Identifier
Values
[1,0] => [1] => [1] => ([],1) => 1
[1,0,1,0] => [1,2] => [2,1] => ([(0,1)],2) => 1
[1,1,0,0] => [2,1] => [1,2] => ([(0,1)],2) => 1
[1,0,1,0,1,0] => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,0,1,1,0,0] => [1,3,2] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,1,0,0,1,0] => [2,1,3] => [3,2,1] => ([(0,2),(2,1)],3) => 1
[1,1,0,1,0,0] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3) => 1
[1,1,1,0,0,0] => [3,1,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[1,1,0,1,1,0,0,0] => [2,4,1,3] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
[1,1,1,0,0,0,1,0] => [3,1,2,4] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[1,1,1,0,0,1,0,0] => [3,1,4,2] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 1
[1,1,1,0,1,0,0,0] => [3,4,1,2] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[1,1,1,1,0,0,0,0] => [4,1,2,3] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 1
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => 1
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => 1
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 1
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8) => 1
[1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => [3,4,5,2,1] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9) => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [2,3,4,5,6,1] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10) => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [2,1,3,4,5,6] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10) => 1
[1,1,0,0,1,1,0,1,0,1,0,0] => [2,1,4,5,6,3] => [3,2,1,4,5,6] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12) => 1
[1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[1,1,1,0,1,0,1,0,1,0,0,0] => [3,4,5,6,1,2] => [6,1,2,3,4,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10) => 1
[1,1,1,1,1,0,0,0,0,0,1,0] => [5,1,2,3,4,6] => [3,4,5,6,2,1] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12) => 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 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
Description
The number of minimal elements in a poset.
Map
pattern poset
Description
The pattern poset of a permutation.
This is the poset of all non-empty permutations that occur in the given permutation as a pattern, ordered by pattern containment.
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
Map
Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $\pi^{-1}c$ where $c = (1,\ldots,n)$ is the long cycle.