Identifier
Values
[1,0] => [1,0] => [[1],[2]] => [2] => 1
[1,0,1,0] => [1,0,1,0] => [[1,3],[2,4]] => [2,2] => 2
[1,1,0,0] => [1,1,0,0] => [[1,2],[3,4]] => [3,1] => 1
[1,0,1,0,1,0] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => [2,2,2] => 3
[1,0,1,1,0,0] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => [2,3,1] => 2
[1,1,0,0,1,0] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => [3,3] => 2
[1,1,0,1,0,0] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => [2,3,1] => 2
[1,1,1,0,0,0] => [1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => [4,2] => 1
[1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => [2,2,2,2] => 4
[1,0,1,0,1,1,0,0] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => [2,2,3,1] => 3
[1,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => [2,3,3] => 3
[1,0,1,1,0,1,0,0] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => [2,2,3,1] => 3
[1,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => [2,4,2] => 2
[1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => [3,3,2] => 2
[1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => [3,4,1] => 2
[1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => [2,3,3] => 3
[1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => [3,4,1] => 2
[1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => [2,4,2] => 2
[1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => [4,4] => 2
[1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => [3,4,1] => 2
[1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => [2,4,2] => 2
[1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => [[1,2,3,4],[5,6,7,8]] => [5,3] => 1
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
Description
The number of weak records in an integer composition.
A weak record is an element $a_i$ such that $a_i \geq a_j$ for all $j < i$.
Map
valley composition
Description
The composition corresponding to the valley set of a standard tableau.
Let $T$ be a standard tableau of size $n$.
An entry $i$ of $T$ is a descent if $i+1$ is in a lower row (in English notation), otherwise $i$ is an ascent.
An entry $2 \leq i \leq n-1$ is a valley if $i-1$ is a descent and $i$ is an ascent.
This map returns the composition $c_1,\dots,c_k$ of $n$ such that $\{c_1, c_1+c_2,\dots, c_1+\dots+c_k\}$ is the valley set of $T$.
Map
bounce path
Description
Sends a Dyck path $D$ of length $2n$ to its bounce path.
This path is formed by starting at the endpoint $(n,n)$ of $D$ and travelling west until encountering the first vertical step of $D$, then south until hitting the diagonal, then west again to hit $D$, etc. until the point $(0,0)$ is reached.
This map is the first part of the zeta map Mp00030zeta map.
Map
to two-row standard tableau
Description
Return a standard tableau of shape $(n,n)$ where $n$ is the semilength of the Dyck path.
Given a Dyck path $D$, its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second row.