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Name for a Broad Class of Permutation Statistics

4

1

$begingroup$

Let $X$ be a set of positive integers. A permutation of $X$ is just an ordering of the elements of $X$, which we can write as a word. So the permutations of ${1,2,3}$ are $123, 132, 213, 231, 312, 321$. A decreasing binary plane tree on $X$ is a rooted tree labeled with the elements of $X$ in which each vertex has at most two children, each child is either a right or a left child, the labels are distinct, and every nonroot vertex has a label that is strictly smaller than its parent's label. There is a well-known bijection $T$ between permutations of $X$ and decreasing binary plane trees on $X$. If $pi$ is a permutation, we can write $pi=LnR$, where $n$ is the largest entry in the permutation. Then let $T(pi)$ be the decreasing binary plane tree whose root is labeled with $n$ and whose left and right subtrees are $T(L)$ and $T(R)$, respectively.

We can consider the shape of a decreasing binary plane tree $T$, which is simply the unlabeled binary plane tree obtained by deleting the labels of $T$. Let us say a permutation statistic $f$ is good if $f(pi)$ only depends on the shape of $T(pi)$. For example, the number of descents of $pi$ is the number of right edges in $T(pi)$, so the number of descents is a good statistic. The number of peaks of $pi$ is the number of vertices in $T(pi)$ that have two children, so the number of peaks is also good. Of course, there are also permutation statistics (such as the position of the minimum entry) that are not good.

My question is simply whether there is an actual name appearing already in the literature for "good" permutation statistics.

combinatorics reference-request permutations

share|cite|improve this question

edited Mar 26 at 21:26

Colin Defant

asked Mar 25 at 23:51

Colin DefantColin Defant

717312

$endgroup$

add a comment | 

4

1

$begingroup$

Let $X$ be a set of positive integers. A permutation of $X$ is just an ordering of the elements of $X$, which we can write as a word. So the permutations of ${1,2,3}$ are $123, 132, 213, 231, 312, 321$. A decreasing binary plane tree on $X$ is a rooted tree labeled with the elements of $X$ in which each vertex has at most two children, each child is either a right or a left child, the labels are distinct, and every nonroot vertex has a label that is strictly smaller than its parent's label. There is a well-known bijection $T$ between permutations of $X$ and decreasing binary plane trees on $X$. If $pi$ is a permutation, we can write $pi=LnR$, where $n$ is the largest entry in the permutation. Then let $T(pi)$ be the decreasing binary plane tree whose root is labeled with $n$ and whose left and right subtrees are $T(L)$ and $T(R)$, respectively.

We can consider the shape of a decreasing binary plane tree $T$, which is simply the unlabeled binary plane tree obtained by deleting the labels of $T$. Let us say a permutation statistic $f$ is good if $f(pi)$ only depends on the shape of $T(pi)$. For example, the number of descents of $pi$ is the number of right edges in $T(pi)$, so the number of descents is a good statistic. The number of peaks of $pi$ is the number of vertices in $T(pi)$ that have two children, so the number of peaks is also good. Of course, there are also permutation statistics (such as the position of the minimum entry) that are not good.

My question is simply whether there is an actual name appearing already in the literature for "good" permutation statistics.

combinatorics reference-request permutations

share|cite|improve this question

edited Mar 26 at 21:26

Colin Defant

asked Mar 25 at 23:51

Colin DefantColin Defant

717312

$endgroup$

add a comment | 

$begingroup$

Let $X$ be a set of positive integers. A permutation of $X$ is just an ordering of the elements of $X$, which we can write as a word. So the permutations of ${1,2,3}$ are $123, 132, 213, 231, 312, 321$. A decreasing binary plane tree on $X$ is a rooted tree labeled with the elements of $X$ in which each vertex has at most two children, each child is either a right or a left child, the labels are distinct, and every nonroot vertex has a label that is strictly smaller than its parent's label. There is a well-known bijection $T$ between permutations of $X$ and decreasing binary plane trees on $X$. If $pi$ is a permutation, we can write $pi=LnR$, where $n$ is the largest entry in the permutation. Then let $T(pi)$ be the decreasing binary plane tree whose root is labeled with $n$ and whose left and right subtrees are $T(L)$ and $T(R)$, respectively.

We can consider the shape of a decreasing binary plane tree $T$, which is simply the unlabeled binary plane tree obtained by deleting the labels of $T$. Let us say a permutation statistic $f$ is good if $f(pi)$ only depends on the shape of $T(pi)$. For example, the number of descents of $pi$ is the number of right edges in $T(pi)$, so the number of descents is a good statistic. The number of peaks of $pi$ is the number of vertices in $T(pi)$ that have two children, so the number of peaks is also good. Of course, there are also permutation statistics (such as the position of the minimum entry) that are not good.

My question is simply whether there is an actual name appearing already in the literature for "good" permutation statistics.

combinatorics reference-request permutations

share|cite|improve this question

edited Mar 26 at 21:26

Colin Defant

asked Mar 25 at 23:51

Colin DefantColin Defant

717312

$endgroup$

share|cite|improve this question

edited Mar 26 at 21:26

Colin Defant

asked Mar 25 at 23:51

Colin DefantColin Defant

717312

share|cite|improve this question

edited Mar 26 at 21:26

Colin Defant

asked Mar 25 at 23:51

Colin DefantColin Defant

717312

asked Mar 25 at 23:51

Colin DefantColin Defant

717312

asked Mar 25 at 23:51

Colin DefantColin Defant

717312