (〈3, 3, 3, 3, 1〉).
![Page 73: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/73.jpg)
More on unimodality
I Let c(S) be the smallest value of k for which P(S i ) is rankunimodal for all i ≥ k.
I Example What is c(2, 3)? The rank gf is(1 + z)(1 + z2 + z4) = 1 + z + z2 + z3 + z4 + z5.
I Ranks of 2, 3, 2:1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 2 2 2 1 1 1
I Ranks of 2, 3, 2, 3 are1 1 1 2 2 2 1 1 1
1 1 1 2 2 2 1 1 11 1 1 2 2 2 1 1 1
1 1 1 2 3 3 2 3 4 3 2 3 3 2 1 1 1which is not unimodal.
I Continuing, P((2, 3)5) is not unimodal, but P((2, 3)k) appearsto be unimodal for k ≥ 6 (checked up to k = 200).
![Page 74: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/74.jpg)
More on unimodality
I Let c(S) be the smallest value of k for which P(S i ) is rankunimodal for all i ≥ k. Well defined?
I Example What is c(2, 3)? The rank gf is(1 + z)(1 + z2 + z4) = 1 + z + z2 + z3 + z4 + z5.
I Ranks of 2, 3, 2:1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 2 2 2 1 1 1
I Ranks of 2, 3, 2, 3 are1 1 1 2 2 2 1 1 1
1 1 1 2 2 2 1 1 11 1 1 2 2 2 1 1 1
1 1 1 2 3 3 2 3 4 3 2 3 3 2 1 1 1which is not unimodal.
I Continuing, P((2, 3)5) is not unimodal, but P((2, 3)k) appearsto be unimodal for k ≥ 6 (checked up to k = 200).
![Page 75: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/75.jpg)
More on unimodality
I Let c(S) be the smallest value of k for which P(S i ) is rankunimodal for all i ≥ k. Well defined?
I Example What is c(2, 3)? The rank gf is(1 + z)(1 + z2 + z4) = 1 + z + z2 + z3 + z4 + z5.
I Ranks of 2, 3, 2:1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 2 2 2 1 1 1
I Ranks of 2, 3, 2, 3 are1 1 1 2 2 2 1 1 1
1 1 1 2 2 2 1 1 11 1 1 2 2 2 1 1 1
1 1 1 2 3 3 2 3 4 3 2 3 3 2 1 1 1which is not unimodal.
I Continuing, P((2, 3)5) is not unimodal, but P((2, 3)k) appearsto be unimodal for k ≥ 6 (checked up to k = 200).
![Page 76: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/76.jpg)
More on unimodality
I Let c(S) be the smallest value of k for which P(S i ) is rankunimodal for all i ≥ k. Well defined?
I Example What is c(2, 3)? The rank gf is(1 + z)(1 + z2 + z4) = 1 + z + z2 + z3 + z4 + z5.
I Ranks of 2, 3, 2:1 1 1 1 1 1◦ ◦ ◦ 1 1 1 1 1 1
1 1 1 2 2 2 1 1 1
I Ranks of 2, 3, 2, 3 are1 1 1 2 2 2 1 1 1
1 1 1 2 2 2 1 1 11 1 1 2 2 2 1 1 1
1 1 1 2 3 3 2 3 4 3 2 3 3 2 1 1 1which is not unimodal.
I Continuing, P((2, 3)5) is not unimodal, but P((2, 3)k) appearsto be unimodal for k ≥ 6 (checked up to k = 200).
![Page 77: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/77.jpg)
More on unimodality
I Let c(S) be the smallest value of k for which P(S i ) is rankunimodal for all i ≥ k. Well defined?
I Example What is c(2, 3)? The rank gf is(1 + z)(1 + z2 + z4) = 1 + z + z2 + z3 + z4 + z5.
I Ranks of 2, 3, 2:1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 2 2 2 1 1 1
I Ranks of 2, 3, 2, 3 are1 1 1 2 2 2 1 1 1◦ ◦ ◦ ◦ 1 1 1 2 2 2 1 1 1
◦ ◦ ◦ ◦ 1 1 1 2 2 2 1 1 1
1 1 1 2 3 3 2 3 4 3 2 3 3 2 1 1 1which is not unimodal.
I Continuing, P((2, 3)5) is not unimodal, but P((2, 3)k) appearsto be unimodal for k ≥ 6 (checked up to k = 200).
