Identifier
Values
[1,2] => [1,0,1,0] => [[1,1],[]] => ([(0,1)],2) => 1
[2,1] => [1,1,0,0] => [[2],[]] => ([(0,1)],2) => 1
[1,2,3] => [1,0,1,0,1,0] => [[1,1,1],[]] => ([(0,2),(2,1)],3) => 2
[1,3,2] => [1,0,1,1,0,0] => [[2,1],[]] => ([(0,1),(0,2)],3) => 1
[2,1,3] => [1,1,0,0,1,0] => [[2,2],[1]] => ([(0,2),(1,2)],3) => 1
[2,3,1] => [1,1,0,1,0,0] => [[3],[]] => ([(0,2),(2,1)],3) => 2
[3,1,2] => [1,1,1,0,0,0] => [[2,2],[]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[3,2,1] => [1,1,1,0,0,0] => [[2,2],[]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,2,3,4] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => ([(0,3),(2,1),(3,2)],4) => 3
[1,2,4,3] => [1,0,1,0,1,1,0,0] => [[2,1,1],[]] => ([(0,2),(0,3),(3,1)],4) => 2
[1,3,2,4] => [1,0,1,1,0,0,1,0] => [[2,2,1],[1]] => ([(0,3),(1,2),(1,3)],4) => 2
[1,3,4,2] => [1,0,1,1,0,1,0,0] => [[3,1],[]] => ([(0,2),(0,3),(3,1)],4) => 2
[1,4,2,3] => [1,0,1,1,1,0,0,0] => [[2,2,1],[]] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 3
[1,4,3,2] => [1,0,1,1,1,0,0,0] => [[2,2,1],[]] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 3
[2,1,3,4] => [1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]] => ([(0,3),(1,2),(2,3)],4) => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0] => [[3,2],[1]] => ([(0,3),(1,2),(1,3)],4) => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0] => [[3,3],[2]] => ([(0,3),(1,2),(2,3)],4) => 2
[2,3,4,1] => [1,1,0,1,0,1,0,0] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => 3
[2,4,1,3] => [1,1,0,1,1,0,0,0] => [[3,3],[1]] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => 3
[2,4,3,1] => [1,1,0,1,1,0,0,0] => [[3,3],[1]] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => 3
[3,1,2,4] => [1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => 3
[3,1,4,2] => [1,1,1,0,0,1,0,0] => [[3,2],[]] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 3
[3,2,1,4] => [1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => 3
[3,2,4,1] => [1,1,1,0,0,1,0,0] => [[3,2],[]] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 3
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => 3
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => ([(0,4),(1,2),(1,4),(2,3)],5) => 3
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [[3,1,1],[]] => ([(0,3),(0,4),(3,2),(4,1)],5) => 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => ([(0,3),(1,2),(1,4),(3,4)],5) => 3
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => [[3,2,1],[1]] => ([(0,3),(0,4),(1,2),(1,4)],5) => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => [[3,3,1],[2]] => ([(0,4),(1,2),(1,3),(3,4)],5) => 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => [[4,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => 3
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 3
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0] => [[3,2,2],[1,1]] => ([(0,4),(1,2),(1,3),(3,4)],5) => 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0] => [[3,3,2],[2,1]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0] => [[4,2],[1]] => ([(0,4),(1,2),(1,4),(2,3)],5) => 3
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => [[3,3,3],[2,2]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 3
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0] => [[4,3],[2]] => ([(0,3),(1,2),(1,4),(3,4)],5) => 3
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 3
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
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
Description
The number of points of the poset minus the width of the poset.
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
Map
cell poset
Description
The Young diagram of a skew partition regarded as a poset.
This is the poset on the cells of the Young diagram, such that a cell $d$ is greater than a cell $c$ if the entry in $d$ must be larger than the entry of $c$ in any standard Young tableau on the skew partition.
Map
skew partition
Description
The parallelogram polyomino corresponding to a Dyck path, interpreted as a skew partition.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
This map returns the skew partition definded by the diagram of $\gamma(D)$.