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Your data matches 664 different statistics following compositions of up to 3 maps.
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Matching statistic: St001114
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001114: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001114: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0
[2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => 0
[2,1,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,3,2] => 0
[3,2,1] => [1,3,2] => 0
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => 1
[4,1,2,3] => [1,4,3,2] => 1
[4,2,1,3] => [1,4,3,2] => 1
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,5,4] => [1,2,3,4,5] => 0
[2,1,4,3,5] => [1,2,3,4,5] => 0
[2,1,4,5,3] => [1,2,3,4,5] => 0
[2,3,1,4,5] => [1,2,3,4,5] => 0
[2,3,1,5,4] => [1,2,3,4,5] => 0
[2,3,4,1,5] => [1,2,3,4,5] => 0
[2,3,4,5,1] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => 0
Description
The number of odd descents of a permutation.
Matching statistic: St000891
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000891: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000891: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 2 = 0 + 2
[2,1] => [1,2] => 2 = 0 + 2
[1,2,3] => [1,2,3] => 2 = 0 + 2
[1,3,2] => [1,2,3] => 2 = 0 + 2
[2,1,3] => [1,2,3] => 2 = 0 + 2
[2,3,1] => [1,2,3] => 2 = 0 + 2
[3,1,2] => [1,3,2] => 2 = 0 + 2
[3,2,1] => [1,3,2] => 2 = 0 + 2
[1,2,3,4] => [1,2,3,4] => 2 = 0 + 2
[1,2,4,3] => [1,2,3,4] => 2 = 0 + 2
[1,3,2,4] => [1,2,3,4] => 2 = 0 + 2
[1,3,4,2] => [1,2,3,4] => 2 = 0 + 2
[1,4,2,3] => [1,2,4,3] => 3 = 1 + 2
[1,4,3,2] => [1,2,4,3] => 3 = 1 + 2
[2,1,3,4] => [1,2,3,4] => 2 = 0 + 2
[2,1,4,3] => [1,2,3,4] => 2 = 0 + 2
[2,3,1,4] => [1,2,3,4] => 2 = 0 + 2
[2,3,4,1] => [1,2,3,4] => 2 = 0 + 2
[2,4,1,3] => [1,2,4,3] => 3 = 1 + 2
[2,4,3,1] => [1,2,4,3] => 3 = 1 + 2
[4,1,2,3] => [1,4,3,2] => 3 = 1 + 2
[4,2,1,3] => [1,4,3,2] => 3 = 1 + 2
[1,2,3,4,5] => [1,2,3,4,5] => 2 = 0 + 2
[1,2,3,5,4] => [1,2,3,4,5] => 2 = 0 + 2
[1,2,4,3,5] => [1,2,3,4,5] => 2 = 0 + 2
[1,2,4,5,3] => [1,2,3,4,5] => 2 = 0 + 2
[1,3,2,4,5] => [1,2,3,4,5] => 2 = 0 + 2
[1,3,2,5,4] => [1,2,3,4,5] => 2 = 0 + 2
[1,3,4,2,5] => [1,2,3,4,5] => 2 = 0 + 2
[1,3,4,5,2] => [1,2,3,4,5] => 2 = 0 + 2
[2,1,3,4,5] => [1,2,3,4,5] => 2 = 0 + 2
[2,1,3,5,4] => [1,2,3,4,5] => 2 = 0 + 2
[2,1,4,3,5] => [1,2,3,4,5] => 2 = 0 + 2
[2,1,4,5,3] => [1,2,3,4,5] => 2 = 0 + 2
[2,3,1,4,5] => [1,2,3,4,5] => 2 = 0 + 2
[2,3,1,5,4] => [1,2,3,4,5] => 2 = 0 + 2
[2,3,4,1,5] => [1,2,3,4,5] => 2 = 0 + 2
[2,3,4,5,1] => [1,2,3,4,5] => 2 = 0 + 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,2,3,4,6,5] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,2,3,5,4,6] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,2,3,5,6,4] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,2,4,3,5,6] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,2,4,3,6,5] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,2,4,5,3,6] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,2,4,5,6,3] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,3,2,4,5,6] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,3,2,4,6,5] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,3,2,5,4,6] => [1,2,3,4,5,6] => 2 = 0 + 2
[1,3,2,5,6,4] => [1,2,3,4,5,6] => 2 = 0 + 2
Description
The number of distinct diagonal sums of a permutation matrix.
For example, the sums of the diagonals of the matrix $$\left(\begin{array}{rrrr}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0
\end{array}\right)$$
are $(1,0,1,0,2,0)$, so the statistic is $3$.