![Page 78: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/78.jpg)
c(n, m)2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2 1 6 1 7 1 5 1 7 1 5 1 5 1 5 1 5 1 5 13 7 6 8 8 8 a 8 a 9 a b a b a c c c c c4 4 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 15 6 1 3 4 6 5 6 5 6 6 7 6 7 6 7 6 7 7 76 1 1 1 3 1 1 1 3 1 3 1 3 1 3 1 3 1 3 17 3 1 1 2 4 4 4 4 4 4 5 5 5 6 5 6 5 6 58 3 1 1 3 1 3 1 1 1 3 1 1 1 1 1 1 1 1 19 4 1 1 1 3 3 3 3 4 4 4 4 4 4 4 4 4 4 510 3 1 1 1 1 1 1 3 1 1 1 3 1 1 1 3 1 3 111 3 1 3 1 1 2 4 4 4 4 4 4 4 4 4 4 4 4 412 1 1 1 1 1 3 1 3 1 3 1 3 1 3 1 1 1 3 113 4 1 3 1 3 1 2 2 3 3 3 3 3 3 4 4 4 4 414 4 1 3 3 1 3 1 3 1 3 1 3 1 1 1 3 1 1 115 1 4 4 4 1 4 4 2 4 4 4 4 4 4 4 4 4 4 416 4 1 3 3 1 1 1 3 1 3 1 3 1 3 1 3 1 3 117 4 1 3 3 1 1 3 1 3 2 3 3 3 3 3 3 3 3 318 4 1 1 1 1 1 1 3 1 3 1 3 1 3 1 3 1 3 119 4 1 4 3 1 1 1 1 1 2 3 2 3 3 3 3 4 4 420 3 1 3 3 1 3 1 1 1 3 1 3 1 3 1 3 1 3 1
![Page 79: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/79.jpg)
Remarks about c(m, n)
I Only the 2,2 entry is proven.
I Most entries are checked out to k = 30. The numbers arehuge; i.e., the underlying poset has (nm)30 elements.
I Some conjectures:I For odd m, limn→∞ c(m, n) = cm. In particular, c3 = 16 (the
value 16 first occurs for k = 90).I For even m with an odd factor, for large enough n, if n even
then c(m, n) = 1 and if n is odd then c(m, n) = 3.I If m = 2d then c(m, n) = 1 for n ≥ 2d+1.I For m ≥ 25, c(m, 3) = 1 if 5 | m; otherwise, c(m, 3) = 4.I For m ≥ 15, c(m, 3) = 1, 3, 4 if
m = ({0}, {5, 10}, other) mod 15.
![Page 80: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/80.jpg)
Remarks about c(m, n)
I Only the 2,2 entry is proven.
I Most entries are checked out to k = 30. The numbers arehuge; i.e., the underlying poset has (nm)30 elements.
I Some conjectures:I For odd m, limn→∞ c(m, n) = cm. In particular, c3 = 16 (the
value 16 first occurs for k = 90).I For even m with an odd factor, for large enough n, if n even
then c(m, n) = 1 and if n is odd then c(m, n) = 3.I If m = 2d then c(m, n) = 1 for n ≥ 2d+1.I For m ≥ 25, c(m, 3) = 1 if 5 | m; otherwise, c(m, 3) = 4.I For m ≥ 15, c(m, 3) = 1, 3, 4 if
m = ({0}, {5, 10}, other) mod 15.
![Page 81: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/81.jpg)
Remarks about c(m, n)
I Only the 2,2 entry is proven.
I Most entries are checked out to k = 30. The numbers arehuge; i.e., the underlying poset has (nm)30 elements.
I Some conjectures:I For odd m, limn→∞ c(m, n) = cm. In particular, c3 = 16 (the
value 16 first occurs for k = 90).I For even m with an odd factor, for large enough n, if n even
then c(m, n) = 1 and if n is odd then c(m, n) = 3.I If m = 2d then c(m, n) = 1 for n ≥ 2d+1.I For m ≥ 25, c(m, 3) = 1 if 5 | m; otherwise, c(m, 3) = 4.I For m ≥ 15, c(m, 3) = 1, 3, 4 if
m = ({0}, {5, 10}, other) mod 15.
![Page 82: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/82.jpg)
Remarks about c(m, n)
I Only the 2,2 entry is proven.
I Most entries are checked out to k = 30. The numbers arehuge; i.e., the underlying poset has (nm)30 elements.
