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
asked Jan 4 at 20:23
JAskgaardJAskgaard
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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 Jan 4 at 20:23
JAskgaardJAskgaard
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 Jan 4 at 20:23
JAskgaardJAskgaard
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 Jan 4 at 20:23
JAskgaardJAskgaard
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 Jan 4 at 20:23
JAskgaardJAskgaard
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 Jan 4 at 20:23
JAskgaardJAskgaard
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 Jan 4 at 20:23
JAskgaardJAskgaard
1015
asked Jan 4 at 20:23
JAskgaardJAskgaard
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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