Mixing
Mathematical Structure and Objects Induced by Pairs of Disjoint Subsets
Clash Royale CLAN TAG#URR8PPP
up vote
3
down vote
favorite
Let $mathcalS$ be a finite, discrete and non-empty set, i.e.,
$$beginalign cardleft(mathcalSright) & =:ninmathbbN^+\ V& := vneemptyset\
E &:= ucap v=emptyset\ \
implies card(V) & = 2^n-2 \
card(E)&= fracsum_k=1^n-1binomnkleft(2^n-k-1right)2\ & =frac3^n+12-2^nendalign $$
Question:
has the structure induced by the non-empty, disjoint proper subsets of a finite set already appeared explicitly or implicitly in publications?
The reason for asking is that I encountered that structure when trying classify weighted $K_4$ graphs via addition and subtraction of edgeweights; that again in hope of finding criteria for convexity of those weighted $K_4$ that admit an isometric planar embedding.
From the idea, that equality of the weight-sums of disjoint edge-sets represents critical configurations on basis of which exhaustive classification of planar euclidean $K_4$ could be possible, I was lead to the disjoint-subset structure, which generated further ideas for investigation, which I present here as appetizers for others as well:
Have the graphs $G(V,E)$ with $V$ and $E$ defined as above, already been encountered, studied or named?
Can the disjoint-subset structure be meaningfully be defined for discrete or continous sets with transfinite cardinality?
if the elements of the finite set correspond to the coordinates of a cartesian space, i.e. $mathcalS=x^i $, then the set of hyperplanes from the mapping $lbrace(u,v)rbracemapsto lbracesum_x^iin ualpha_i x^i-sum_x^jin valpha_jx^j=0rbrace$ defines a partitioning $mathcalP$of $mathbbR^n$, that is symmetric in the $x^i$, but the connected components of $xinmathbbR^nquad wedgequad |x|=1quad wedgequad sum_x^iin ualpha_i x^i-sum_x^jin valpha_jx^jne 0 forall left(u,vright)in E$ are not all of equal size, leading to the question about the sizes of those connected components
the intersection of $n-1$ of the hyperplanes, as defined above, with the $n-1$ unit-sphere yields a set of points, whose convex hull in turn defines a polytopes; have any of those already been encountered for certain $n>2$
the polytopes as defined provides yield a collection of $m$-dimensional cells, $0le mle n-1$; do their "center points" provide a "useful" discretization of $mathbbR^n$, i.e. one on which proofs can be based?
Here is an example for the hyperplanes generated in $mathbbR^3$ for the values of the variables $x,y,z$ resulting in the planes $x-y=0, y-z=0, z-x=0, x+y-z=0, y+z-x=0, z+x-y=0$:
graph-theory convex-polytopes
edited 36 mins ago
Andrés E. Caicedo
25k595167
asked 3 hours ago
Manfred Weis
4,17321036
add a comment |Â
up vote
3
down vote
favorite
Let $mathcalS$ be a finite, discrete and non-empty set, i.e.,
$$beginalign cardleft(mathcalSright) & =:ninmathbbN^+\ V& := vneemptyset\
E &:= ucap v=emptyset\ \
implies card(V) & = 2^n-2 \
card(E)&= fracsum_k=1^n-1binomnkleft(2^n-k-1right)2\ & =frac3^n+12-2^nendalign $$
Question:
has the structure induced by the non-empty, disjoint proper subsets of a finite set already appeared explicitly or implicitly in publications?
The reason for asking is that I encountered that structure when trying classify weighted $K_4$ graphs via addition and subtraction of edgeweights; that again in hope of finding criteria for convexity of those weighted $K_4$ that admit an isometric planar embedding.
From the idea, that equality of the weight-sums of disjoint edge-sets represents critical configurations on basis of which exhaustive classification of planar euclidean $K_4$ could be possible, I was lead to the disjoint-subset structure, which generated further ideas for investigation, which I present here as appetizers for others as well:
Have the graphs $G(V,E)$ with $V$ and $E$ defined as above, already been encountered, studied or named?