I Some conjectures:I For odd m, limn→∞ c(m, n) = cm. In particular, c3 = 16 (the
value 16 first occurs for k = 90).I For even m with an odd factor, for large enough n, if n even
then c(m, n) = 1 and if n is odd then c(m, n) = 3.I If m = 2d then c(m, n) = 1 for n ≥ 2d+1.I For m ≥ 25, c(m, 3) = 1 if 5 | m; otherwise, c(m, 3) = 4.I For m ≥ 15, c(m, 3) = 1, 3, 4 if
m = ({0}, {5, 10}, other) mod 15.
![Page 83: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/83.jpg)
Remarks about c(m, n)
I Only the 2,2 entry is proven.
I Most entries are checked out to k = 30. The numbers arehuge; i.e., the underlying poset has (nm)30 elements.
I Some conjectures:I For odd m, limn→∞ c(m, n) = cm. In particular, c3 = 16 (the
value 16 first occurs for k = 90).I For even m with an odd factor, for large enough n, if n even
then c(m, n) = 1 and if n is odd then c(m, n) = 3.I If m = 2d then c(m, n) = 1 for n ≥ 2d+1.I For m ≥ 25, c(m, 3) = 1 if 5 | m; otherwise, c(m, 3) = 4.I For m ≥ 15, c(m, 3) = 1, 3, 4 if
m = ({0}, {5, 10}, other) mod 15.
![Page 84: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/84.jpg)
Remarks about c(m, n)
I Only the 2,2 entry is proven.
I Most entries are checked out to k = 30. The numbers arehuge; i.e., the underlying poset has (nm)30 elements.
I Some conjectures:I For odd m, limn→∞ c(m, n) = cm. In particular, c3 = 16 (the
value 16 first occurs for k = 90).I For even m with an odd factor, for large enough n, if n even
then c(m, n) = 1 and if n is odd then c(m, n) = 3.I If m = 2d then c(m, n) = 1 for n ≥ 2d+1.I For m ≥ 25, c(m, 3) = 1 if 5 | m; otherwise, c(m, 3) = 4.I For m ≥ 15, c(m, 3) = 1, 3, 4 if
m = ({0}, {5, 10}, other) mod 15.
![Page 85: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/85.jpg)
Remarks about c(m, n)
I Only the 2,2 entry is proven.
I Most entries are checked out to k = 30. The numbers arehuge; i.e., the underlying poset has (nm)30 elements.
I Some conjectures:I For odd m, limn→∞ c(m, n) = cm. In particular, c3 = 16 (the
value 16 first occurs for k = 90).I For even m with an odd factor, for large enough n, if n even
then c(m, n) = 1 and if n is odd then c(m, n) = 3.I If m = 2d then c(m, n) = 1 for n ≥ 2d+1.I For m ≥ 25, c(m, 3) = 1 if 5 | m; otherwise, c(m, 3) = 4.I For m ≥ 15, c(m, 3) = 1, 3, 4 if
m = ({0}, {5, 10}, other) mod 15.
![Page 86: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/86.jpg)
Remarks about c(m, n)
I Only the 2,2 entry is proven.
I Most entries are checked out to k = 30. The numbers arehuge; i.e., the underlying poset has (nm)30 elements.
I Some conjectures:I For odd m, limn→∞ c(m, n) = cm. In particular, c3 = 16 (the
value 16 first occurs for k = 90).I For even m with an odd factor, for large enough n, if n even
then c(m, n) = 1 and if n is odd then c(m, n) = 3.I If m = 2d then c(m, n) = 1 for n ≥ 2d+1.I For m ≥ 25, c(m, 3) = 1 if 5 | m; otherwise, c(m, 3) = 4.I For m ≥ 15, c(m, 3) = 1, 3, 4 if
m = ({0}, {5, 10}, other) mod 15.
![Page 87: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/87.jpg)
Open Problems
I Is the Gray code necessary condition sufficient?
I Remove the BC 6= 6 from a previous theorem.
I Gray codes for general column-convex polyominoes?
I Gray codes for convex polyominoes?
I Hexominoes?
I Prove some of the unimodality results. Perhaps the numbersare asymptotically normal?
![Page 88: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/88.jpg)
Open Problems
I Is the Gray code necessary condition sufficient?
I Remove the BC 6= 6 from a previous theorem.
I Gray codes for general column-convex polyominoes?
I Gray codes for convex polyominoes?
I Hexominoes?
I Prove some of the unimodality results. Perhaps the numbersare asymptotically normal?
![Page 89: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/89.jpg)
Open Problems
I Is the Gray code necessary condition sufficient?
I Remove the BC 6= 6 from a previous theorem.
I Gray codes for general column-convex polyominoes?
I Gray codes for convex polyominoes?
