Identifier
-
Mp00230:
Integer partitions
—parallelogram polyomino⟶
Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000461: Permutations ⟶ ℤ
Values
[2] => [1,0,1,0] => [1,2] => [1,2] => 2
[1,1] => [1,1,0,0] => [2,1] => [1,2] => 2
[3] => [1,0,1,0,1,0] => [1,2,3] => [1,2,3] => 3
[2,1] => [1,0,1,1,0,0] => [1,3,2] => [1,2,3] => 3
[1,1,1] => [1,1,0,1,0,0] => [2,3,1] => [1,2,3] => 3
[4] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => 4
[3,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,3,4] => 4
[2,2] => [1,1,1,0,0,0] => [3,2,1] => [1,3,2] => 1
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,2,3,4] => 4
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => [1,2,3,4] => 4
[5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => 5
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,4,5] => 5
[3,2] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => [1,2,4,3] => 1
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,3,4,5] => 5
[2,2,1] => [1,1,1,0,0,1,0,0] => [3,2,4,1] => [1,3,4,2] => 1
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,2,3,4,5] => 5
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [1,2,3,4,5] => 5
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [1,2,3,4,5,6] => 6
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [1,2,3,5,4] => 1
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [1,2,3,4,5,6] => 6
[3,3] => [1,1,1,0,1,0,0,0] => [4,2,3,1] => [1,4,2,3] => 2
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [1,2,4,5,3] => 1
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [1,2,3,4,5,6] => 6
[2,2,2] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => [1,4,2,3] => 2
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [1,3,4,5,2] => 1
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [1,2,3,4,5,6] => 6
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 6
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 7
[6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => 7
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => [1,2,3,4,6,5] => 1
[5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,4,6,7,5] => [1,2,3,4,5,6,7] => 7
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,5,3,4,2] => [1,2,5,3,4] => 2
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => [1,2,3,5,6,4] => 1
[4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,3,5,6,7,4] => [1,2,3,4,5,6,7] => 7
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [4,2,3,5,1] => [1,4,5,2,3] => 2
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [1,2,5,3,4] => 2
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => [1,2,4,5,6,3] => 1
[3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,2,4,5,6,7,3] => [1,2,3,4,5,6,7] => 7
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [1,4,5,2,3] => 2
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [3,2,4,5,6,1] => [1,3,4,5,6,2] => 1
[2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,7,2] => [1,2,3,4,5,6,7] => 7
[1,1,1,1,1,1,1] => [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] => 7
[6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,4,7,6,5] => [1,2,3,4,5,7,6] => 1
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,6,4,5,3] => [1,2,3,6,4,5] => 2
[5,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,3,6,5,7,4] => [1,2,3,4,6,7,5] => 1
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1] => [1,5,2,3,4] => 3
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,5,3,4,6,2] => [1,2,5,6,3,4] => 2
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => [1,2,3,6,4,5] => 2
[4,2,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [1,2,5,4,6,7,3] => [1,2,3,5,6,7,4] => 1
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => [1,5,2,3,4] => 3
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [4,2,3,5,6,1] => [1,4,5,6,2,3] => 2
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,4,3,6,2] => [1,2,5,6,3,4] => 2
[3,2,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [1,4,3,5,6,7,2] => [1,2,4,5,6,7,3] => 1
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => [1,5,2,3,4] => 3
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [4,3,2,5,6,1] => [1,4,5,6,2,3] => 2
[2,2,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [3,2,4,5,6,7,1] => [1,3,4,5,6,7,2] => 1
[6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,3,7,5,6,4] => [1,2,3,4,7,5,6] => 2
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,6,3,4,5,2] => [1,2,6,3,4,5] => 3
[5,3,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [1,2,6,4,5,7,3] => [1,2,3,6,7,4,5] => 2
[5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,3,7,6,5,4] => [1,2,3,4,7,5,6] => 2
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [5,2,3,4,6,1] => [1,5,6,2,3,4] => 3
