Slice: Difference between revisions
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For <math>n\in{\Bbb Z}_+</math> and <math>(a,b,c)</math> in the triangular grid | |||
:<math>\Delta_n := \{ (a,b,c) \in {\Bbb Z}_+^3: a+b+c = n \},</math> | :<math>\Delta_n := \{ (a,b,c) \in {\Bbb Z}_+^3: a+b+c = n \},</math> | ||
the '''slice''' <math>\Gamma_{a,b,c}</math> is the set of all strings in <math>[3]^n</math> with <math>a</math> occurences of <math>1</math>, <math>b</math> occurences of <math>2</math>, and <math>c</math> occurences of <math>3</math>. Thus, for instance, | |||
:<math>\Gamma_{1,2,0} = \{ 011, 101, 101 \}.</math> | :<math>\Gamma_{1,2,0} = \{ 011, 101, 101 \}.</math> | ||
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The cube <math>[3]^n</math> has <math>\frac{(n+1)(n+2)}{2}</math> slices, and each slice <math>\Gamma_{a,b,c}</math> has cardinality <math>\frac{n!}{a!b!c!}</math>. | The cube <math>[3]^n</math> has <math>\frac{(n+1)(n+2)}{2}</math> slices, and each slice <math>\Gamma_{a,b,c}</math> has cardinality <math>\frac{n!}{a!b!c!}</math>. | ||
The slice <math>\Gamma_{a,b,c}</math> is incident to <math>2^a+2^b+2^c-3</math> [[line|combinatorial lines]]. | |||
The [[equal-slices measure]] gives each slice a total measure of 1. | The [[equal-slices measure]] on <math>[3]^n</math> gives each slice a total measure of <math>1</math>. | ||
The k=2 analogue of a slice | The <math>k=2</math> analogue of a slice, sometimes called a ''layer'', plays a role in the proof of [[Sperner's theorem]]. | ||
A [[combinatorial line]] must touch three slices <math>\Gamma_{a+r,b,c}, \Gamma_{a,b+r,c}, \Gamma_{a,b,c+r}</math> | A [[combinatorial line]] must touch three slices <math>\Gamma_{a+r,b,c}, \Gamma_{a,b+r,c}, \Gamma_{a,b,c+r}</math> corresponding to an equilateral triangle in <math>\Delta_n</math>. Conversely, one might hope that any sufficiently "rich" subsets of three slices <math>\Gamma_{a+r,b,c}, \Gamma_{a,b+r,c}, \Gamma_{a,b,c+r}</math> in an equilateral triangle will have many combinatorial lines between them. However there appear to be quite a lot of obstructions to this hope, consider e.g. the subset of <math>\Gamma_{a+r,b,c}</math> of strings whose first digit is <math>1</math>, and the subsets of <math>\Gamma_{a,b+r,c}, \Gamma_{a,b,c+r}</math> of strings whose first digit is <math>2</math>. | ||
Latest revision as of 12:47, 21 February 2009
For [math]\displaystyle{ n\in{\Bbb Z}_+ }[/math] and [math]\displaystyle{ (a,b,c) }[/math] in the triangular grid
- [math]\displaystyle{ \Delta_n := \{ (a,b,c) \in {\Bbb Z}_+^3: a+b+c = n \}, }[/math]
the slice [math]\displaystyle{ \Gamma_{a,b,c} }[/math] is the set of all strings in [math]\displaystyle{ [3]^n }[/math] with [math]\displaystyle{ a }[/math] occurences of [math]\displaystyle{ 1 }[/math], [math]\displaystyle{ b }[/math] occurences of [math]\displaystyle{ 2 }[/math], and [math]\displaystyle{ c }[/math] occurences of [math]\displaystyle{ 3 }[/math]. Thus, for instance,
- [math]\displaystyle{ \Gamma_{1,2,0} = \{ 011, 101, 101 \}. }[/math]
The cube [math]\displaystyle{ [3]^n }[/math] has [math]\displaystyle{ \frac{(n+1)(n+2)}{2} }[/math] slices, and each slice [math]\displaystyle{ \Gamma_{a,b,c} }[/math] has cardinality [math]\displaystyle{ \frac{n!}{a!b!c!} }[/math].
The slice [math]\displaystyle{ \Gamma_{a,b,c} }[/math] is incident to [math]\displaystyle{ 2^a+2^b+2^c-3 }[/math] combinatorial lines.
The equal-slices measure on [math]\displaystyle{ [3]^n }[/math] gives each slice a total measure of [math]\displaystyle{ 1 }[/math].
The [math]\displaystyle{ k=2 }[/math] analogue of a slice, sometimes called a layer, plays a role in the proof of Sperner's theorem.
A combinatorial line must touch three slices [math]\displaystyle{ \Gamma_{a+r,b,c}, \Gamma_{a,b+r,c}, \Gamma_{a,b,c+r} }[/math] corresponding to an equilateral triangle in [math]\displaystyle{ \Delta_n }[/math]. Conversely, one might hope that any sufficiently "rich" subsets of three slices [math]\displaystyle{ \Gamma_{a+r,b,c}, \Gamma_{a,b+r,c}, \Gamma_{a,b,c+r} }[/math] in an equilateral triangle will have many combinatorial lines between them. However there appear to be quite a lot of obstructions to this hope, consider e.g. the subset of [math]\displaystyle{ \Gamma_{a+r,b,c} }[/math] of strings whose first digit is [math]\displaystyle{ 1 }[/math], and the subsets of [math]\displaystyle{ \Gamma_{a,b+r,c}, \Gamma_{a,b,c+r} }[/math] of strings whose first digit is [math]\displaystyle{ 2 }[/math].
