Identifier
Values
[1,0] => [[1],[2]] => [2,1] => [2,1] => 0
[1,0,1,0] => [[1,3],[2,4]] => [2,4,1,3] => [3,4,1,2] => 0
[1,1,0,0] => [[1,2],[3,4]] => [3,4,1,2] => [2,4,1,3] => 0
[1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => [2,4,6,1,3,5] => [3,5,6,1,2,4] => 0
[1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => [3,4,6,1,2,5] => 0
[1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => [3,4,6,1,2,5] => [2,5,6,1,3,4] => 0
[1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => [3,5,6,1,2,4] => [2,4,6,1,3,5] => 0
[1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [2,3,6,1,4,5] => 0
[1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => [2,4,6,8,1,3,5,7] => [3,5,7,8,1,2,4,6] => 0
[1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => [2,4,7,8,1,3,5,6] => [3,5,6,8,1,2,4,7] => 0
[1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => [2,5,6,8,1,3,4,7] => [3,4,7,8,1,2,5,6] => 0
[1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => [2,5,7,8,1,3,4,6] => [3,4,6,8,1,2,5,7] => 0
[1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => [3,4,6,8,1,2,5,7] => [2,5,7,8,1,3,4,6] => 0
[1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => [3,4,7,8,1,2,5,6] => [2,5,6,8,1,3,4,7] => 0
[1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => [3,5,6,8,1,2,4,7] => [2,4,7,8,1,3,5,6] => 0
[1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => [3,5,7,8,1,2,4,6] => [2,4,6,8,1,3,5,7] => 0
[1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => [2,4,6,8,10,1,3,5,7,9] => [3,5,7,9,10,1,2,4,6,8] => 0
[1,0,1,0,1,0,1,1,0,0] => [[1,3,5,7,8],[2,4,6,9,10]] => [2,4,6,9,10,1,3,5,7,8] => [3,5,7,8,10,1,2,4,6,9] => 0
[1,0,1,0,1,1,0,0,1,0] => [[1,3,5,6,9],[2,4,7,8,10]] => [2,4,7,8,10,1,3,5,6,9] => [3,5,6,9,10,1,2,4,7,8] => 0
[1,0,1,0,1,1,0,1,0,0] => [[1,3,5,6,8],[2,4,7,9,10]] => [2,4,7,9,10,1,3,5,6,8] => [3,5,6,8,10,1,2,4,7,9] => 0
[1,0,1,1,0,0,1,0,1,0] => [[1,3,4,7,9],[2,5,6,8,10]] => [2,5,6,8,10,1,3,4,7,9] => [3,4,7,9,10,1,2,5,6,8] => 0
[1,0,1,1,0,0,1,1,0,0] => [[1,3,4,7,8],[2,5,6,9,10]] => [2,5,6,9,10,1,3,4,7,8] => [3,4,7,8,10,1,2,5,6,9] => 0
[1,0,1,1,0,1,0,0,1,0] => [[1,3,4,6,9],[2,5,7,8,10]] => [2,5,7,8,10,1,3,4,6,9] => [3,4,6,9,10,1,2,5,7,8] => 0
[1,0,1,1,0,1,0,1,0,0] => [[1,3,4,6,8],[2,5,7,9,10]] => [2,5,7,9,10,1,3,4,6,8] => [3,4,6,8,10,1,2,5,7,9] => 0
[1,1,0,0,1,0,1,0,1,0] => [[1,2,5,7,9],[3,4,6,8,10]] => [3,4,6,8,10,1,2,5,7,9] => [2,5,7,9,10,1,3,4,6,8] => 0
[1,1,0,0,1,0,1,1,0,0] => [[1,2,5,7,8],[3,4,6,9,10]] => [3,4,6,9,10,1,2,5,7,8] => [2,5,7,8,10,1,3,4,6,9] => 0
[1,1,0,0,1,1,0,0,1,0] => [[1,2,5,6,9],[3,4,7,8,10]] => [3,4,7,8,10,1,2,5,6,9] => [2,5,6,9,10,1,3,4,7,8] => 0
[1,1,0,0,1,1,0,1,0,0] => [[1,2,5,6,8],[3,4,7,9,10]] => [3,4,7,9,10,1,2,5,6,8] => [2,5,6,8,10,1,3,4,7,9] => 0
[1,1,0,1,0,0,1,0,1,0] => [[1,2,4,7,9],[3,5,6,8,10]] => [3,5,6,8,10,1,2,4,7,9] => [2,4,7,9,10,1,3,5,6,8] => 0
