Identifier
-
Mp00024:
Dyck paths
—to 321-avoiding permutation⟶
Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St001665: Permutations ⟶ ℤ
Values
[1,0] => [1] => {{1}} => [1] => 0
[1,0,1,0] => [2,1] => {{1,2}} => [2,1] => 1
[1,1,0,0] => [1,2] => {{1},{2}} => [1,2] => 0
[1,0,1,0,1,0] => [2,1,3] => {{1,2},{3}} => [2,1,3] => 1
[1,0,1,1,0,0] => [2,3,1] => {{1,2,3}} => [2,3,1] => 1
[1,1,0,0,1,0] => [3,1,2] => {{1,2,3}} => [2,3,1] => 1
[1,1,0,1,0,0] => [1,3,2] => {{1},{2,3}} => [1,3,2] => 1
[1,1,1,0,0,0] => [1,2,3] => {{1},{2},{3}} => [1,2,3] => 0
[1,0,1,0,1,0,1,0] => [2,1,4,3] => {{1,2},{3,4}} => [2,1,4,3] => 2
[1,0,1,0,1,1,0,0] => [2,4,1,3] => {{1,2,3,4}} => [2,3,4,1] => 1
[1,0,1,1,0,0,1,0] => [2,1,3,4] => {{1,2},{3},{4}} => [2,1,3,4] => 1
[1,0,1,1,0,1,0,0] => [2,3,1,4] => {{1,2,3},{4}} => [2,3,1,4] => 1
[1,0,1,1,1,0,0,0] => [2,3,4,1] => {{1,2,3,4}} => [2,3,4,1] => 1
[1,1,0,0,1,0,1,0] => [3,1,4,2] => {{1,2,3,4}} => [2,3,4,1] => 1
[1,1,0,0,1,1,0,0] => [3,4,1,2] => {{1,3},{2,4}} => [3,4,1,2] => 1
[1,1,0,1,0,0,1,0] => [3,1,2,4] => {{1,2,3},{4}} => [2,3,1,4] => 1
[1,1,0,1,0,1,0,0] => [1,3,2,4] => {{1},{2,3},{4}} => [1,3,2,4] => 1
[1,1,0,1,1,0,0,0] => [1,3,4,2] => {{1},{2,3,4}} => [1,3,4,2] => 1
[1,1,1,0,0,0,1,0] => [4,1,2,3] => {{1,2,3,4}} => [2,3,4,1] => 1
[1,1,1,0,0,1,0,0] => [1,4,2,3] => {{1},{2,3,4}} => [1,3,4,2] => 1
[1,1,1,0,1,0,0,0] => [1,2,4,3] => {{1},{2},{3,4}} => [1,2,4,3] => 1
[1,1,1,1,0,0,0,0] => [1,2,3,4] => {{1},{2},{3},{4}} => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,5] => {{1,2},{3,4},{5}} => [2,1,4,3,5] => 2
[1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => {{1,2,3,4},{5}} => [2,3,4,1,5] => 1
[1,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3] => {{1,2},{3,4,5}} => [2,1,4,5,3] => 2
[1,0,1,0,1,1,0,1,0,0] => [2,4,1,5,3] => {{1,2,3,4,5}} => [2,3,4,5,1] => 1
[1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => {{1,2,4},{3,5}} => [2,4,5,1,3] => 1
[1,0,1,1,0,0,1,0,1,0] => [2,1,5,3,4] => {{1,2},{3,4,5}} => [2,1,4,5,3] => 2
[1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => {{1,2,3,4,5}} => [2,3,4,5,1] => 1
[1,0,1,1,0,1,0,0,1,0] => [2,1,3,5,4] => {{1,2},{3},{4,5}} => [2,1,3,5,4] => 2
[1,0,1,1,0,1,0,1,0,0] => [2,3,1,5,4] => {{1,2,3},{4,5}} => [2,3,1,5,4] => 2
[1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4] => {{1,2,3,4,5}} => [2,3,4,5,1] => 1
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => {{1,2},{3},{4},{5}} => [2,1,3,4,5] => 1
[1,0,1,1,1,0,0,1,0,0] => [2,3,1,4,5] => {{1,2,3},{4},{5}} => [2,3,1,4,5] => 1
[1,0,1,1,1,0,1,0,0,0] => [2,3,4,1,5] => {{1,2,3,4},{5}} => [2,3,4,1,5] => 1
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => {{1,2,3,4,5}} => [2,3,4,5,1] => 1
[1,1,0,0,1,0,1,0,1,0] => [3,1,4,2,5] => {{1,2,3,4},{5}} => [2,3,4,1,5] => 1
[1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,5] => {{1,3},{2,4},{5}} => [3,4,1,2,5] => 1
[1,1,0,0,1,1,0,0,1,0] => [3,1,4,5,2] => {{1,2,3,4,5}} => [2,3,4,5,1] => 1
[1,1,0,0,1,1,0,1,0,0] => [3,4,1,5,2] => {{1,3},{2,4,5}} => [3,4,1,5,2] => 1
