- FindStat

Identifier

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000374: Permutations ⟶ ℤ

Values

[1,0] => [1] => [1] => 0

[1,0,1,0] => [1,2] => [1,2] => 0

[1,1,0,0] => [2,1] => [2,1] => 1

[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => 0

[1,0,1,1,0,0] => [1,3,2] => [1,3,2] => 1

[1,1,0,0,1,0] => [2,1,3] => [2,1,3] => 1

[1,1,0,1,0,0] => [2,3,1] => [3,1,2] => 2

[1,1,1,0,0,0] => [3,2,1] => [3,2,1] => 1

[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => 0

[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,4,3] => 1

[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2,4] => 1

[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,4,2,3] => 2

[1,0,1,1,1,0,0,0] => [1,4,3,2] => [1,4,3,2] => 1

[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => 1

[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,4,3] => 2

[1,1,0,1,0,0,1,0] => [2,3,1,4] => [3,1,2,4] => 2

[1,1,0,1,0,1,0,0] => [2,3,4,1] => [4,1,2,3] => 3

[1,1,0,1,1,0,0,0] => [2,4,3,1] => [4,1,3,2] => 2

[1,1,1,0,0,0,1,0] => [3,2,1,4] => [3,2,1,4] => 1

[1,1,1,0,0,1,0,0] => [3,2,4,1] => [4,2,1,3] => 2

[1,1,1,0,1,0,0,0] => [4,2,3,1] => [4,2,3,1] => 1

[1,1,1,1,0,0,0,0] => [4,3,2,1] => [4,3,2,1] => 1

[1,0,1,0,1,0,1,0,1,0] => [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] => [1,2,3,5,4] => 1

[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [1,2,4,3,5] => 1

[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,5,3,4] => 2

[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [1,2,5,4,3] => 1

[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,3,2,4,5] => 1

[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,3,2,5,4] => 2

[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,4,2,3,5] => 2

[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,5,2,3,4] => 3

[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [1,5,2,4,3] => 2

[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [1,4,3,2,5] => 1

[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [1,5,3,2,4] => 2

[1,0,1,1,1,0,1,0,0,0] => [1,5,3,4,2] => [1,5,3,4,2] => 1

[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [1,5,4,3,2] => 1

[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [2,1,3,4,5] => 1

[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [2,1,3,5,4] => 2

[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [2,1,4,3,5] => 2

[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [2,1,5,3,4] => 3

[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [2,1,5,4,3] => 2

[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [3,1,2,4,5] => 2

[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [3,1,2,5,4] => 3

[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [4,1,2,3,5] => 3

[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => 4

[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [5,1,2,4,3] => 3

[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [4,1,3,2,5] => 2

[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [5,1,3,2,4] => 3

[1,1,0,1,1,0,1,0,0,0] => [2,5,3,4,1] => [5,1,3,4,2] => 2

[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [5,1,4,3,2] => 2

[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [3,2,1,4,5] => 1

[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [3,2,1,5,4] => 2

[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [4,2,1,3,5] => 2

[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [5,2,1,3,4] => 3

[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [5,2,1,4,3] => 2

[1,1,1,0,1,0,0,0,1,0] => [4,2,3,1,5] => [4,2,3,1,5] => 1

[1,1,1,0,1,0,0,1,0,0] => [4,2,3,5,1] => [5,2,3,1,4] => 2

[1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1] => [5,2,3,4,1] => 1

[1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => [5,2,4,3,1] => 1

[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [4,3,2,1,5] => 1

[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [5,3,2,1,4] => 2

[1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1] => [5,3,2,4,1] => 1

[1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => [5,4,2,3,1] => 1

[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [5,4,3,2,1] => 1

[1,0,1,0,1,0,1,0,1,0,1,0] => [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] => [1,2,3,4,6,5] => 1

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => 1

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [1,2,3,6,4,5] => 2

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => 1

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => 1

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => 2

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [1,2,5,3,4,6] => 2

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => 3

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => [1,2,6,3,5,4] => 2

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => 1

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => [1,2,6,4,3,5] => 2

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => 1

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => 1

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => 1

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => 2

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => 2

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [1,3,2,6,4,5] => 3

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => [1,3,2,6,5,4] => 2

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [1,4,2,3,5,6] => 2

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [1,4,2,3,6,5] => 3

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [1,5,2,3,4,6] => 3

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [1,6,2,3,4,5] => 4

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => [1,6,2,3,5,4] => 3

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => [1,5,2,4,3,6] => 2

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => [1,6,2,4,3,5] => 3

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,6,4,5,2] => [1,6,2,4,5,3] => 2

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => [1,6,2,5,4,3] => 2

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => [1,4,3,2,5,6] => 1

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => [1,4,3,2,6,5] => 2

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => [1,5,3,2,4,6] => 2

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => [1,6,3,2,4,5] => 3

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => [1,6,3,2,5,4] => 2

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,5,3,4,2,6] => [1,5,3,4,2,6] => 1

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,5,3,4,6,2] => [1,6,3,4,2,5] => 2

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,6,3,4,5,2] => [1,6,3,4,5,2] => 1

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,6,3,5,4,2] => [1,6,3,5,4,2] => 1

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Description

The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also St000213The number of weak exceedances (also weak excedences) of a permutation. and St000119The number of occurrences of the pattern 321 in a permutation..

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..

Map

inverse

Description

Sends a permutation to its inverse.