I Hexominoes?
I Prove some of the unimodality results. Perhaps the numbersare asymptotically normal?
![Page 90: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/90.jpg)
Open Problems
I Is the Gray code necessary condition sufficient?
I Remove the BC 6= 6 from a previous theorem.
I Gray codes for general column-convex polyominoes?
I Gray codes for convex polyominoes?
I Hexominoes?
I Prove some of the unimodality results. Perhaps the numbersare asymptotically normal?
![Page 91: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/91.jpg)
Open Problems
I Is the Gray code necessary condition sufficient?
I Remove the BC 6= 6 from a previous theorem.
I Gray codes for general column-convex polyominoes?
I Gray codes for convex polyominoes?
I Hexominoes?
I Prove some of the unimodality results. Perhaps the numbersare asymptotically normal?
![Page 92: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/92.jpg)
Open Problems
I Is the Gray code necessary condition sufficient?
I Remove the BC 6= 6 from a previous theorem.
I Gray codes for general column-convex polyominoes?
I Gray codes for convex polyominoes?
I Hexominoes?
I Prove some of the unimodality results. Perhaps the numbersare asymptotically normal?
![Page 93: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/93.jpg)
Thanks for still being here!
And please let me know if you have seen that poset before...
And on to the Venn diagrams.
![Page 94: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/94.jpg)
Definition - Venn diagram
I Made from simple closedcurves C1,C2, . . . ,Cn.
I Infinite intersection notallowed (usually, but nothere!).
I Let Xi denote the interioror the exterior of the curveCi and considerX1 ∩ X2 ∩ · · · ∩ Xn.
I Euler diagram if each suchintersection is connected.
I Venn diagram if Euler andno intersection is empty.
I Independent family if nointersection is empty.
![Page 95: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/95.jpg)
Definition - Venn diagram
I Made from simple closedcurves C1,C2, . . . ,Cn.
I Infinite intersection notallowed (usually, but nothere!).
I Let Xi denote the interioror the exterior of the curveCi and considerX1 ∩ X2 ∩ · · · ∩ Xn.
I Euler diagram if each suchintersection is connected.
I Venn diagram if Euler andno intersection is empty.
I Independent family if nointersection is empty.
![Page 96: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/96.jpg)
Definition - Venn diagram
I Made from simple closedcurves C1,C2, . . . ,Cn.
I Infinite intersection notallowed (usually, but nothere!).
I Let Xi denote the interioror the exterior of the curveCi and considerX1 ∩ X2 ∩ · · · ∩ Xn.
I Euler diagram if each suchintersection is connected.
I Venn diagram if Euler andno intersection is empty.
I Independent family if nointersection is empty.
![Page 97: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/97.jpg)
Definition - Venn diagram
I Made from simple closedcurves C1,C2, . . . ,Cn.
I Infinite intersection notallowed (usually, but nothere!).
I Let Xi denote the interioror the exterior of the curveCi and considerX1 ∩ X2 ∩ · · · ∩ Xn.
I Euler diagram if each suchintersection is connected.
I Venn diagram if Euler andno intersection is empty.
I Independent family if nointersection is empty.
![Page 98: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/98.jpg)
Definition - Venn diagram
I Made from simple closedcurves C1,C2, . . . ,Cn.
I Infinite intersection notallowed (usually, but nothere!).
I Let Xi denote the interioror the exterior of the curveCi and considerX1 ∩ X2 ∩ · · · ∩ Xn.
I Euler diagram if each suchintersection is connected.
I Venn diagram if Euler andno intersection is empty.
I Independent family if nointersection is empty.
![Page 99: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/99.jpg)
Definition - Venn diagram
I Made from simple closedcurves C1,C2, . . . ,Cn.
I Infinite intersection notallowed (usually, but nothere!).
I Let Xi denote the interioror the exterior of the curveCi and considerX1 ∩ X2 ∩ · · · ∩ Xn.
I Euler diagram if each suchintersection is connected.
I Venn diagram if Euler andno intersection is empty.
I Independent family if nointersection is empty.
![Page 100: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/100.jpg)
“polyVenn” diagrams
I Minimum area Venn diagrams.
I Curves are the boundaries of a polyomino.I Each interior/exterior intersection is a square.
I Total bounded area is 2n − 1.I Each polyomino is a 2n−1-omino.
I Introduced by Mark Thompson on a recreational math webpage.
