Identifier
-
Mp00119:
Dyck paths
—to 321-avoiding permutation (Krattenthaler)⟶
Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000673: Permutations ⟶ ℤ
Values
[1,0,1,0] => [1,2] => {{1},{2}} => [1,2] => 0
[1,1,0,0] => [2,1] => {{1,2}} => [2,1] => 2
[1,0,1,0,1,0] => [1,2,3] => {{1},{2},{3}} => [1,2,3] => 0
[1,0,1,1,0,0] => [1,3,2] => {{1},{2,3}} => [1,3,2] => 2
[1,1,0,0,1,0] => [2,1,3] => {{1,2},{3}} => [2,1,3] => 2
[1,1,0,1,0,0] => [2,3,1] => {{1,2,3}} => [2,3,1] => 3
[1,1,1,0,0,0] => [3,1,2] => {{1,2,3}} => [2,3,1] => 3
[1,0,1,0,1,0,1,0] => [1,2,3,4] => {{1},{2},{3},{4}} => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0] => [1,2,4,3] => {{1},{2},{3,4}} => [1,2,4,3] => 2
[1,0,1,1,0,0,1,0] => [1,3,2,4] => {{1},{2,3},{4}} => [1,3,2,4] => 2
[1,0,1,1,0,1,0,0] => [1,3,4,2] => {{1},{2,3,4}} => [1,3,4,2] => 3
[1,0,1,1,1,0,0,0] => [1,4,2,3] => {{1},{2,3,4}} => [1,3,4,2] => 3
[1,1,0,0,1,0,1,0] => [2,1,3,4] => {{1,2},{3},{4}} => [2,1,3,4] => 2
[1,1,0,0,1,1,0,0] => [2,1,4,3] => {{1,2},{3,4}} => [2,1,4,3] => 4
[1,1,0,1,0,0,1,0] => [2,3,1,4] => {{1,2,3},{4}} => [2,3,1,4] => 3
[1,1,0,1,0,1,0,0] => [2,3,4,1] => {{1,2,3,4}} => [2,3,4,1] => 4
[1,1,0,1,1,0,0,0] => [2,4,1,3] => {{1,2,3,4}} => [2,3,4,1] => 4
[1,1,1,0,0,0,1,0] => [3,1,2,4] => {{1,2,3},{4}} => [2,3,1,4] => 3
[1,1,1,0,0,1,0,0] => [3,1,4,2] => {{1,2,3,4}} => [2,3,4,1] => 4
[1,1,1,0,1,0,0,0] => [3,4,1,2] => {{1,3},{2,4}} => [3,4,1,2] => 4
[1,1,1,1,0,0,0,0] => [4,1,2,3] => {{1,2,3,4}} => [2,3,4,1] => 4
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}} => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => {{1},{2},{3},{4,5}} => [1,2,3,5,4] => 2
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => {{1},{2},{3,4},{5}} => [1,2,4,3,5] => 2
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => {{1},{2},{3,4,5}} => [1,2,4,5,3] => 3
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => {{1},{2},{3,4,5}} => [1,2,4,5,3] => 3
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => {{1},{2,3},{4},{5}} => [1,3,2,4,5] => 2
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => {{1},{2,3},{4,5}} => [1,3,2,5,4] => 4
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => {{1},{2,3,4},{5}} => [1,3,4,2,5] => 3
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => {{1},{2,3,4,5}} => [1,3,4,5,2] => 4
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4] => {{1},{2,3,4,5}} => [1,3,4,5,2] => 4
[1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => {{1},{2,3,4},{5}} => [1,3,4,2,5] => 3
[1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => {{1},{2,3,4,5}} => [1,3,4,5,2] => 4
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => {{1},{2,4},{3,5}} => [1,4,5,2,3] => 4
[1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => {{1},{2,3,4,5}} => [1,3,4,5,2] => 4
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => {{1,2},{3},{4},{5}} => [2,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => {{1,2},{3},{4,5}} => [2,1,3,5,4] => 4
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => {{1,2},{3,4},{5}} => [2,1,4,3,5] => 4
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => {{1,2},{3,4,5}} => [2,1,4,5,3] => 5
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,3,4] => {{1,2},{3,4,5}} => [2,1,4,5,3] => 5
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => {{1,2,3},{4},{5}} => [2,3,1,4,5] => 3
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => {{1,2,3},{4,5}} => [2,3,1,5,4] => 5
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => {{1,2,3,4},{5}} => [2,3,4,1,5] => 4
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => {{1,2,3,4,5}} => [2,3,4,5,1] => 5
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4] => {{1,2,3,4,5}} => [2,3,4,5,1] => 5
[1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,5] => {{1,2,3,4},{5}} => [2,3,4,1,5] => 4
[1,1,0,1,1,0,0,1,0,0] => [2,4,1,5,3] => {{1,2,3,4,5}} => [2,3,4,5,1] => 5
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,1,3] => {{1,2,4},{3,5}} => [2,4,5,1,3] => 5
[1,1,0,1,1,1,0,0,0,0] => [2,5,1,3,4] => {{1,2,3,4,5}} => [2,3,4,5,1] => 5
[1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => {{1,2,3},{4},{5}} => [2,3,1,4,5] => 3
[1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => {{1,2,3},{4,5}} => [2,3,1,5,4] => 5
[1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => {{1,2,3,4},{5}} => [2,3,4,1,5] => 4
[1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => {{1,2,3,4,5}} => [2,3,4,5,1] => 5
[1,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4] => {{1,2,3,4,5}} => [2,3,4,5,1] => 5
[1,1,1,0,1,0,0,0,1,0] => [3,4,1,2,5] => {{1,3},{2,4},{5}} => [3,4,1,2,5] => 4
[1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => {{1,3},{2,4,5}} => [3,4,1,5,2] => 5
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => {{1,2,3,4,5}} => [2,3,4,5,1] => 5
[1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => {{1,3},{2,4,5}} => [3,4,1,5,2] => 5
[1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => {{1,2,3,4},{5}} => [2,3,4,1,5] => 4
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => {{1,2,3,4,5}} => [2,3,4,5,1] => 5
[1,1,1,1,0,0,1,0,0,0] => [4,1,5,2,3] => {{1,2,4},{3,5}} => [2,4,5,1,3] => 5
