Identifier
Values
([2],3) => [2] => [1,1,0,0,1,0] => [3,1,2] => 2
([1,1],3) => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 2
([3,1],3) => [3,1] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 4
([2,1,1],3) => [2,1,1] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 4
([4,2],3) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => 6
([3,1,1],3) => [3,1,1] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => 5
([2,2,1,1],3) => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => 6
([5,3,1],3) => [5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => [6,4,2,1,3,5] => 9
([4,2,1,1],3) => [4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => 8
([3,2,2,1,1],3) => [3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => 9
([5,3,1,1],3) => [5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0] => [6,4,2,3,1,5] => 10
([4,2,2,1,1],3) => [4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => 10
([2],4) => [2] => [1,1,0,0,1,0] => [3,1,2] => 2
([1,1],4) => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 2
([3],4) => [3] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 3
([2,1],4) => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 3
([1,1,1],4) => [1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 3
([4,1],4) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => 5
([2,2],4) => [2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 4
([3,1,1],4) => [3,1,1] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => 5
([2,1,1,1],4) => [2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => 5
([5,2],4) => [5,2] => [1,1,1,1,0,0,1,0,0,0,1,0] => [6,3,1,2,4,5] => 7
([4,1,1],4) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => 6
([3,2,1],4) => [3,2,1] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => 6
([3,1,1,1],4) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => 6
([2,2,1,1,1],4) => [2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => 7
([6,3],4) => [6,3] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0] => [7,4,1,2,3,5,6] => 9
([5,2,1],4) => [5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => [6,3,2,1,4,5] => 8
([4,1,1,1],4) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 7
([4,2,2],4) => [4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => 8
([3,3,1,1],4) => [3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => 8
([3,2,1,1,1],4) => [3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => 8
([2,2,2,1,1,1],4) => [2,2,2,1,1,1] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [3,4,5,2,6,7,1] => 9
([2],5) => [2] => [1,1,0,0,1,0] => [3,1,2] => 2
([1,1],5) => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 2
([3],5) => [3] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 3
([2,1],5) => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 3
([1,1,1],5) => [1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 3
([4],5) => [4] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 4
([3,1],5) => [3,1] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 4
([2,2],5) => [2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 4
([2,1,1],5) => [2,1,1] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 4
([1,1,1,1],5) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 4
([5,1],5) => [5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [6,2,1,3,4,5] => 6
([3,2],5) => [3,2] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => 5
([4,1,1],5) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => 6
([2,2,1],5) => [2,2,1] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => 5
([3,1,1,1],5) => [3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => 6
([2,1,1,1,1],5) => [2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => [3,2,4,5,6,1] => 6
([6,2],5) => [6,2] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0] => [7,3,1,2,4,5,6] => 8
([5,1,1],5) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [6,2,3,1,4,5] => 7
([3,3],5) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 6
([4,2,1],5) => [4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => 7
([4,1,1,1],5) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 7
([2,2,2],5) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 6
([3,2,1,1],5) => [3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => 7
([3,1,1,1,1],5) => [3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => [4,2,3,5,6,1] => 7
([2,2,1,1,1,1],5) => [2,2,1,1,1,1] => [1,0,1,1,1,1,0,1,1,0,0,0,0,0] => [3,4,2,5,6,7,1] => 8
([2],6) => [2] => [1,1,0,0,1,0] => [3,1,2] => 2
([1,1],6) => [1,1] => [1,0,1,1,0,0] => [2,3,1] => 2
([3],6) => [3] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 3
([2,1],6) => [2,1] => [1,0,1,0,1,0] => [3,2,1] => 3
([1,1,1],6) => [1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 3
([4],6) => [4] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 4
([3,1],6) => [3,1] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 4
([2,2],6) => [2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 4
([2,1,1],6) => [2,1,1] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 4
([1,1,1,1],6) => [1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 4
([5],6) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => 5
([4,1],6) => [4,1] => [1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => 5
([3,2],6) => [3,2] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => 5
([3,1,1],6) => [3,1,1] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => 5
([2,2,1],6) => [2,2,1] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => 5
([2,1,1,1],6) => [2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => 5
([1,1,1,1,1],6) => [1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 5
([6,1],6) => [6,1] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [7,2,1,3,4,5,6] => 7
([4,2],6) => [4,2] => [1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => 6
([5,1,1],6) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [6,2,3,1,4,5] => 7
([3,3],6) => [3,3] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 6
([3,2,1],6) => [3,2,1] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => 6
([4,1,1,1],6) => [4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 7
([2,2,2],6) => [2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 6
([2,2,1,1],6) => [2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => 6
([3,1,1,1,1],6) => [3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => [4,2,3,5,6,1] => 7
([2,1,1,1,1,1],6) => [2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [3,2,4,5,6,7,1] => 7
([7,2],6) => [7,2] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0] => [8,3,1,2,4,5,6,7] => 9
([6,1,1],6) => [6,1,1] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [7,2,3,1,4,5,6] => 8
([4,3],6) => [4,3] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 7
([5,2,1],6) => [5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => [6,3,2,1,4,5] => 8
([5,1,1,1],6) => [5,1,1,1] => [1,1,0,1,1,1,0,0,0,0,1,0] => [6,2,3,4,1,5] => 8
([3,3,1],6) => [3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => 7
([3,2,2],6) => [3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => 7
([4,2,1,1],6) => [4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => 8
([4,1,1,1,1],6) => [4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [5,2,3,4,6,1] => 8
([2,2,2,1],6) => [2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => 7
([3,2,1,1,1],6) => [3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => 8
([3,1,1,1,1,1],6) => [3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [4,2,3,5,6,7,1] => 8
([2,2,1,1,1,1,1],6) => [2,2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [3,4,2,5,6,7,8,1] => 9
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 inversions of a permutation.
This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
to partition
Description
Considers a core as a partition.
This embedding is graded and injective but not surjective on $k$-cores for a given parameter $k$, while it is surjective and neither graded nor injective on the collection of all cores.