Identifier
-
Mp00230:
Integer partitions
—parallelogram polyomino⟶
Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001549: Permutations ⟶ ℤ
Values
[1] => [1,0] => [1] => [1] => 0
[2] => [1,0,1,0] => [2,1] => [1,2] => 0
[1,1] => [1,1,0,0] => [1,2] => [1,2] => 0
[3] => [1,0,1,0,1,0] => [2,3,1] => [1,2,3] => 0
[2,1] => [1,0,1,1,0,0] => [2,1,3] => [1,3,2] => 0
[1,1,1] => [1,1,0,1,0,0] => [3,1,2] => [1,2,3] => 0
[4] => [1,0,1,0,1,0,1,0] => [2,3,4,1] => [1,2,3,4] => 0
[3,1] => [1,0,1,0,1,1,0,0] => [2,3,1,4] => [1,4,2,3] => 0
[2,2] => [1,1,1,0,0,0] => [1,2,3] => [1,2,3] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [2,4,1,3] => [1,3,2,4] => 0
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [3,4,1,2] => [1,2,3,4] => 0
[5] => [1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => [1,5,2,3,4] => 0
[3,2] => [1,0,1,1,1,0,0,0] => [2,1,3,4] => [1,3,4,2] => 0
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => [1,4,2,3,5] => 0
[2,2,1] => [1,1,1,0,0,1,0,0] => [1,4,2,3] => [1,4,2,3] => 0
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => [1,3,2,4,5] => 0
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => [1,2,3,4,5] => 0
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6] => [1,6,2,3,4,5] => 0
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => [1,4,5,2,3] => 1
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [2,3,4,6,1,5] => [1,5,2,3,4,6] => 0
[3,3] => [1,1,1,0,1,0,0,0] => [4,1,2,3] => [1,2,3,4] => 0
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [2,1,5,3,4] => [1,5,2,3,4] => 0
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [2,3,5,6,1,4] => [1,4,2,3,5,6] => 0
[2,2,2] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 0
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => [1,4,5,2,3] => 1
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [2,4,5,6,1,3] => [1,3,2,4,5,6] => 0
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [3,4,5,6,1,2] => [1,2,3,4,5,6] => 0
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => 0
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,6] => [1,5,6,2,3,4] => 1
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => [1,3,4,2,5] => 0
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [2,3,1,6,4,5] => [1,6,2,3,4,5] => 0
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => [1,5,2,3,4] => 0
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => [1,3,4,5,2] => 0
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [2,1,5,6,3,4] => [1,5,6,2,3,4] => 1
[3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [2,3,5,6,7,1,4] => [1,4,2,3,5,6,7] => 0
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [1,4,5,6,2,3] => [1,4,5,6,2,3] => 2
[2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [2,4,5,6,7,1,3] => [1,3,2,4,5,6,7] => 0
[1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [3,4,5,6,7,1,2] => [1,2,3,4,5,6,7] => 0
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [2,3,6,1,4,5] => [1,4,5,2,3,6] => 1
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => [1,2,3,4,5] => 0
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [2,5,1,6,3,4] => [1,6,2,5,3,4] => 0
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6] => [1,4,5,6,2,3] => 2
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => [1,2,3,5,4] => 0
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [4,1,5,6,2,3] => [1,5,6,2,3,4] => 1
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [2,1,3,6,4,5] => [1,3,6,2,4,5] => 0
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => [1,2,3,4,5] => 0
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,2,5,6,3,4] => [1,2,5,6,3,4] => 1
[2,2,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [1,4,5,6,7,2,3] => [1,4,5,6,7,2,3] => 3
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [2,5,6,1,3,4] => [1,3,4,2,5,6] => 0
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [4,5,1,6,2,3] => [1,6,2,3,4,5] => 0
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [2,5,1,3,4,6] => [1,3,4,6,2,5] => 0
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,2,6,3,5] => [1,2,6,3,5,4] => 0
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [2,6,1,3,4,5] => [1,3,4,5,2,6] => 0
[3,2,2,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0] => [2,1,3,6,7,4,5] => [1,3,6,7,2,4,5] => 1
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [5,1,2,6,3,4] => [1,2,6,3,4,5] => 0
[2,2,2,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [1,2,5,6,7,3,4] => [1,2,5,6,7,3,4] => 2
