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Matching statistic: St000374
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(load all 13 compositions to match this statistic)
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 1 = 2 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 1 = 2 - 1
[3,1,2] => 2 = 3 - 1
[3,2,1] => 1 = 2 - 1
[1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => 1 = 2 - 1
[1,3,2,4] => 1 = 2 - 1
[1,3,4,2] => 1 = 2 - 1
[1,4,2,3] => 2 = 3 - 1
[1,4,3,2] => 1 = 2 - 1
[2,1,3,4] => 1 = 2 - 1
[2,1,4,3] => 2 = 3 - 1
[2,3,1,4] => 1 = 2 - 1
[2,3,4,1] => 1 = 2 - 1
[2,4,1,3] => 2 = 3 - 1
[2,4,3,1] => 1 = 2 - 1
[3,1,2,4] => 2 = 3 - 1
[3,1,4,2] => 2 = 3 - 1
[3,2,1,4] => 1 = 2 - 1
[3,2,4,1] => 1 = 2 - 1
[3,4,1,2] => 2 = 3 - 1
[3,4,2,1] => 1 = 2 - 1
[4,1,2,3] => 3 = 4 - 1
[4,1,3,2] => 2 = 3 - 1
[4,2,1,3] => 2 = 3 - 1
[4,2,3,1] => 1 = 2 - 1
[4,3,1,2] => 2 = 3 - 1
[4,3,2,1] => 1 = 2 - 1
[1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => 1 = 2 - 1
[1,2,4,3,5] => 1 = 2 - 1
[1,2,4,5,3] => 1 = 2 - 1
[1,2,5,3,4] => 2 = 3 - 1
[1,2,5,4,3] => 1 = 2 - 1
[1,3,2,4,5] => 1 = 2 - 1
[1,3,2,5,4] => 2 = 3 - 1
[1,3,4,2,5] => 1 = 2 - 1
[1,3,4,5,2] => 1 = 2 - 1
[1,3,5,2,4] => 2 = 3 - 1
[1,3,5,4,2] => 1 = 2 - 1
[1,4,2,3,5] => 2 = 3 - 1
[1,4,2,5,3] => 2 = 3 - 1
[1,4,3,2,5] => 1 = 2 - 1
[1,4,3,5,2] => 1 = 2 - 1
[1,4,5,2,3] => 2 = 3 - 1
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 [[St000213]] and [[St000119]].
Matching statistic: St000996
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 1 = 2 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 1 = 2 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 2 = 3 - 1
[3,1,2] => 1 = 2 - 1
[3,2,1] => 1 = 2 - 1
[1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => 1 = 2 - 1
[1,3,2,4] => 1 = 2 - 1
[1,3,4,2] => 2 = 3 - 1
[1,4,2,3] => 1 = 2 - 1
[1,4,3,2] => 1 = 2 - 1
[2,1,3,4] => 1 = 2 - 1
[2,1,4,3] => 2 = 3 - 1
[2,3,1,4] => 2 = 3 - 1
[2,3,4,1] => 3 = 4 - 1
[2,4,1,3] => 2 = 3 - 1
[2,4,3,1] => 2 = 3 - 1
[3,1,2,4] => 1 = 2 - 1
[3,1,4,2] => 2 = 3 - 1
[3,2,1,4] => 1 = 2 - 1
[3,2,4,1] => 2 = 3 - 1
[3,4,1,2] => 2 = 3 - 1
[3,4,2,1] => 2 = 3 - 1
[4,1,2,3] => 1 = 2 - 1
[4,1,3,2] => 1 = 2 - 1
[4,2,1,3] => 1 = 2 - 1
[4,2,3,1] => 1 = 2 - 1
[4,3,1,2] => 1 = 2 - 1
[4,3,2,1] => 1 = 2 - 1
[1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => 1 = 2 - 1
[1,2,4,3,5] => 1 = 2 - 1
[1,2,4,5,3] => 2 = 3 - 1
[1,2,5,3,4] => 1 = 2 - 1
[1,2,5,4,3] => 1 = 2 - 1
[1,3,2,4,5] => 1 = 2 - 1
[1,3,2,5,4] => 2 = 3 - 1
[1,3,4,2,5] => 2 = 3 - 1
[1,3,4,5,2] => 3 = 4 - 1
[1,3,5,2,4] => 2 = 3 - 1
[1,3,5,4,2] => 2 = 3 - 1
[1,4,2,3,5] => 1 = 2 - 1
[1,4,2,5,3] => 2 = 3 - 1
[1,4,3,2,5] => 1 = 2 - 1
[1,4,3,5,2] => 2 = 3 - 1
[1,4,5,2,3] => 2 = 3 - 1
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St000015
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
Description
The number of peaks of a Dyck path.