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => {{1,2,3,4,5}} => [2,3,4,5,1] => 1
[1,1,0,1,0,0,1,0,1,0] => [3,1,5,2,4] => {{1,2,3,4,5}} => [2,3,4,5,1] => 1
[1,1,0,1,0,0,1,1,0,0] => [3,5,1,2,4] => {{1,3},{2,4,5}} => [3,4,1,5,2] => 1
[1,1,0,1,0,1,0,0,1,0] => [3,1,2,5,4] => {{1,2,3},{4,5}} => [2,3,1,5,4] => 2
[1,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4] => {{1},{2,3},{4,5}} => [1,3,2,5,4] => 2
[1,1,0,1,0,1,1,0,0,0] => [1,3,5,2,4] => {{1},{2,3,4,5}} => [1,3,4,5,2] => 1
[1,1,0,1,1,0,0,0,1,0] => [3,1,2,4,5] => {{1,2,3},{4},{5}} => [2,3,1,4,5] => 1
[1,1,0,1,1,0,0,1,0,0] => [1,3,2,4,5] => {{1},{2,3},{4},{5}} => [1,3,2,4,5] => 1
[1,1,0,1,1,0,1,0,0,0] => [1,3,4,2,5] => {{1},{2,3,4},{5}} => [1,3,4,2,5] => 1
[1,1,0,1,1,1,0,0,0,0] => [1,3,4,5,2] => {{1},{2,3,4,5}} => [1,3,4,5,2] => 1
[1,1,1,0,0,0,1,0,1,0] => [4,1,5,2,3] => {{1,2,4},{3,5}} => [2,4,5,1,3] => 1
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => {{1,2,3,4,5}} => [2,3,4,5,1] => 1
[1,1,1,0,0,1,0,0,1,0] => [4,1,2,5,3] => {{1,2,3,4,5}} => [2,3,4,5,1] => 1
[1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,3] => {{1},{2,3,4,5}} => [1,3,4,5,2] => 1
[1,1,1,0,0,1,1,0,0,0] => [1,4,5,2,3] => {{1},{2,4},{3,5}} => [1,4,5,2,3] => 1
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,3,5] => {{1,2,3,4},{5}} => [2,3,4,1,5] => 1
[1,1,1,0,1,0,0,1,0,0] => [1,4,2,3,5] => {{1},{2,3,4},{5}} => [1,3,4,2,5] => 1
[1,1,1,0,1,0,1,0,0,0] => [1,2,4,3,5] => {{1},{2},{3,4},{5}} => [1,2,4,3,5] => 1
[1,1,1,0,1,1,0,0,0,0] => [1,2,4,5,3] => {{1},{2},{3,4,5}} => [1,2,4,5,3] => 1
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => {{1,2,3,4,5}} => [2,3,4,5,1] => 1
[1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,4] => {{1},{2,3,4,5}} => [1,3,4,5,2] => 1
[1,1,1,1,0,0,1,0,0,0] => [1,2,5,3,4] => {{1},{2},{3,4,5}} => [1,2,4,5,3] => 1
[1,1,1,1,0,1,0,0,0,0] => [1,2,3,5,4] => {{1},{2},{3},{4,5}} => [1,2,3,5,4] => 1
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}} => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5] => {{1,2},{3,4},{5,6}} => [2,1,4,3,6,5] => 3
[1,0,1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,6,5] => {{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => 2
[1,0,1,0,1,0,1,1,0,0,1,0] => [2,1,4,6,3,5] => {{1,2},{3,4,5,6}} => [2,1,4,5,6,3] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [2,4,1,6,3,5] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => {{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => {{1,2},{3,4},{5},{6}} => [2,1,4,3,5,6] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [2,4,1,3,5,6] => {{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => {{1,2},{3,4,5},{6}} => [2,1,4,5,3,6] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [2,4,1,5,3,6] => {{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => {{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => 1