[1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => {{1,2,3,4,5}} => [2,3,4,5,1] => 5
[1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => {{1,2,3,4,5}} => [2,3,4,5,1] => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => {{1},{2},{3},{4},{5},{6}} => [1,2,3,4,5,6] => 0
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => {{1},{2},{3},{4},{5,6}} => [1,2,3,4,6,5] => 2
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => {{1},{2},{3},{4,5},{6}} => [1,2,3,5,4,6] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => {{1},{2},{3},{4,5,6}} => [1,2,3,5,6,4] => 3
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => {{1},{2},{3},{4,5,6}} => [1,2,3,5,6,4] => 3
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => {{1},{2},{3,4},{5},{6}} => [1,2,4,3,5,6] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => {{1},{2},{3,4},{5,6}} => [1,2,4,3,6,5] => 4
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => {{1},{2},{3,4,5},{6}} => [1,2,4,5,3,6] => 3
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => {{1},{2},{3,4,5,6}} => [1,2,4,5,6,3] => 4
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,3,5] => {{1},{2},{3,4,5,6}} => [1,2,4,5,6,3] => 4
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,3,4,6] => {{1},{2},{3,4,5},{6}} => [1,2,4,5,3,6] => 3
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,3,6,4] => {{1},{2},{3,4,5,6}} => [1,2,4,5,6,3] => 4
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,3,4] => {{1},{2},{3,5},{4,6}} => [1,2,5,6,3,4] => 4
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => {{1},{2},{3,4,5,6}} => [1,2,4,5,6,3] => 4
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => {{1},{2,3},{4},{5},{6}} => [1,3,2,4,5,6] => 2
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => {{1},{2,3},{4},{5,6}} => [1,3,2,4,6,5] => 4
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => {{1},{2,3},{4,5},{6}} => [1,3,2,5,4,6] => 4
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => {{1},{2,3},{4,5,6}} => [1,3,2,5,6,4] => 5
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,4,5] => {{1},{2,3},{4,5,6}} => [1,3,2,5,6,4] => 5
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => {{1},{2,3,4},{5},{6}} => [1,3,4,2,5,6] => 3
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => {{1},{2,3,4},{5,6}} => [1,3,4,2,6,5] => 5
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => {{1},{2,3,4,5},{6}} => [1,3,4,5,2,6] => 4
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => {{1},{2,3,4,5,6}} => [1,3,4,5,6,2] => 5
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,2,5] => {{1},{2,3,4,5,6}} => [1,3,4,5,6,2] => 5
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,2,4,6] => {{1},{2,3,4,5},{6}} => [1,3,4,5,2,6] => 4
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,2,6,4] => {{1},{2,3,4,5,6}} => [1,3,4,5,6,2] => 5
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,2,4] => {{1},{2,3,5},{4,6}} => [1,3,5,6,2,4] => 5
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,2,4,5] => {{1},{2,3,4,5,6}} => [1,3,4,5,6,2] => 5
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,2,3,5,6] => {{1},{2,3,4},{5},{6}} => [1,3,4,2,5,6] => 3
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,2,3,6,5] => {{1},{2,3,4},{5,6}} => [1,3,4,2,6,5] => 5
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3,6] => {{1},{2,3,4,5},{6}} => [1,3,4,5,2,6] => 4
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,6,3] => {{1},{2,3,4,5,6}} => [1,3,4,5,6,2] => 5
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,2,6,3,5] => {{1},{2,3,4,5,6}} => [1,3,4,5,6,2] => 5
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,2,3,6] => {{1},{2,4},{3,5},{6}} => [1,4,5,2,3,6] => 4
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,2,6,3] => {{1},{2,4},{3,5,6}} => [1,4,5,2,6,3] => 5
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,2,3] => {{1},{2,3,4,5,6}} => [1,3,4,5,6,2] => 5
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,2,3,5] => {{1},{2,4},{3,5,6}} => [1,4,5,2,6,3] => 5
[1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,2,3,4,6] => {{1},{2,3,4,5},{6}} => [1,3,4,5,2,6] => 4
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Description
The number of non-fixed points of a permutation.
In other words, this statistic is $n$ minus the number of fixed points (St000022The number of fixed points of a permutation.) of $\pi$.
In other words, this statistic is $n$ minus the number of fixed points (St000022The number of fixed points of a permutation.) of $\pi$.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
to cycle type
Description
Let $\pi=c_1\dots c_r$ a permutation of size $n$ decomposed in its cyclic parts. The associated set partition of $[n]$ then is $S=S_1\cup\dots\cup S_r$ such that $S_i$ is the set of integers in the cycle $c_i$.
A permutation is cyclic [1] if and only if its cycle type is a hook partition [2].
A permutation is cyclic [1] if and only if its cycle type is a hook partition [2].
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.
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.
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