![Page 101: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/101.jpg)
PolyVenn diagram with congruent curves
B
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
DBA A
AAA
AB
B
B
B
B
B
B
B
B
B B
D
EE
EEEE
E E
E E
EE
B
B
B
(d)
(c)
E
DDDD
D D D D
D D
DD
D DC
E E E
A
B
B
B
B
B
B
B
B
C C C C
C C C C
D
A
A AB B
(a)
A
A
A
A
A
A
AD
C
C
(b)
A A
AA
A A
A A A
C
DD
DD
DDA A
A
A
B
B B
B
C
![Page 102: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/102.jpg)
A CB
D E F
![Page 103: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/103.jpg)
A CB
D E F
A A A
A
A A A
A
A
A A
AA A
A
AA
A A A
A
A A
A
A
AA
AAAAA
![Page 104: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/104.jpg)
A CB
D E F
B B B B
BB B B
B
B
BB
BB
B
B
B
B B B B B
B
B
BBBBBBBB
A A A
A
A A A
A
A
A A
AA A
A
AA
A A A
A
A A
A
A
AA
AAAAA
![Page 105: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/105.jpg)
A CB
D E F
C C C
C
C C
C CCCCC
C
CC
C C C C C C
C
C C
C
C
CC CCC
C
B B B B
BB B B
B
B
BB
BB
B
B
B
B B B B B
B
B
BBBBBBBB
A A A
A
A A A
A
A
A A
AA A
A
AA
A A A
A
A A
A
A
AA
AAAAA
![Page 106: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/106.jpg)
A CB
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D
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![Page 107: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/107.jpg)
A CB
D E F
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A A A
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![Page 108: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/108.jpg)
A CB
D E F
FF
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A A A
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![Page 109: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/109.jpg)
A CB
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Top figure colored bycardinality of the underlyingset. Red = 1, yellow = 2, etc.
![Page 110: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/110.jpg)
A
B
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n = 7.
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In a minimum bounding box
E
E
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![Page 112: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/112.jpg)
A naıve way to construct a polyVenn diagram for any n.
DECECDBEBDBCAEADACABEDCB
ABC
BCDE ACDE ABDE ABCE ABCD CDE BDE BCE BCD ADE ACE ACD ABE ABD
A ABCDE
I A 5-Venn diagram with curve A highlighted.
I Area in general is 322n.
![Page 113: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/113.jpg)
Symmetric chain decomposition of the Boolean lattice
.
ABC
BCACAB
CA
(a)
AD
ADACBCAB
CDBDABD
D
ACD
C
BCD
B
ABC
A
$\emptyset$
ABCD
(b)
$\emptyset$
D
BD
B
ABCD
BCDACDABD
CD
D
AD
ABD
ABCD
A B
BCAB
ABC BCD
ACD
(c)
BD CD
C
AC
I A partition of all 2n
subsets into chains of theform
x1 ⊂ x2 ⊂ · · · ⊂ xt
where |xi | = n − |xt−i+1|and |xi | = |xi−1 − 1|.
I Various algorithms known:
I De Bruijn, vanEbbenhorstTengbergen, andKruyswijk.
I Aigner.
![Page 114: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/114.jpg)
.
ACE
CDE BCE
ABE
ABCE
DEADE
ABDE
ABD
ACDEBCDE
AB
ABC
ABCD
ABCDE
BD
BDE
BECD
E
AE
D
AD
C
AC
ACD
B
BC
BCD
A
CE
I Another 5-Venn diagram withcurve A highlighted.
I Based on symmetric chaindecomposition.
I Area is 2n +( nn/2
).
![Page 115: Polyominoes, Gray Codes, and Venn Diagrams](https://reader033.fdocuments.in/reader033/viewer/2022041507/62525d9012f1c109126140cb/html5/thumbnails/115.jpg)
I Are there congruent n-polyVenns for n ≥ 6? We saw earlierthat they exist for n = 2, 3, 4, 5.
I Is there a 5-polyVenn whose curves are convex polyominoes?
I Are there minimum bounding box n-polyVenns for n ≥ 6? Wesaw earlier that they exist for n = 2, 3, 4, 5.
I Are there minimum area n-polyVenns for n ≥ 8? We sawearlier examples for n = 6, 7.
I One problem for which we have not attempted solutions is theconstruction of n-polyVenns that fill an w × h box, wherewh = 2n − 1. Of course, a necessary condition is that 2n − 1not be a Mersenne prime. For example, is there are4-polyVenn that fits in a 3× 5 rectangle or a 6-polyVenn thatfits in a 7× 9 or 3× 27 rectangle?
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Thanks for coming!