Matching statistic: St000023
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [3,1,2] => 0
[3,2,1] => [1,3,2] => [3,1,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[4,1,2,3] => [1,4,3,2] => [4,1,3,2] => 1
[4,2,1,3] => [1,4,3,2] => [4,1,3,2] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
Description
The number of inner peaks of a permutation.
The number of peaks including the boundary is [[St000092]].
Matching statistic: St000214
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [3,1,2] => 0
[3,2,1] => [1,3,2] => [3,1,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[4,1,2,3] => [1,4,3,2] => [4,1,3,2] => 1
[4,2,1,3] => [1,4,3,2] => [4,1,3,2] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
Description
The number of adjacencies of a permutation.
An adjacency of a permutation $\pi$ is an index $i$ such that $\pi(i)-1 = \pi(i+1)$. Adjacencies are also known as ''small descents''.
This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
Matching statistic: St000215
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000215: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000215: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [3,1,2] => 0
[3,2,1] => [1,3,2] => [3,1,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[4,1,2,3] => [1,4,3,2] => [4,1,3,2] => 1
[4,2,1,3] => [1,4,3,2] => [4,1,3,2] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
Description
The number of adjacencies of a permutation, zero appended.
An adjacency is a descent of the form $(e+1,e)$ in the word corresponding to the permutation in one-line notation. This statistic, $\operatorname{adj_0}$, counts adjacencies in the word with a zero appended.
$(\operatorname{adj_0}, \operatorname{des})$ and $(\operatorname{fix}, \operatorname{exc})$ are equidistributed, see [1].
Matching statistic: St000317
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000317: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00310: Permutations —toric promotion⟶ Permutations
St000317: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,2,3] => [3,2,1] => 0
[2,1,3] => [1,2,3] => [3,2,1] => 0
[2,3,1] => [1,2,3] => [3,2,1] => 0
[3,1,2] => [1,3,2] => [2,3,1] => 0
[3,2,1] => [1,3,2] => [2,3,1] => 0
[1,2,3,4] => [1,2,3,4] => [4,2,3,1] => 0
[1,2,4,3] => [1,2,3,4] => [4,2,3,1] => 0
[1,3,2,4] => [1,2,3,4] => [4,2,3,1] => 0
[1,3,4,2] => [1,2,3,4] => [4,2,3,1] => 0
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => 1
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => 1
[2,1,3,4] => [1,2,3,4] => [4,2,3,1] => 0
[2,1,4,3] => [1,2,3,4] => [4,2,3,1] => 0
[2,3,1,4] => [1,2,3,4] => [4,2,3,1] => 0
[2,3,4,1] => [1,2,3,4] => [4,2,3,1] => 0
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => 1
[4,1,2,3] => [1,4,3,2] => [3,1,2,4] => 1
[4,2,1,3] => [1,4,3,2] => [3,1,2,4] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,2,3,4,1] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [5,2,3,4,1] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [5,2,3,4,1] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [5,2,3,4,1] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [5,2,3,4,1] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [5,2,3,4,1] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [5,2,3,4,1] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [5,2,3,4,1] => 0
[2,1,3,4,5] => [1,2,3,4,5] => [5,2,3,4,1] => 0
[2,1,3,5,4] => [1,2,3,4,5] => [5,2,3,4,1] => 0
[2,1,4,3,5] => [1,2,3,4,5] => [5,2,3,4,1] => 0
[2,1,4,5,3] => [1,2,3,4,5] => [5,2,3,4,1] => 0
[2,3,1,4,5] => [1,2,3,4,5] => [5,2,3,4,1] => 0
[2,3,1,5,4] => [1,2,3,4,5] => [5,2,3,4,1] => 0
[2,3,4,1,5] => [1,2,3,4,5] => [5,2,3,4,1] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [5,2,3,4,1] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6,2,3,4,5,1] => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [6,2,3,4,5,1] => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [6,2,3,4,5,1] => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [6,2,3,4,5,1] => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [6,2,3,4,5,1] => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [6,2,3,4,5,1] => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [6,2,3,4,5,1] => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [6,2,3,4,5,1] => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [6,2,3,4,5,1] => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [6,2,3,4,5,1] => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [6,2,3,4,5,1] => 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [6,2,3,4,5,1] => 0
Description
The cycle descent number of a permutation.
Let $(i_1,\ldots,i_k)$ be a cycle of a permutation $\pi$ such that $i_1$ is its smallest element. A **cycle descent** of $(i_1,\ldots,i_k)$ is an $i_a$ for $1 \leq a < k$ such that $i_a > i_{a+1}$. The **cycle descent set** of $\pi$ is then the set of descents in all the cycles of $\pi$, and the **cycle descent number** is its cardinality.
