Identifier
Values
[1] => [1,0,1,0] => [3,1,2] => [3,1,2] => 2
[2] => [1,1,0,0,1,0] => [2,4,1,3] => [4,2,1,3] => 2
[1,1] => [1,0,1,1,0,0] => [3,1,4,2] => [4,3,1,2] => 2
[3] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [5,2,3,1,4] => 2
[2,1] => [1,0,1,0,1,0] => [4,1,2,3] => [4,1,2,3] => 2
[1,1,1] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [5,3,1,4,2] => 2
[4] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [6,2,3,4,1,5] => 2
[3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [3,1,5,2,4] => 2
[2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [5,2,4,1,3] => 2
[2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [4,5,1,2,3] => 2
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [6,3,1,4,5,2] => 2
[5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => [7,2,3,4,5,1,6] => 2
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [4,1,6,3,2,5] => 2
[3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [5,2,1,3,4] => 2
[3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [5,3,1,2,4] => 2
[2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [5,1,4,2,3] => 2
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [5,6,1,2,4,3] => 2
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [4,2,1,6,3,5] => 2
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [6,3,4,1,2,5] => 2
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [6,2,3,5,1,4] => 2
[3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [5,1,2,3,4] => 2
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [5,3,1,2,6,4] => 2
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [6,2,4,1,5,3] => 2
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [6,4,5,1,2,3] => 3
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [6,2,3,1,4,5] => 2
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [4,1,2,6,3,5] => 2
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [6,3,1,4,2,5] => 2
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [3,1,6,5,2,4] => 3
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [5,2,6,1,3,4] => 2
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [5,6,1,2,3,4] => 2
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [6,1,4,2,5,3] => 3
[6,1,1] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [6,3,4,5,1,8,2,7] => [8,5,6,3,4,1,2,7] => 2
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [3,1,6,2,4,5] => 2
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [6,2,4,1,3,5] => 2
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [4,6,1,2,3,5] => 2
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [6,2,1,5,3,4] => 2
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [6,3,1,5,2,4] => 3
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [5,1,6,2,3,4] => 2
[5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,4,7,1,5,6] => [7,2,3,4,1,5,6] => 2
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [6,2,1,3,4,5] => 2
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [6,3,1,2,4,5] => 2
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [6,1,4,2,3,5] => 2
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [6,1,2,5,3,4] => 2
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => 2
[5,4,1,1] => [1,1,0,1,1,0,0,0,1,0,1,0] => [4,3,1,7,2,5,6] => [7,3,4,1,2,5,6] => 2
[4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [6,1,7,2,3,4,5] => [7,6,1,2,3,4,5] => 2
[5,4,3] => [1,1,1,0,0,0,1,0,1,0,1,0] => [2,3,7,1,4,5,6] => [7,2,3,1,4,5,6] => 2
[4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [7,1,2,3,6,4,5] => [6,1,2,7,3,4,5] => 2
[5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => [7,2,1,3,4,5,6] => 2
[5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [3,1,7,2,4,5,6] => [7,3,1,2,4,5,6] => 2
[6,5,4] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [2,3,4,8,1,5,6,7] => [8,2,3,4,1,5,6,7] => 2
[6,3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0,1,0] => [3,1,6,5,2,8,4,7] => [8,3,1,5,6,2,4,7] => 3
[5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => 2
[5,4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [8,1,7,2,3,4,5,6] => [7,8,1,2,3,4,5,6] => 2
[] => [] => [1] => [1] => 1
[6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => 2
[6,4,4,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [4,1,2,6,3,8,5,7] => [8,1,4,2,6,3,5,7] => 3
[6,4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,0,1,0] => [8,1,2,6,3,4,5,7] => [6,1,8,2,3,4,5,7] => 2
[6,5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [3,1,8,2,4,5,6,7] => [8,3,1,2,4,5,6,7] => 2
[5,5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [3,1,7,2,4,5,8,6] => [8,3,1,2,4,7,5,6] => 3
[6,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [2,8,1,3,4,5,6,7] => [8,2,1,3,4,5,6,7] => 2
[6,5,4,3] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => [2,3,8,1,4,5,6,7] => [8,2,3,1,4,5,6,7] => 2
[7,6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [9,1,2,3,4,5,6,7,8] => [9,1,2,3,4,5,6,7,8] => 2
[7,6,5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [3,1,9,2,4,5,6,7,8] => [9,3,1,2,4,5,6,7,8] => 2
[6,5,4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [9,1,8,2,3,4,5,6,7] => [8,9,1,2,3,4,5,6,7] => 2
[7,6,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,9,1,3,4,5,6,7,8] => [9,2,1,3,4,5,6,7,8] => 2
[7,6,5,4,3] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => [2,3,9,1,4,5,6,7,8] => [9,2,3,1,4,5,6,7,8] => 2
[8,7,6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [10,1,2,3,4,5,6,7,8,9] => [10,1,2,3,4,5,6,7,8,9] => 2
[7,6,5,4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [9,1,10,2,3,4,5,6,7,8] => [10,9,1,2,3,4,5,6,7,8] => 2
[9,8,7,6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [11,1,2,3,4,5,6,7,8,9,10] => [11,1,2,3,4,5,6,7,8,9,10] => 2
[8,7,6,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,10,1,3,4,5,6,7,8,9] => [10,2,1,3,4,5,6,7,8,9] => 2
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 parts of the shifted shape of a permutation.
The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled.
This statistic records the number of parts of the shifted shape.
Map
Clarke-Steingrimsson-Zeng
Description
The Clarke-Steingrimsson-Zeng map sending descents to excedances.
This is the map $\Phi$ in [1, sec.3]. In particular, it satisfies
$$ (des, Dbot, Ddif, Res)\pi = (exc, Ebot, Edif, Ine)\Phi(\pi), $$
where
  • $des$ is the number of descents, St000021The number of descents of a permutation.,
  • $exc$ is the number of (strict) excedances, St000155The number of exceedances (also excedences) of a permutation.,
  • $Dbot$ is the sum of the descent bottoms, St000154The sum of the descent bottoms of a permutation.,
  • $Ebot$ is the sum of the excedance bottoms,
  • $Ddif$ is the sum of the descent differences, St000030The sum of the descent differences of a permutations.,
  • $Edif$ is the sum of the excedance differences (or depth), St000029The depth of a permutation.,
  • $Res$ is the sum of the (right) embracing numbers,
  • $Ine$ is the sum of the side numbers.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.