Nomenclature for descending and ascending sets defined by a permutation
I have a question mostly about nomenclature.
Let $pi: mathbb{N} to mathbb{N}$ be a permutation of the natural numbers. For $x in mathbb{N}$, I have studied the sets
$ S_x = left{ x' in mathbb{N}| x' < x + frac{1}{2} , pi(x') > x + frac{1}{2} right} , \
T_x = left{ x' in mathbb{N} | x' > x + frac{1}{2} , pi(x') < x + frac{1}{2} right}$
where the permution ascends above $x$ in for the elements in $S_x$, and descends below $x$ for the elements in $T_x$.
My questions are:
1) Do the sets $S_x, T_x$ have a name in math literature? Are there any books or articles with results about them? (Obviously, $# S_x = # T_x$, but anything more interesting?)
2) I need a symbol for $pi( S_x)$ and $pi(T_x)$, what should I use? I lean towards $S_x^* := pi( T_x )$ and $T_x^* := pi( S_x )$.
permutations
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JAskgaard
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I have a question mostly about nomenclature.
Let $pi: mathbb{N} to mathbb{N}$ be a permutation of the natural numbers. For $x in mathbb{N}$, I have studied the sets
$ S_x = left{ x' in mathbb{N}| x' < x + frac{1}{2} , pi(x') > x + frac{1}{2} right} , \
T_x = left{ x' in mathbb{N} | x' > x + frac{1}{2} , pi(x') < x + frac{1}{2} right}$
where the permution ascends above $x$ in for the elements in $S_x$, and descends below $x$ for the elements in $T_x$.
My questions are:
1) Do the sets $S_x, T_x$ have a name in math literature? Are there any books or articles with results about them? (Obviously, $# S_x = # T_x$, but anything more interesting?)
2) I need a symbol for $pi( S_x)$ and $pi(T_x)$, what should I use? I lean towards $S_x^* := pi( T_x )$ and $T_x^* := pi( S_x )$.
permutations
asked yesterday
JAskgaard
1015
New contributor
JAskgaard is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
I have a question mostly about nomenclature.
Let $pi: mathbb{N} to mathbb{N}$ be a permutation of the natural numbers. For $x in mathbb{N}$, I have studied the sets
$ S_x = left{ x' in mathbb{N}| x' < x + frac{1}{2} , pi(x') > x + frac{1}{2} right} , \
T_x = left{ x' in mathbb{N} | x' > x + frac{1}{2} , pi(x') < x + frac{1}{2} right}$
where the permution ascends above $x$ in for the elements in $S_x$, and descends below $x$ for the elements in $T_x$.
My questions are:
1) Do the sets $S_x, T_x$ have a name in math literature? Are there any books or articles with results about them? (Obviously, $# S_x = # T_x$, but anything more interesting?)
2) I need a symbol for $pi( S_x)$ and $pi(T_x)$, what should I use? I lean towards $S_x^* := pi( T_x )$ and $T_x^* := pi( S_x )$.
permutations
asked yesterday
JAskgaard
1015
New contributor
JAskgaard is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I have a question mostly about nomenclature.
Let $pi: mathbb{N} to mathbb{N}$ be a permutation of the natural numbers. For $x in mathbb{N}$, I have studied the sets
$ S_x = left{ x' in mathbb{N}| x' < x + frac{1}{2} , pi(x') > x + frac{1}{2} right} , \
T_x = left{ x' in mathbb{N} | x' > x + frac{1}{2} , pi(x') < x + frac{1}{2} right}$
where the permution ascends above $x$ in for the elements in $S_x$, and descends below $x$ for the elements in $T_x$.
My questions are:
1) Do the sets $S_x, T_x$ have a name in math literature? Are there any books or articles with results about them? (Obviously, $# S_x = # T_x$, but anything more interesting?)
2) I need a symbol for $pi( S_x)$ and $pi(T_x)$, what should I use? I lean towards $S_x^* := pi( T_x )$ and $T_x^* := pi( S_x )$.
permutations
permutations
asked yesterday
JAskgaard
1015
New contributor
JAskgaard is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
JAskgaard
1015
New contributor
JAskgaard is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
JAskgaard
1015
New contributor
JAskgaard is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
JAskgaard
1015
asked yesterday
JAskgaard
1015
1015
New contributor
JAskgaard is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
JAskgaard is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
JAskgaard is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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