Matching statistic: St000325
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [1,2] => [2,1] => 2
[2,1] => [2,1] => [1,2] => 1
[1,2,3] => [1,3,2] => [2,1,3] => 2
[1,3,2] => [1,3,2] => [2,1,3] => 2
[2,1,3] => [2,1,3] => [3,2,1] => 3
[2,3,1] => [2,3,1] => [3,1,2] => 2
[3,1,2] => [3,1,2] => [1,3,2] => 2
[3,2,1] => [3,2,1] => [1,2,3] => 1
[1,2,3,4] => [1,4,3,2] => [2,1,3,4] => 2
[1,2,4,3] => [1,4,3,2] => [2,1,3,4] => 2
[1,3,2,4] => [1,4,3,2] => [2,1,3,4] => 2
[1,3,4,2] => [1,4,3,2] => [2,1,3,4] => 2
[1,4,2,3] => [1,4,3,2] => [2,1,3,4] => 2
[1,4,3,2] => [1,4,3,2] => [2,1,3,4] => 2
[2,1,3,4] => [2,1,4,3] => [3,2,1,4] => 3
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 3
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 3
[2,3,4,1] => [2,4,3,1] => [3,1,2,4] => 2
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 3
[2,4,3,1] => [2,4,3,1] => [3,1,2,4] => 2
[3,1,2,4] => [3,1,4,2] => [4,2,1,3] => 3
[3,1,4,2] => [3,1,4,2] => [4,2,1,3] => 3
[3,2,1,4] => [3,2,1,4] => [4,3,2,1] => 4
[3,2,4,1] => [3,2,4,1] => [4,3,1,2] => 3
[3,4,1,2] => [3,4,1,2] => [4,1,3,2] => 3
[3,4,2,1] => [3,4,2,1] => [4,1,2,3] => 2
[4,1,2,3] => [4,1,3,2] => [1,3,2,4] => 2
[4,1,3,2] => [4,1,3,2] => [1,3,2,4] => 2
[4,2,1,3] => [4,2,1,3] => [1,4,3,2] => 3
[4,2,3,1] => [4,2,3,1] => [1,4,2,3] => 2
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 2
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 1
[1,2,3,4,5] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,2,3,5,4] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,2,4,3,5] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,2,4,5,3] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,2,5,3,4] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,2,5,4,3] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,3,2,4,5] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,3,2,5,4] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,3,4,2,5] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,3,4,5,2] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,3,5,2,4] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,3,5,4,2] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,4,2,3,5] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,4,2,5,3] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,4,3,2,5] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,4,3,5,2] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,4,5,2,3] => [1,5,4,3,2] => [2,1,3,4,5] => 2
Description
The width of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The width of the tree is given by the number of leaves of this tree.
Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]].
See also [[St000308]] for the height of this tree.