Matching statistic: St000356
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [3,1,2] => 0
[3,2,1] => [1,3,2] => [3,1,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[4,1,2,3] => [1,4,3,2] => [4,1,3,2] => 1
[4,2,1,3] => [1,4,3,2] => [4,1,3,2] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
Description
The number of occurrences of the pattern 13-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.
Matching statistic: St000463
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000463: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000463: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [3,1,2] => 0
[3,2,1] => [1,3,2] => [3,1,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[4,1,2,3] => [1,4,3,2] => [4,1,3,2] => 1
[4,2,1,3] => [1,4,3,2] => [4,1,3,2] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
Description
The number of admissible inversions of a permutation.
Let $w = w_1,w_2,\dots,w_k$ be a word of length $k$ with distinct letters from $[n]$.
An admissible inversion of $w$ is a pair $(w_i,w_j)$ such that $1\leq i < j\leq k$ and $w_i > w_j$ that satisfies either of the following conditions:
$1 < i$ and $w_{i−1} < w_i$ or there is some $l$ such that $i < l < j$ and $w_i < w_l$.
Matching statistic: St000534
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000534: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000534: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [3,1,2] => 0
[3,2,1] => [1,3,2] => [3,1,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[4,1,2,3] => [1,4,3,2] => [4,1,3,2] => 1
[4,2,1,3] => [1,4,3,2] => [4,1,3,2] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
Description
The number of 2-rises of a permutation.
A 2-rise of a permutation $\pi$ is an index $i$ such that $\pi(i)+2 = \pi(i+1)$.
For 1-rises, or successions, see [[St000441]].
Matching statistic: St000624
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000624: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St000624: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [3,1,2] => 0
[3,2,1] => [1,3,2] => [3,1,2] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[4,1,2,3] => [1,4,3,2] => [4,1,3,2] => 1
[4,2,1,3] => [1,4,3,2] => [4,1,3,2] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
Description
The normalized sum of the minimal distances to a greater element.
Set $\pi_0 = \pi_{n+1} = n+1$, then this statistic is
$$
\sum_{i=1}^n \min_d(\pi_{i-1-d}>\pi_i\text{ or }\pi_{i+1+d}>\pi_i)
$$
A closely related statistic appears in [1].
The generating function for the sequence of maximal values attained on $\mathfrak S_r$, $r\geq 0$ apparently satisfies the functional equation
$$
(x-1)^2 (x+1)^3 f(x^2) - (x-1)^2 (x+1) f(x) + x^3 = 0.
$$
The following 654 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000646The number of big ascents of a permutation. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000871The number of very big ascents of a permutation. St000921The number of internal inversions of a binary word. St000934The 2-degree of an integer partition. St001061The number of indices that are both descents and recoils of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001394The genus of a permutation. St001470The cyclic holeyness of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001731The factorization defect of a permutation. St001777The number of weak descents in an integer composition. St001964The interval resolution global dimension of a poset. St000099The number of valleys of a permutation, including the boundary. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000619The number of cyclic descents of a permutation. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000651The maximal size of a rise in a permutation. St000652The maximal difference between successive positions of a permutation. St000883The number of longest increasing subsequences of a permutation. St000897The number of different multiplicities of parts of an integer partition. St000904The maximal number of repetitions of an integer composition. St000959The number of strong Bruhat factorizations of a permutation. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001735The number of permutations with the same set of runs. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000893The number of distinct diagonal sums of an alternating sign matrix. St001642The Prague dimension of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000089The absolute variation of a composition. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000142The number of even parts of a partition. St000150The floored half-sum of the multiplicities of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000217The number of occurrences of the pattern 312 in a permutation. St000222The number of alignments in the permutation. St000223The number of nestings in the permutation. St000237The number of small exceedances. St000241The number of cyclical small excedances. St000242The number of indices that are not cyclical small weak excedances. St000257The number of distinct parts of a partition that occur at least twice. St000292The number of ascents of a binary word. St000295The length of the border of a binary word. St000348The non-inversion sum of a binary word. St000352The Elizalde-Pak rank of a permutation. St000353The number of inner valleys of a permutation. St000355The number of occurrences of the pattern 21-3. St000358The number of occurrences of the pattern 31-2. St000359The number of occurrences of the pattern 23-1. St000360The number of occurrences of the pattern 32-1. St000366The number of double descents of a permutation. St000367The number of simsun double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000441The number of successions of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000486The number of cycles of length at least 3 of a permutation. St000538The number of even inversions 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. St000546The number of global descents of a permutation. St000552The number of cut vertices of a graph. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000556The number of occurrences of the pattern {{1},{2,3}} in a set partition. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000628The balance of a binary word. St000647The number of big descents of a permutation. St000648The number of 2-excedences of a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000665The number of rafts of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000682The Grundy value of Welter's game on a binary word. St000709The number of occurrences of 14-2-3 or 14-3-2. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000731The number of double exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000766The number of inversions of an integer composition. St000779The tier of a permutation. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St000850The number of 1/2-balanced pairs in a poset. St000872The number of very big descents of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000989The number of final rises of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001115The number of even descents of a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001153The number of blocks with even minimum in a set partition. St001172The number of 1-rises at odd height of a Dyck path. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001214The aft of an integer partition. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001307The number of induced stars on four vertices in a graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001323The independence gap of a graph. St001388The number of non-attacking neighbors of a permutation. St001403The number of vertical separators in a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001424The number of distinct squares in a binary word. St001469The holeyness of a permutation. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001578The minimal number of edges to add or remove to make a graph a line graph. St001584The area statistic between a Dyck path and its bounce path. St001587Half of the largest even part of an integer partition. St001638The book thickness of a graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001673The degree of asymmetry of an integer composition. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001689The number of celebrities in a graph. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000001The number of reduced words for a permutation. St000021The number of descents of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000060The greater neighbor of the maximum. St000092The number of outer peaks of a permutation. St000124The cardinality of the preimage of the Simion-Schmidt map. St000153The number of adjacent cycles of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000216The absolute length of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000277The number of ribbon shaped standard tableaux. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000307The number of rowmotion orbits of a poset. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000354The number of recoils of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000387The matching number of a graph. St000390The number of runs of ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000402Half the size of the symmetry class of a permutation. St000482The (zero)-forcing number of a graph. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000537The cutwidth of a graph. St000542The number of left-to-right-minima of a permutation. St000547The number of even non-empty partial sums of an integer partition. St000631The number of distinct palindromic decompositions of a binary word. St000654The first descent of a permutation. St000662The staircase size of the code of a permutation. St000669The number of permutations obtained by switching ascents or descents of size 2. St000670The reversal length of a permutation. St000701The protection number of a binary tree. St000703The number of deficiencies of a permutation. St000706The product of the factorials of the multiplicities of an integer partition. St000742The number of big ascents of a permutation after prepending zero. St000744The length of the path to the largest entry in a standard Young tableau. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000767The number of runs in an integer composition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000778The metric dimension of a graph. St000783The side length of the largest staircase partition fitting into a partition. St000785The number of distinct colouring schemes of a graph. St000809The reduced reflection length of the permutation. St000816The number of standard composition tableaux of the composition. St000820The number of compositions obtained by rotating the composition. St000829The Ulam distance of a permutation to the identity permutation. St000834The number of right outer peaks of a permutation. St000847The number of standard Young tableaux whose descent set is the binary word. St000862The number of parts of the shifted shape of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000886The number of permutations with the same antidiagonal sums. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000899The maximal number of repetitions of an integer composition. St000903The number of different parts of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000913The number of ways to refine the partition into singletons. St000919The number of maximal left branches of a binary tree. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St000988The orbit size of a permutation under Foata's bijection. St000990The first ascent of a permutation. St000993The multiplicity of the largest part of an integer partition. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001081The number of minimal length factorizations of a permutation into star transpositions. St001090The number of pop-stack-sorts needed to sort a permutation. St001111The weak 2-dynamic chromatic number of a graph. St001220The width of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001270The bandwidth of a graph. St001313The number of Dyck paths above the lattice path given by a binary word. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001375The pancake length of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001432The order dimension of the partition. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001593This is the number of standard Young tableaux of the given shifted shape. St001644The dimension of a graph. St001652The length of a longest interval of consecutive numbers. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001716The 1-improper chromatic number of a graph. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001884The number of borders of a binary word. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001928The number of non-overlapping descents in a permutation. St001933The largest multiplicity of a part in an integer partition. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000007The number of saliances of the permutation. St000058The order of a permutation. St000312The number of leaves in a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000325The width of the tree associated to a permutation. St000451The length of the longest pattern of the form k 1 2. St000469The distinguishing number of a graph. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000636The hull number of a graph. St000638The number of up-down runs of a permutation. St000702The number of weak deficiencies of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St000219The number of occurrences of the pattern 231 in a permutation. St001557The number of inversions of the second entry of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001060The distinguishing index of a graph. St000741The Colin de Verdière graph invariant. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000456The monochromatic index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001498The normalised height of a Nakayama algebra with magnitude 1. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001645The pebbling number of a connected graph. 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 $eA$. St001330The hat guessing number of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000264The girth of a graph, which is not a tree. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000929The constant term of the character polynomial of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001556The number of inversions of the third entry of a permutation. St001520The number of strict 3-descents. St001960The number of descents of a permutation minus one if its first entry is not one. St000958The number of Bruhat factorizations of a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001856The number of edges in the reduced word graph of a permutation. St001948The number of augmented double ascents of a permutation. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001487The number of inner corners of a skew partition. St001569The maximal modular displacement of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000495The number of inversions of distance at most 2 of a permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001863The number of weak excedances of a signed permutation. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000928The sum of the coefficients of the character polynomial of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000477The weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000997The even-odd crank of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000679The pruning number of an ordered tree. St001624The breadth of a lattice. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000022The number of fixed points of a permutation. St001625The Möbius invariant of a lattice. St001621The number of atoms of a lattice. St001618The cardinality of the Frattini sublattice of a lattice. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001309The number of four-cliques in a graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000911The number of maximal antichains of maximal size in a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000907The number of maximal antichains of minimal length in a poset. St000717The number of ordinal summands of a poset. St001857The number of edges in the reduced word graph of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St000657The smallest part of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000084The number of subtrees. St000328The maximum number of child nodes in a tree. St000077The number of boxed and circled entries. St000102The charge of a semistandard tableau. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000268The number of strongly connected orientations of a graph. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000323The minimal crossing number of a graph. St000338The number of pixed points of a permutation. St000344The number of strongly connected outdegree sequences of a graph. St000351The determinant of the adjacency matrix of a graph. St000365The number of double ascents of a permutation. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000462The major index minus the number of excedences of a permutation. St000496The rcs statistic of a set partition. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000516The number of stretching pairs of a permutation. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000623The number of occurrences of the pattern 52341 in a permutation. St000637The length of the longest cycle in a graph. St000666The number of right tethers of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000943The number of spots the most unlucky car had to go further in a parking function. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000962The 3-shifted major index of a permutation. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001082The number of boxed occurrences of 123 in a permutation. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001305The number of induced cycles on four vertices in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001354The number of series nodes in the modular decomposition of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001381The fertility of a permutation. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001402The number of separators in a permutation. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001736The total number of cycles in a graph. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001781The interlacing number of a set partition. St001793The difference between the clique number and the chromatic number of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001797The number of overfull subgraphs of a graph. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001847The number of occurrences of the pattern 1432 in a permutation. St001851The number of Hecke atoms of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001903The number of fixed points of a parking function. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000056The decomposition (or block) number of a permutation. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000253The crossing number of a set partition. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000272The treewidth of a graph. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000544The cop number of a graph. St000570The Edelman-Greene number of a permutation. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000640The rank of the largest boolean interval in a poset. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000694The number of affine bounded permutations that project to a given permutation. St000729The minimal arc length of a set partition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000864The number of circled entries of the shifted recording tableau of a permutation. St000873The aix statistic of a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000948The chromatic discriminant of a graph. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001162The minimum jump of a permutation. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001271The competition number of a graph. St001277The degeneracy of a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001344The neighbouring number of a permutation. St001353The number of prime nodes in the modular decomposition of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001358The largest degree of a regular subgraph of a graph. St001363The Euler characteristic of a graph according to Knill. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001546The number of monomials in the Tutte polynomial of a graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001592The maximal number of simple paths between any two different vertices of a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001806The upper middle entry of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001889The size of the connectivity set of a signed permutation. St000061The number of nodes on the left branch of a binary tree. St000105The number of blocks in the set partition. St000251The number of nonsingleton blocks of a set partition. St000487The length of the shortest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000633The size of the automorphism group of a poset. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000846The maximal number of elements covering an element of a poset. St001029The size of the core of a graph. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001119The length of a shortest maximal path in a graph. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001316The domatic number of a graph. St001399The distinguishing number of a poset. St001494The Alon-Tarsi number of a graph. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001580The acyclic chromatic number of a graph. St001623The number of doubly irreducible elements of a lattice. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001724The 2-packing differential of a graph. St001826The maximal number of leaves on a vertex of a graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000171The degree of the graph. St000172The Grundy number of a graph. St001108The 2-dynamic chromatic number of a graph. St001112The 3-weak dynamic number of a graph. St001118The acyclic chromatic index of a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001963The tree-depth of a graph. St001110The 3-dynamic chromatic number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001746The coalition number of a graph. St001871The number of triconnected components of a graph. St000782The indicator function of whether a given perfect matching is an L & P matching.
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