Can the disjoint-subset structure be meaningfully be defined for discrete or continous sets with transfinite cardinality?
if the elements of the finite set correspond to the coordinates of a cartesian space, i.e. $mathcalS=x^i $, then the set of hyperplanes from the mapping $lbrace(u,v)rbracemapsto lbracesum_x^iin ualpha_i x^i-sum_x^jin valpha_jx^j=0rbrace$ defines a partitioning $mathcalP$of $mathbbR^n$, that is symmetric in the $x^i$, but the connected components of $xinmathbbR^nquad wedgequad |x|=1quad wedgequad sum_x^iin ualpha_i x^i-sum_x^jin valpha_jx^jne 0 forall left(u,vright)in E$ are not all of equal size, leading to the question about the sizes of those connected components
the intersection of $n-1$ of the hyperplanes, as defined above, with the $n-1$ unit-sphere yields a set of points, whose convex hull in turn defines a polytopes; have any of those already been encountered for certain $n>2$
the polytopes as defined provides yield a collection of $m$-dimensional cells, $0le mle n-1$; do their "center points" provide a "useful" discretization of $mathbbR^n$, i.e. one on which proofs can be based?
Here is an example for the hyperplanes generated in $mathbbR^3$ for the values of the variables $x,y,z$ resulting in the planes $x-y=0, y-z=0, z-x=0, x+y-z=0, y+z-x=0, z+x-y=0$:
graph-theory convex-polytopes
edited 36 mins ago
Andrés E. Caicedo
25k595167
asked 3 hours ago
Manfred Weis
4,17321036
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Let $mathcalS$ be a finite, discrete and non-empty set, i.e.,
$$beginalign cardleft(mathcalSright) & =:ninmathbbN^+\ V& := vneemptyset\
E &:= ucap v=emptyset\ \
implies card(V) & = 2^n-2 \
card(E)&= fracsum_k=1^n-1binomnkleft(2^n-k-1right)2\ & =frac3^n+12-2^nendalign $$
Question:
has the structure induced by the non-empty, disjoint proper subsets of a finite set already appeared explicitly or implicitly in publications?
The reason for asking is that I encountered that structure when trying classify weighted $K_4$ graphs via addition and subtraction of edgeweights; that again in hope of finding criteria for convexity of those weighted $K_4$ that admit an isometric planar embedding.
From the idea, that equality of the weight-sums of disjoint edge-sets represents critical configurations on basis of which exhaustive classification of planar euclidean $K_4$ could be possible, I was lead to the disjoint-subset structure, which generated further ideas for investigation, which I present here as appetizers for others as well:
Have the graphs $G(V,E)$ with $V$ and $E$ defined as above, already been encountered, studied or named?
Can the disjoint-subset structure be meaningfully be defined for discrete or continous sets with transfinite cardinality?
if the elements of the finite set correspond to the coordinates of a cartesian space, i.e. $mathcalS=x^i $, then the set of hyperplanes from the mapping $lbrace(u,v)rbracemapsto lbracesum_x^iin ualpha_i x^i-sum_x^jin valpha_jx^j=0rbrace$ defines a partitioning $mathcalP$of $mathbbR^n$, that is symmetric in the $x^i$, but the connected components of $xinmathbbR^nquad wedgequad |x|=1quad wedgequad sum_x^iin ualpha_i x^i-sum_x^jin valpha_jx^jne 0 forall left(u,vright)in E$ are not all of equal size, leading to the question about the sizes of those connected components
the intersection of $n-1$ of the hyperplanes, as defined above, with the $n-1$ unit-sphere yields a set of points, whose convex hull in turn defines a polytopes; have any of those already been encountered for certain $n>2$
the polytopes as defined provides yield a collection of $m$-dimensional cells, $0le mle n-1$; do their "center points" provide a "useful" discretization of $mathbbR^n$, i.e. one on which proofs can be based?