[6,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [2,3,6,7,1,4,5] => [1,4,5,2,3,6,7] => 1
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [4,5,6,1,2,3] => [1,2,3,4,5,6] => 0
[5,3,2] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [2,3,6,1,4,5,7] => [1,4,5,7,2,3,6] => 2
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [4,5,1,2,3,6] => [1,2,3,6,4,5] => 0
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,1,3,4,5,6] => [1,3,4,5,6,2] => 0
[4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [2,5,1,3,7,4,6] => [1,3,7,2,5,4,6] => 0
[4,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [2,3,7,1,4,5,6] => [1,4,5,6,2,3,7] => 2
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,2,3,6,4,5] => [1,2,3,6,4,5] => 0
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [4,6,1,2,3,5] => [1,2,3,5,4,6] => 0
[3,3,2,1,1] => [1,1,1,0,1,1,0,0,0,1,0,1,0,0] => [4,1,2,6,7,3,5] => [1,2,6,7,3,5,4] => 1
[3,2,2,2,1] => [1,0,1,1,1,1,0,1,0,0,0,1,0,0] => [2,6,1,3,7,4,5] => [1,3,7,2,6,4,5] => 0
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [5,6,1,2,3,4] => [1,2,3,4,5,6] => 0
[2,2,2,2,1,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0] => [5,1,2,6,7,3,4] => [1,2,6,7,3,4,5] => 1
[6,5] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [2,5,6,7,1,3,4] => [1,3,4,2,5,6,7] => 0
[5,4,2] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [2,5,6,1,3,4,7] => [1,3,4,7,2,5,6] => 0
[5,3,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,1,4,5,6,7] => [1,4,5,6,7,2,3] => 3
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [4,1,2,3,5,6] => [1,2,3,5,6,4] => 0
[4,4,2,1] => [1,1,1,0,1,0,1,1,0,0,0,1,0,0] => [4,5,1,2,7,3,6] => [1,2,7,3,6,4,5] => 0
[4,3,3,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [2,1,3,4,7,5,6] => [1,3,4,7,2,5,6] => 0
[4,3,2,2] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [2,5,7,1,3,4,6] => [1,3,4,6,2,5,7] => 0
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,6,2,3,4,5] => [1,6,2,3,4,5] => 0
[3,3,3,1,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0] => [1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => 1
[3,3,2,2,1] => [1,1,1,0,1,1,0,1,0,0,0,1,0,0] => [4,6,1,2,7,3,5] => [1,2,7,3,5,4,6] => 0
[3,2,2,2,2] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [2,6,7,1,3,4,5] => [1,3,4,5,2,6,7] => 0
[2,2,2,2,2,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0] => [5,6,1,2,7,3,4] => [1,2,7,3,4,5,6] => 0
[6,6] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [4,5,6,7,1,2,3] => [1,2,3,4,5,6,7] => 0
[5,5,2] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => [4,5,6,1,2,3,7] => [1,2,3,7,4,5,6] => 0
[5,4,3] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [2,5,1,3,4,6,7] => [1,3,4,6,7,2,5] => 1
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [6,1,2,3,4,5] => [1,2,3,4,5,6] => 0
[4,4,3,1] => [1,1,1,0,1,1,1,0,0,0,0,1,0,0] => [4,1,2,3,7,5,6] => [1,2,3,7,4,5,6] => 0
[4,4,2,2] => [1,1,1,0,1,0,1,1,0,1,0,0,0,0] => [4,5,7,1,2,3,6] => [1,2,3,6,4,5,7] => 0
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[3,3,2,2,2] => [1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [4,6,7,1,2,3,5] => [1,2,3,5,4,6,7] => 0
[2,2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [5,6,7,1,2,3,4] => [1,2,3,4,5,6,7] => 0
[5,5,3] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [4,5,1,2,3,6,7] => [1,2,3,6,7,4,5] => 1
[5,4,4] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [2,7,1,3,4,5,6] => [1,3,4,5,6,2,7] => 0
[4,4,4,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0] => [6,1,2,3,7,4,5] => [1,2,3,7,4,5,6] => 0
[4,3,3,3] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,1,3,4,5,6,7] => [1,3,4,5,6,7,2] => 0
[3,3,3,3,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => 0
[5,5,4] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [4,7,1,2,3,5,6] => [1,2,3,5,6,4,7] => 0
[4,4,3,3] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [4,1,2,3,5,6,7] => [1,2,3,5,6,7,4] => 0
>>> Load all 106 entries. <<<
search for individual values
searching the database for the individual values of this statistic
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search for generating function
searching the database for statistics with the same generating function
Description
The number of restricted non-inversions between exceedances.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \}.$$
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \}.$$
Map
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
searching the database
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