[1,1,0,1,0,0,1,1,0,0] => [[1,2,4,7,8],[3,5,6,9,10]] => [3,5,6,9,10,1,2,4,7,8] => [2,4,7,8,10,1,3,5,6,9] => 0
[1,1,0,1,0,1,0,0,1,0] => [[1,2,4,6,9],[3,5,7,8,10]] => [3,5,7,8,10,1,2,4,6,9] => [2,4,6,9,10,1,3,5,7,8] => 0
[1,1,0,1,0,1,0,1,0,0] => [[1,2,4,6,8],[3,5,7,9,10]] => [3,5,7,9,10,1,2,4,6,8] => [2,4,6,8,10,1,3,5,7,9] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => [2,4,6,8,10,12,1,3,5,7,9,11] => [3,5,7,9,11,12,1,2,4,6,8,10] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [[1,3,5,7,9,10],[2,4,6,8,11,12]] => [2,4,6,8,11,12,1,3,5,7,9,10] => [3,5,7,9,10,12,1,2,4,6,8,11] => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [[1,3,5,7,8,11],[2,4,6,9,10,12]] => [2,4,6,9,10,12,1,3,5,7,8,11] => [3,5,7,8,11,12,1,2,4,6,9,10] => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [[1,3,5,7,8,10],[2,4,6,9,11,12]] => [2,4,6,9,11,12,1,3,5,7,8,10] => [3,5,7,8,10,12,1,2,4,6,9,11] => 0
[1,0,1,0,1,1,0,0,1,0,1,0] => [[1,3,5,6,9,11],[2,4,7,8,10,12]] => [2,4,7,8,10,12,1,3,5,6,9,11] => [3,5,6,9,11,12,1,2,4,7,8,10] => 0
[1,0,1,0,1,1,0,0,1,1,0,0] => [[1,3,5,6,9,10],[2,4,7,8,11,12]] => [2,4,7,8,11,12,1,3,5,6,9,10] => [3,5,6,9,10,12,1,2,4,7,8,11] => 0
[1,0,1,0,1,1,0,1,0,0,1,0] => [[1,3,5,6,8,11],[2,4,7,9,10,12]] => [2,4,7,9,10,12,1,3,5,6,8,11] => [3,5,6,8,11,12,1,2,4,7,9,10] => 0
[1,0,1,0,1,1,0,1,0,1,0,0] => [[1,3,5,6,8,10],[2,4,7,9,11,12]] => [2,4,7,9,11,12,1,3,5,6,8,10] => [3,5,6,8,10,12,1,2,4,7,9,11] => 0
[1,0,1,1,0,0,1,0,1,0,1,0] => [[1,3,4,7,9,11],[2,5,6,8,10,12]] => [2,5,6,8,10,12,1,3,4,7,9,11] => [3,4,7,9,11,12,1,2,5,6,8,10] => 0
[1,0,1,1,0,0,1,0,1,1,0,0] => [[1,3,4,7,9,10],[2,5,6,8,11,12]] => [2,5,6,8,11,12,1,3,4,7,9,10] => [3,4,7,9,10,12,1,2,5,6,8,11] => 0
[1,0,1,1,0,0,1,1,0,0,1,0] => [[1,3,4,7,8,11],[2,5,6,9,10,12]] => [2,5,6,9,10,12,1,3,4,7,8,11] => [3,4,7,8,11,12,1,2,5,6,9,10] => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [[1,3,4,7,8,10],[2,5,6,9,11,12]] => [2,5,6,9,11,12,1,3,4,7,8,10] => [3,4,7,8,10,12,1,2,5,6,9,11] => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [[1,3,4,6,9,11],[2,5,7,8,10,12]] => [2,5,7,8,10,12,1,3,4,6,9,11] => [3,4,6,9,11,12,1,2,5,7,8,10] => 0
[1,0,1,1,0,1,0,0,1,1,0,0] => [[1,3,4,6,9,10],[2,5,7,8,11,12]] => [2,5,7,8,11,12,1,3,4,6,9,10] => [3,4,6,9,10,12,1,2,5,7,8,11] => 0
[1,0,1,1,0,1,0,1,0,0,1,0] => [[1,3,4,6,8,11],[2,5,7,9,10,12]] => [2,5,7,9,10,12,1,3,4,6,8,11] => [3,4,6,8,11,12,1,2,5,7,9,10] => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [[1,3,4,6,8,10],[2,5,7,9,11,12]] => [2,5,7,9,11,12,1,3,4,6,8,10] => [3,4,6,8,10,12,1,2,5,7,9,11] => 0
[1,1,0,0,1,0,1,0,1,0,1,0] => [[1,2,5,7,9,11],[3,4,6,8,10,12]] => [3,4,6,8,10,12,1,2,5,7,9,11] => [2,5,7,9,11,12,1,3,4,6,8,10] => 0
[1,1,0,0,1,0,1,0,1,1,0,0] => [[1,2,5,7,9,10],[3,4,6,8,11,12]] => [3,4,6,8,11,12,1,2,5,7,9,10] => [2,5,7,9,10,12,1,3,4,6,8,11] => 0
[1,1,0,0,1,0,1,1,0,0,1,0] => [[1,2,5,7,8,11],[3,4,6,9,10,12]] => [3,4,6,9,10,12,1,2,5,7,8,11] => [2,5,7,8,11,12,1,3,4,6,9,10] => 0