Here is an example for the hyperplanes generated in $mathbbR^3$ for the values of the variables $x,y,z$ resulting in the planes $x-y=0, y-z=0, z-x=0, x+y-z=0, y+z-x=0, z+x-y=0$:
graph-theory convex-polytopes
edited 36 mins ago
Andrés E. Caicedo
25k595167
asked 3 hours ago
Manfred Weis
4,17321036
Let $mathcalS$ be a finite, discrete and non-empty set, i.e.,
$$beginalign cardleft(mathcalSright) & =:ninmathbbN^+\ V& := vneemptyset\
E &:= ucap v=emptyset\ \
implies card(V) & = 2^n-2 \
card(E)&= fracsum_k=1^n-1binomnkleft(2^n-k-1right)2\ & =frac3^n+12-2^nendalign $$
Question:
has the structure induced by the non-empty, disjoint proper subsets of a finite set already appeared explicitly or implicitly in publications?
The reason for asking is that I encountered that structure when trying classify weighted $K_4$ graphs via addition and subtraction of edgeweights; that again in hope of finding criteria for convexity of those weighted $K_4$ that admit an isometric planar embedding.
From the idea, that equality of the weight-sums of disjoint edge-sets represents critical configurations on basis of which exhaustive classification of planar euclidean $K_4$ could be possible, I was lead to the disjoint-subset structure, which generated further ideas for investigation, which I present here as appetizers for others as well:
Have the graphs $G(V,E)$ with $V$ and $E$ defined as above, already been encountered, studied or named?
Can the disjoint-subset structure be meaningfully be defined for discrete or continous sets with transfinite cardinality?
if the elements of the finite set correspond to the coordinates of a cartesian space, i.e. $mathcalS=x^i $, then the set of hyperplanes from the mapping $lbrace(u,v)rbracemapsto lbracesum_x^iin ualpha_i x^i-sum_x^jin valpha_jx^j=0rbrace$ defines a partitioning $mathcalP$of $mathbbR^n$, that is symmetric in the $x^i$, but the connected components of $xinmathbbR^nquad wedgequad |x|=1quad wedgequad sum_x^iin ualpha_i x^i-sum_x^jin valpha_jx^jne 0 forall left(u,vright)in E$ are not all of equal size, leading to the question about the sizes of those connected components
the intersection of $n-1$ of the hyperplanes, as defined above, with the $n-1$ unit-sphere yields a set of points, whose convex hull in turn defines a polytopes; have any of those already been encountered for certain $n>2$
the polytopes as defined provides yield a collection of $m$-dimensional cells, $0le mle n-1$; do their "center points" provide a "useful" discretization of $mathbbR^n$, i.e. one on which proofs can be based?
Here is an example for the hyperplanes generated in $mathbbR^3$ for the values of the variables $x,y,z$ resulting in the planes $x-y=0, y-z=0, z-x=0, x+y-z=0, y+z-x=0, z+x-y=0$:
graph-theory convex-polytopes
graph-theory convex-polytopes
edited 36 mins ago
Andrés E. Caicedo
25k595167
asked 3 hours ago
Manfred Weis
4,17321036
edited 36 mins ago
Andrés E. Caicedo
25k595167
asked 3 hours ago
Manfred Weis
4,17321036
edited 36 mins ago
Andrés E. Caicedo
25k595167
edited 36 mins ago
Andrés E. Caicedo
25k595167
edited 36 mins ago
Andrés E. Caicedo
25k595167
25k595167
asked 3 hours ago
Manfred Weis
4,17321036
asked 3 hours ago
Manfred Weis
4,17321036
asked 3 hours ago
Manfred Weis
4,17321036
4,17321036
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
4
down vote
accepted
Yes, this appears explicitly in, for example
Wieder, T. (2008). The number of certain k-combinations of an n-set. Applied Mathematics E-Notes, 8, 45-52. Link.