Matching statistic: St000470
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [1,2] => [2,1] => 2
[2,1] => [2,1] => [1,2] => 1
[1,2,3] => [1,3,2] => [2,1,3] => 2
[1,3,2] => [1,3,2] => [2,1,3] => 2
[2,1,3] => [2,1,3] => [3,2,1] => 3
[2,3,1] => [2,3,1] => [3,1,2] => 2
[3,1,2] => [3,1,2] => [1,3,2] => 2
[3,2,1] => [3,2,1] => [1,2,3] => 1
[1,2,3,4] => [1,4,3,2] => [2,1,3,4] => 2
[1,2,4,3] => [1,4,3,2] => [2,1,3,4] => 2
[1,3,2,4] => [1,4,3,2] => [2,1,3,4] => 2
[1,3,4,2] => [1,4,3,2] => [2,1,3,4] => 2
[1,4,2,3] => [1,4,3,2] => [2,1,3,4] => 2
[1,4,3,2] => [1,4,3,2] => [2,1,3,4] => 2
[2,1,3,4] => [2,1,4,3] => [3,2,1,4] => 3
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 3
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 3
[2,3,4,1] => [2,4,3,1] => [3,1,2,4] => 2
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 3
[2,4,3,1] => [2,4,3,1] => [3,1,2,4] => 2
[3,1,2,4] => [3,1,4,2] => [4,2,1,3] => 3
[3,1,4,2] => [3,1,4,2] => [4,2,1,3] => 3
[3,2,1,4] => [3,2,1,4] => [4,3,2,1] => 4
[3,2,4,1] => [3,2,4,1] => [4,3,1,2] => 3
[3,4,1,2] => [3,4,1,2] => [4,1,3,2] => 3
[3,4,2,1] => [3,4,2,1] => [4,1,2,3] => 2
[4,1,2,3] => [4,1,3,2] => [1,3,2,4] => 2
[4,1,3,2] => [4,1,3,2] => [1,3,2,4] => 2
[4,2,1,3] => [4,2,1,3] => [1,4,3,2] => 3
[4,2,3,1] => [4,2,3,1] => [1,4,2,3] => 2
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 2
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 1
[1,2,3,4,5] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,2,3,5,4] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,2,4,3,5] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,2,4,5,3] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,2,5,3,4] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,2,5,4,3] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,3,2,4,5] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,3,2,5,4] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,3,4,2,5] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,3,4,5,2] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,3,5,2,4] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,3,5,4,2] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,4,2,3,5] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,4,2,5,3] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,4,3,2,5] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,4,3,5,2] => [1,5,4,3,2] => [2,1,3,4,5] => 2
[1,4,5,2,3] => [1,5,4,3,2] => [2,1,3,4,5] => 2
Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
Matching statistic: St000740
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [1,2] => [1,2] => 2
[2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,3,2] => [1,3,2] => 2
[1,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => [2,1,3] => 3
[2,3,1] => [2,3,1] => [1,3,2] => 2
[3,1,2] => [3,1,2] => [3,1,2] => 2
[3,2,1] => [3,2,1] => [3,2,1] => 1
[1,2,3,4] => [1,4,3,2] => [1,4,3,2] => 2
[1,2,4,3] => [1,4,3,2] => [1,4,3,2] => 2
[1,3,2,4] => [1,4,3,2] => [1,4,3,2] => 2
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 2
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,4,3] => [2,1,4,3] => 3
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 3
[2,3,1,4] => [2,4,1,3] => [2,4,1,3] => 3
[2,3,4,1] => [2,4,3,1] => [1,4,3,2] => 2
[2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 3
[2,4,3,1] => [2,4,3,1] => [1,4,3,2] => 2
[3,1,2,4] => [3,1,4,2] => [2,1,4,3] => 3
[3,1,4,2] => [3,1,4,2] => [2,1,4,3] => 3
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 4
[3,2,4,1] => [3,2,4,1] => [2,1,4,3] => 3
[3,4,1,2] => [3,4,1,2] => [2,4,1,3] => 3
[3,4,2,1] => [3,4,2,1] => [1,4,3,2] => 2
[4,1,2,3] => [4,1,3,2] => [4,1,3,2] => 2
[4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 2
[4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 3
[4,2,3,1] => [4,2,3,1] => [4,1,3,2] => 2
[4,3,1,2] => [4,3,1,2] => [4,3,1,2] => 2
