Identifier
Values
[1] => [1,0] => [1,0] => [[1],[]] => 0
[1,2] => [1,0,1,0] => [1,1,0,0] => [[2],[]] => 0
[2,1] => [1,1,0,0] => [1,0,1,0] => [[1,1],[]] => 0
[1,2,3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => [[2,2],[]] => 0
[1,3,2] => [1,0,1,1,0,0] => [1,0,1,1,0,0] => [[2,1],[]] => 0
[2,1,3] => [1,1,0,0,1,0] => [1,1,0,1,0,0] => [[3],[]] => 0
[2,3,1] => [1,1,0,1,0,0] => [1,1,0,0,1,0] => [[2,2],[1]] => 1
[3,1,2] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => [[1,1,1],[]] => 0
[3,2,1] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => [[1,1,1],[]] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0] => [[2,2,1],[]] => 0
[1,3,4,2] => [1,0,1,1,0,1,0,0] => [1,0,1,1,0,1,0,0] => [[3,1],[]] => 0
[1,4,2,3] => [1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => [[2,1,1],[]] => 0
[1,4,3,2] => [1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,0,0] => [[2,1,1],[]] => 0
[2,1,3,4] => [1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,0] => [[3,3],[1]] => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => [[3,2],[1]] => 1
[2,3,1,4] => [1,1,0,1,0,0,1,0] => [1,1,0,1,0,1,0,0] => [[4],[]] => 0
[2,3,4,1] => [1,1,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0] => [[3,3],[2]] => 2
[2,4,1,3] => [1,1,0,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]] => 2
[2,4,3,1] => [1,1,0,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]] => 2
[3,1,2,4] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => [[3,2],[]] => 0
[3,1,4,2] => [1,1,1,0,0,1,0,0] => [1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => 1
[3,2,1,4] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => [[3,2],[]] => 0
[3,2,4,1] => [1,1,1,0,0,1,0,0] => [1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => 1
[3,4,1,2] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => [[2,2,1],[1]] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => [[2,2,1],[1]] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 0
[4,1,3,2] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 0
[4,2,1,3] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 0
[4,2,3,1] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 0
[4,3,1,2] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 0
[4,3,2,1] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => 0
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => [1,0,1,1,0,1,0,1,0,0] => [[4,1],[]] => 0
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,0,0] => [[3,2,1],[1]] => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,0,0] => [[3,2,1],[1]] => 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => [[3,1,1],[]] => 0
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => [[3,1,1],[]] => 0
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => 0
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => 0
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => 0
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => 0
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => 0
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => 0
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => [[4,2],[1]] => 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,1,0,0] => [[3,2,2],[1,1]] => 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,1,0,0] => [[3,2,2],[1,1]] => 2
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => [[4,3],[2]] => 2
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [[5],[]] => 0
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => 3
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => [[3,3,3],[2,2]] => 4
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => [[3,3,3],[2,2]] => 4
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,1,0,0,1,0] => [[3,3,2],[2,1]] => 3
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,1,0,0,1,0] => [[3,3,2],[2,1]] => 3
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => 3
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => 3
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => 3
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => 3
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => 3
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => 3
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => [[3,3,1],[2]] => 2
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => [[3,3,1],[2]] => 2
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => 2
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => 2
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => 2
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => 2
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => 2
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => 2
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => 1
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => 1
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => 1
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => 1
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => 1
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => 1
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 0
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 0
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 0
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 0
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 0
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 0
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 0
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 0
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 0
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 0
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 0
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 0
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 0
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 0
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 0
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 0
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 0
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 0
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 0
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 0
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 0
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 0
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 0
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => 0
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 missing boxes of a skew partition.
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
swap returns and last descent
Description
Return a Dyck path with number of returns and length of the last descent interchanged.
This is the specialisation of the map $\Phi$ in [1] to Dyck paths. It is characterised by the fact that the number of up steps before a down step that is neither a return nor part of the last descent is preserved.
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 $\gamma(D)$.