[1,1,0,0,1,0,1,1,0,1,0,0] => [[1,2,5,7,8,10],[3,4,6,9,11,12]] => [3,4,6,9,11,12,1,2,5,7,8,10] => [2,5,7,8,10,12,1,3,4,6,9,11] => 0
[1,1,0,0,1,1,0,0,1,0,1,0] => [[1,2,5,6,9,11],[3,4,7,8,10,12]] => [3,4,7,8,10,12,1,2,5,6,9,11] => [2,5,6,9,11,12,1,3,4,7,8,10] => 0
[1,1,0,0,1,1,0,0,1,1,0,0] => [[1,2,5,6,9,10],[3,4,7,8,11,12]] => [3,4,7,8,11,12,1,2,5,6,9,10] => [2,5,6,9,10,12,1,3,4,7,8,11] => 0
[1,1,0,0,1,1,0,1,0,0,1,0] => [[1,2,5,6,8,11],[3,4,7,9,10,12]] => [3,4,7,9,10,12,1,2,5,6,8,11] => [2,5,6,8,11,12,1,3,4,7,9,10] => 0
[1,1,0,0,1,1,0,1,0,1,0,0] => [[1,2,5,6,8,10],[3,4,7,9,11,12]] => [3,4,7,9,11,12,1,2,5,6,8,10] => [2,5,6,8,10,12,1,3,4,7,9,11] => 0
[1,1,0,1,0,0,1,0,1,0,1,0] => [[1,2,4,7,9,11],[3,5,6,8,10,12]] => [3,5,6,8,10,12,1,2,4,7,9,11] => [2,4,7,9,11,12,1,3,5,6,8,10] => 0
[1,1,0,1,0,0,1,0,1,1,0,0] => [[1,2,4,7,9,10],[3,5,6,8,11,12]] => [3,5,6,8,11,12,1,2,4,7,9,10] => [2,4,7,9,10,12,1,3,5,6,8,11] => 0
[1,1,0,1,0,0,1,1,0,0,1,0] => [[1,2,4,7,8,11],[3,5,6,9,10,12]] => [3,5,6,9,10,12,1,2,4,7,8,11] => [2,4,7,8,11,12,1,3,5,6,9,10] => 0
[1,1,0,1,0,0,1,1,0,1,0,0] => [[1,2,4,7,8,10],[3,5,6,9,11,12]] => [3,5,6,9,11,12,1,2,4,7,8,10] => [2,4,7,8,10,12,1,3,5,6,9,11] => 0
[1,1,0,1,0,1,0,0,1,0,1,0] => [[1,2,4,6,9,11],[3,5,7,8,10,12]] => [3,5,7,8,10,12,1,2,4,6,9,11] => [2,4,6,9,11,12,1,3,5,7,8,10] => 0
[1,1,0,1,0,1,0,0,1,1,0,0] => [[1,2,4,6,9,10],[3,5,7,8,11,12]] => [3,5,7,8,11,12,1,2,4,6,9,10] => [2,4,6,9,10,12,1,3,5,7,8,11] => 0
[1,1,0,1,0,1,0,1,0,0,1,0] => [[1,2,4,6,8,11],[3,5,7,9,10,12]] => [3,5,7,9,10,12,1,2,4,6,8,11] => [2,4,6,8,11,12,1,3,5,7,9,10] => 0
[1,1,0,1,0,1,0,1,0,1,0,0] => [[1,2,4,6,8,10],[3,5,7,9,11,12]] => [3,5,7,9,11,12,1,2,4,6,8,10] => [2,4,6,8,10,12,1,3,5,7,9,11] => 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 difference in Coxeter length of a permutation and its image under the Simion-Schmidt map.
The Simion-Schmidt map takes a permutation and turns each occcurrence of [3,2,1] into an occurrence of [3,1,2], thus reducing the number of inversions of the permutation. This statistic records the difference in length of the permutation and its image.
Apparently, this statistic can be described as the number of occurrences of the mesh pattern ([3,2,1], {(0,3),(0,2)}). Equivalent mesh patterns are ([3,2,1], {(0,2),(1,2)}), ([3,2,1], {(0,3),(1,3)}) and ([3,2,1], {(1,2),(1,3)}).
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
Map
invert Laguerre heap
Description
The permutation obtained by inverting the corresponding Laguerre heap, according to Viennot.
Let $\pi$ be a permutation. Following Viennot [1], we associate to $\pi$ a heap of pieces, by considering each decreasing run $(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of $\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.
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.