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[1,2,3,4,5] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,2,3,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,2,4,3,5] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,2,4,5,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,2,5,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,2,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,3,2,4,5] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,3,2,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,3,4,2,5] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,3,4,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,3,5,2,4] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,4,2,3,5] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,4,2,5,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,4,3,2,5] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,4,3,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St001068
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St000021
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => [2,1] => 1 = 2 - 1
[2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,3,2] => [2,1,3] => 1 = 2 - 1
[1,3,2] => [1,3,2] => [2,1,3] => 1 = 2 - 1
[2,1,3] => [2,1,3] => [3,2,1] => 2 = 3 - 1
[2,3,1] => [2,3,1] => [3,1,2] => 1 = 2 - 1
[3,1,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [1,4,3,2] => [2,1,3,4] => 1 = 2 - 1
[1,2,4,3] => [1,4,3,2] => [2,1,3,4] => 1 = 2 - 1
[1,3,2,4] => [1,4,3,2] => [2,1,3,4] => 1 = 2 - 1
[1,3,4,2] => [1,4,3,2] => [2,1,3,4] => 1 = 2 - 1
[1,4,2,3] => [1,4,3,2] => [2,1,3,4] => 1 = 2 - 1
[1,4,3,2] => [1,4,3,2] => [2,1,3,4] => 1 = 2 - 1
[2,1,3,4] => [2,1,4,3] => [3,2,1,4] => 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 2 = 3 - 1
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 2 = 3 - 1
[2,3,4,1] => [2,4,3,1] => [3,1,2,4] => 1 = 2 - 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 2 = 3 - 1
[2,4,3,1] => [2,4,3,1] => [3,1,2,4] => 1 = 2 - 1
[3,1,2,4] => [3,1,4,2] => [4,2,1,3] => 2 = 3 - 1
[3,1,4,2] => [3,1,4,2] => [4,2,1,3] => 2 = 3 - 1
[3,2,1,4] => [3,2,1,4] => [4,3,2,1] => 3 = 4 - 1
[3,2,4,1] => [3,2,4,1] => [4,3,1,2] => 2 = 3 - 1
[3,4,1,2] => [3,4,1,2] => [4,1,3,2] => 2 = 3 - 1
[3,4,2,1] => [3,4,2,1] => [4,1,2,3] => 1 = 2 - 1
[4,1,2,3] => [4,1,3,2] => [1,3,2,4] => 1 = 2 - 1
[4,1,3,2] => [4,1,3,2] => [1,3,2,4] => 1 = 2 - 1
[4,2,1,3] => [4,2,1,3] => [1,4,3,2] => 2 = 3 - 1
[4,2,3,1] => [4,2,3,1] => [1,4,2,3] => 1 = 2 - 1
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,4,5] => [1,5,4,3,2] => [2,1,3,4,5] => 1 = 2 - 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,1,3,4,5] => 1 = 2 - 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,1,3,4,5] => 1 = 2 - 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,1,3,4,5] => 1 = 2 - 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,1,3,4,5] => 1 = 2 - 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,1,3,4,5] => 1 = 2 - 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,1,3,4,5] => 1 = 2 - 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,1,3,4,5] => 1 = 2 - 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,1,3,4,5] => 1 = 2 - 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,1,3,4,5] => 1 = 2 - 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,1,3,4,5] => 1 = 2 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,1,3,4,5] => 1 = 2 - 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,1,3,4,5] => 1 = 2 - 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,1,3,4,5] => 1 = 2 - 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,1,3,4,5] => 1 = 2 - 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,1,3,4,5] => 1 = 2 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,1,3,4,5] => 1 = 2 - 1
Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000053
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
Description
The number of valleys of the Dyck path.