[1,0,1,0,1,1,1,0,0,0,1,0] => [2,1,4,5,6,3] => {{1,2},{3,4,5,6}} => [2,1,4,5,6,3] => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [2,4,1,5,6,3] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [2,4,5,1,6,3] => {{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,5,6,1,3] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,5,3,6,4] => {{1,2},{3,4,5,6}} => [2,1,4,5,6,3] => 2
[1,0,1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,5,6,3,4] => {{1,2},{3,5},{4,6}} => [2,1,5,6,3,4] => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [2,5,1,6,3,4] => {{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => 1
[1,0,1,1,0,0,1,1,1,0,0,0] => [2,5,6,1,3,4] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 1
[1,0,1,1,0,1,0,0,1,0,1,0] => [2,1,5,3,4,6] => {{1,2},{3,4,5},{6}} => [2,1,4,5,3,6] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [2,5,1,3,4,6] => {{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [2,1,3,5,4,6] => {{1,2},{3},{4,5},{6}} => [2,1,3,5,4,6] => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [2,3,1,5,4,6] => {{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => {{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => 1
[1,0,1,1,0,1,1,0,0,0,1,0] => [2,1,3,5,6,4] => {{1,2},{3},{4,5,6}} => [2,1,3,5,6,4] => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [2,3,1,5,6,4] => {{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => 2
[1,0,1,1,0,1,1,0,1,0,0,0] => [2,3,5,1,6,4] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,3,5,6,1,4] => {{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,6,3,4,5] => {{1,2},{3,4,5,6}} => [2,1,4,5,6,3] => 2
[1,0,1,1,1,0,0,0,1,1,0,0] => [2,6,1,3,4,5] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 1
[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,3,6,4,5] => {{1,2},{3},{4,5,6}} => [2,1,3,5,6,4] => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [2,3,1,6,4,5] => {{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [2,3,6,1,4,5] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 1
[1,0,1,1,1,0,1,0,0,0,1,0] => [2,1,3,4,6,5] => {{1,2},{3},{4},{5,6}} => [2,1,3,4,6,5] => 2
[1,0,1,1,1,0,1,0,0,1,0,0] => [2,3,1,4,6,5] => {{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [2,3,4,1,6,5] => {{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => 2
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,3,4,6,1,5] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => 1
>>> Load all 325 entries. <<<
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 pure excedances of a permutation.
A pure excedance of a permutation $\pi$ is a position $i < \pi_i$ such that there is no $j < i$ with $i\leq \pi_j < \pi_i$.
A pure excedance of a permutation $\pi$ is a position $i < \pi_i$ such that there is no $j < i$ with $i\leq \pi_j < \pi_i$.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
to 321-avoiding permutation
Description
Sends a Dyck path to a 321-avoiding permutation.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.
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].
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!