Plugging in the first few values of $n$ yields OEIS A000392, at which a search for the string "subset" leads to the description (emphasis added):
Let P(A) be the power set of an n-element set A. Then a(n+1) = the number of pairs of elements x,y of P(A) for which x and y are disjoint and for which x is not a subset of y and y is not a subset of x. Wieder calls these "disjoint strict 2-combinations". - Ross La Haye, Jan 11 2008
The name of this OEIS sequence is Stirling numbers of second kind (wiki) and is denoted $S(n,3)$. I imagine that this information could yield significantly more by way of references.
answered 2 hours ago
Benjamin Dickman
4,82523463
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
Yes, this appears explicitly in, for example
Wieder, T. (2008). The number of certain k-combinations of an n-set. Applied Mathematics E-Notes, 8, 45-52. Link.
Plugging in the first few values of $n$ yields OEIS A000392, at which a search for the string "subset" leads to the description (emphasis added):
Let P(A) be the power set of an n-element set A. Then a(n+1) = the number of pairs of elements x,y of P(A) for which x and y are disjoint and for which x is not a subset of y and y is not a subset of x. Wieder calls these "disjoint strict 2-combinations". - Ross La Haye, Jan 11 2008
The name of this OEIS sequence is Stirling numbers of second kind (wiki) and is denoted $S(n,3)$. I imagine that this information could yield significantly more by way of references.
answered 2 hours ago
Benjamin Dickman
4,82523463
add a comment |Â
up vote
4
down vote
accepted
Yes, this appears explicitly in, for example
Wieder, T. (2008). The number of certain k-combinations of an n-set. Applied Mathematics E-Notes, 8, 45-52. Link.
Plugging in the first few values of $n$ yields OEIS A000392, at which a search for the string "subset" leads to the description (emphasis added):
Let P(A) be the power set of an n-element set A. Then a(n+1) = the number of pairs of elements x,y of P(A) for which x and y are disjoint and for which x is not a subset of y and y is not a subset of x. Wieder calls these "disjoint strict 2-combinations". - Ross La Haye, Jan 11 2008
The name of this OEIS sequence is Stirling numbers of second kind (wiki) and is denoted $S(n,3)$. I imagine that this information could yield significantly more by way of references.
answered 2 hours ago
Benjamin Dickman
4,82523463
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
Yes, this appears explicitly in, for example
Wieder, T. (2008). The number of certain k-combinations of an n-set. Applied Mathematics E-Notes, 8, 45-52. Link.
Plugging in the first few values of $n$ yields OEIS A000392, at which a search for the string "subset" leads to the description (emphasis added):
Let P(A) be the power set of an n-element set A. Then a(n+1) = the number of pairs of elements x,y of P(A) for which x and y are disjoint and for which x is not a subset of y and y is not a subset of x. Wieder calls these "disjoint strict 2-combinations". - Ross La Haye, Jan 11 2008
The name of this OEIS sequence is Stirling numbers of second kind (wiki) and is denoted $S(n,3)$. I imagine that this information could yield significantly more by way of references.
answered 2 hours ago
Benjamin Dickman
4,82523463
Yes, this appears explicitly in, for example
Wieder, T. (2008). The number of certain k-combinations of an n-set. Applied Mathematics E-Notes, 8, 45-52. Link.
Plugging in the first few values of $n$ yields OEIS A000392, at which a search for the string "subset" leads to the description (emphasis added):
Let P(A) be the power set of an n-element set A. Then a(n+1) = the number of pairs of elements x,y of P(A) for which x and y are disjoint and for which x is not a subset of y and y is not a subset of x. Wieder calls these "disjoint strict 2-combinations". - Ross La Haye, Jan 11 2008
The name of this OEIS sequence is Stirling numbers of second kind (wiki) and is denoted $S(n,3)$. I imagine that this information could yield significantly more by way of references.
answered 2 hours ago
Benjamin Dickman
4,82523463
answered 2 hours ago
Benjamin Dickman
4,82523463
answered 2 hours ago
Benjamin Dickman
4,82523463
answered 2 hours ago
Benjamin Dickman
4,82523463
4,82523463
add a comment |Â
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f310635%2fmathematical-structure-and-objects-induced-by-pairs-of-disjoint-subsets%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