Matching statistic: St000155
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => [2,1] => 1 = 2 - 1
[2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[1,3,2] => [1,3,2] => [3,1,2] => 1 = 2 - 1
[2,1,3] => [2,1,3] => [2,3,1] => 2 = 3 - 1
[2,3,1] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[3,1,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,2,3,4] => [1,4,3,2] => [4,1,2,3] => 1 = 2 - 1
[1,2,4,3] => [1,4,3,2] => [4,1,2,3] => 1 = 2 - 1
[1,3,2,4] => [1,4,3,2] => [4,1,2,3] => 1 = 2 - 1
[1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 1 = 2 - 1
[1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 1 = 2 - 1
[1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 1 = 2 - 1
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 2 = 3 - 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2 = 3 - 1
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 2 = 3 - 1
[2,3,4,1] => [2,4,3,1] => [3,1,2,4] => 1 = 2 - 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 2 = 3 - 1
[2,4,3,1] => [2,4,3,1] => [3,1,2,4] => 1 = 2 - 1
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => 2 = 3 - 1
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => 2 = 3 - 1
[3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 3 = 4 - 1
[3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 2 = 3 - 1
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 2 = 3 - 1
[3,4,2,1] => [3,4,2,1] => [2,1,3,4] => 1 = 2 - 1
[4,1,2,3] => [4,1,3,2] => [1,4,2,3] => 1 = 2 - 1
[4,1,3,2] => [4,1,3,2] => [1,4,2,3] => 1 = 2 - 1
[4,2,1,3] => [4,2,1,3] => [1,3,4,2] => 2 = 3 - 1
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 2 - 1
[1,2,3,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 2 - 1
[1,2,4,3,5] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 2 - 1
[1,2,4,5,3] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 2 - 1
[1,2,5,3,4] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 2 - 1
[1,2,5,4,3] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 2 - 1
[1,3,2,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 2 - 1
[1,3,2,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 2 - 1
[1,3,4,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 2 - 1
[1,3,5,2,4] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 2 - 1
[1,4,2,3,5] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 2 - 1
[1,4,2,5,3] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 2 - 1
[1,4,3,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 2 - 1
[1,4,3,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 2 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [5,1,2,3,4] => 1 = 2 - 1
Description
The number of exceedances (also excedences) of a permutation.
This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$.
It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
The following 108 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000316The number of non-left-to-right-maxima of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001489The maximum of the number of descents and the number of inverse descents. St000054The first entry of the permutation. St000062The length of the longest increasing subsequence of the permutation. St000105The number of blocks in the set partition. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000291The number of descents of a binary word. St000314The number of left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000390The number of runs of ones in a binary word. St000443The number of long tunnels of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000542The number of left-to-right-minima of a permutation. St000912The number of maximal antichains in a poset. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001497The position of the largest weak excedence of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St000024The number of double up and double down steps of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000120The number of left tunnels of a Dyck path. St000141The maximum drop size of a permutation. St000211The rank of the set partition. St000245The number of ascents of a permutation. St000292The number of ascents of a binary word. St000305The inverse major index of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000340The number of non-final maximal constant sub-paths of length greater than one. St000362The size of a minimal vertex cover of a graph. St000446The disorder of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001176The size of a partition minus its first part. St001180Number of indecomposable injective modules with projective dimension at most 1. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001346The number of parking functions that give the same permutation. St000829The Ulam distance of a permutation to the identity permutation. St000702The number of weak deficiencies of a permutation. St000925The number of topologically connected components of a set partition. St000990The first ascent of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St000288The number of ones in a binary word. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000809The reduced reflection length of the permutation. St001812The biclique partition number of a graph. St000159The number of distinct parts of the integer partition. St000393The number of strictly increasing runs in a binary word. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001427The number of descents of a signed permutation. St000741The Colin de Verdière graph invariant. St001644The dimension of a graph. St000260The radius of a connected graph. St000259The diameter of a connected graph. St000647The number of big descents of a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000264The girth of a graph, which is not a tree. St001645The pebbling number of a connected graph. St001875The number of simple modules with projective dimension at most 1. St001060The distinguishing index of a graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000455The second largest eigenvalue of a graph if it is integral. St000143The largest repeated part of a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000710The number of big deficiencies of a permutation. St000991The number of right-to-left minima of a permutation. St000711The number of big exceedences of a permutation. St001720The minimal length of a chain of small intervals in a lattice. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001863The number of weak excedances of a signed permutation. St001152The number of pairs with even minimum in a perfect matching. St001712The number of natural descents of a standard Young tableau. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice.
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