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,6,3,5,4,2] => [1,2,6,3,4,5] => 3
[4,3,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [1,5,3,4,6,7,2] => [1,2,5,6,7,3,4] => 2
[4,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [1,2,6,5,4,7,3] => [1,2,3,6,7,4,5] => 2
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [1,5,2,4,3] => 1
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,4,3,6,1] => [1,5,6,2,3,4] => 3
[3,3,1,1,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0] => [4,2,3,5,6,7,1] => [1,4,5,6,7,2,3] => 2
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,6,4,5,3,2] => [1,2,6,3,4,5] => 3
[3,2,2,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0] => [1,5,4,3,6,7,2] => [1,2,5,6,7,3,4] => 2
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [5,3,4,2,6,1] => [1,5,6,2,3,4] => 3
[2,2,2,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [4,3,2,5,6,7,1] => [1,4,5,6,7,2,3] => 2
[6,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [1,2,7,4,5,6,3] => [1,2,3,7,4,5,6] => 3
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [6,2,3,4,5,1] => [1,6,2,3,4,5] => 4
[5,4,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => [1,6,3,4,5,7,2] => [1,2,6,7,3,4,5] => 3
[5,3,2] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [1,2,7,4,6,5,3] => [1,2,3,7,4,5,6] => 3
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [6,2,3,5,4,1] => [1,6,2,3,4,5] => 4
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => [1,2,6,3,5,4] => 1
[4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [1,6,3,5,4,7,2] => [1,2,6,7,3,4,5] => 3
[4,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [1,2,7,5,6,4,3] => [1,2,3,7,4,5,6] => 3
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [5,4,3,2,6,1] => [1,5,6,2,4,3] => 1
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [6,2,4,5,3,1] => [1,6,2,3,4,5] => 4
[3,2,2,2,1] => [1,0,1,1,1,1,0,1,0,0,0,1,0,0] => [1,6,4,5,3,7,2] => [1,2,6,7,3,4,5] => 3
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [6,3,4,5,2,1] => [1,6,2,3,4,5] => 4
[6,5] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [1,7,3,4,5,6,2] => [1,2,7,3,4,5,6] => 4
[5,4,2] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [1,7,3,4,6,5,2] => [1,2,7,3,4,5,6] => 4
[5,3,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [1,2,7,6,5,4,3] => [1,2,3,7,4,6,5] => 1
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [6,2,5,4,3,1] => [1,6,2,3,5,4] => 1
[4,3,3,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [1,6,5,4,3,7,2] => [1,2,6,7,3,5,4] => 1
[4,3,2,2] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [1,7,3,5,6,4,2] => [1,2,7,3,4,5,6] => 4
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [6,4,3,5,2,1] => [1,6,2,4,5,3] => 1
[3,2,2,2,2] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [1,7,4,5,6,3,2] => [1,2,7,3,4,5,6] => 4
[5,4,3] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [1,7,3,6,5,4,2] => [1,2,7,3,4,6,5] => 1
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [6,5,3,4,2,1] => [1,6,2,5,3,4] => 2
[4,3,3,2] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [1,7,5,4,6,3,2] => [1,2,7,3,5,6,4] => 1
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [6,5,4,3,2,1] => [1,6,2,5,3,4] => 2
[5,4,4] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [1,7,6,4,5,3,2] => [1,2,7,3,6,4,5] => 2
[4,3,3,3] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,7,6,5,4,3,2] => [1,2,7,3,6,4,5] => 2
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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 rix statistic of a permutation.
This statistic is defined recursively as follows: $rix([]) = 0$, and if $w_i = \max\{w_1, w_2,\dots, w_k\}$, then
$rix(w) := 0$ if $i = 1 < k$,
$rix(w) := 1 + rix(w_1,w_2,\dots,w_{k−1})$ if $i = k$ and
$rix(w) := rix(w_{i+1},w_{i+2},\dots,w_k)$ if $1 < i < k$.
This statistic is defined recursively as follows: $rix([]) = 0$, and if $w_i = \max\{w_1, w_2,\dots, w_k\}$, then
$rix(w) := 0$ if $i = 1 < k$,
$rix(w) := 1 + rix(w_1,w_2,\dots,w_{k−1})$ if $i = k$ and
$rix(w) := rix(w_{i+1},w_{i+2},\dots,w_k)$ if $1 < i < k$.
Map
to non-crossing permutation
Description
Sends a Dyck path $D$ with valley at positions $\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation $\pi$ having descents $\{i_1,\ldots,i_k\}$ and whose inverse has descents $\{j_1,\ldots,j_k\}$.
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to $n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to $n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
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
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
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