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Your data matches 140 different statistics following compositions of up to 3 maps.
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Matching statistic: St000007
(load all 34 compositions to match this statistic)
(load all 34 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => 1
[[2,2]]
=> [1,2] => 1
[[1],[2]]
=> [2,1] => 2
[[1,3]]
=> [1,2] => 1
[[2,3]]
=> [1,2] => 1
[[3,3]]
=> [1,2] => 1
[[1],[3]]
=> [2,1] => 2
[[2],[3]]
=> [2,1] => 2
[[1,1,2]]
=> [1,2,3] => 1
[[1,2,2]]
=> [1,2,3] => 1
[[2,2,2]]
=> [1,2,3] => 1
[[1,1],[2]]
=> [3,1,2] => 2
[[1,2],[2]]
=> [2,1,3] => 1
[[1,4]]
=> [1,2] => 1
[[2,4]]
=> [1,2] => 1
[[3,4]]
=> [1,2] => 1
[[4,4]]
=> [1,2] => 1
[[1],[4]]
=> [2,1] => 2
[[2],[4]]
=> [2,1] => 2
[[3],[4]]
=> [2,1] => 2
[[1,1,3]]
=> [1,2,3] => 1
[[1,2,3]]
=> [1,2,3] => 1
[[1,3,3]]
=> [1,2,3] => 1
[[2,2,3]]
=> [1,2,3] => 1
[[2,3,3]]
=> [1,2,3] => 1
[[3,3,3]]
=> [1,2,3] => 1
[[1,1],[3]]
=> [3,1,2] => 2
[[1,2],[3]]
=> [3,1,2] => 2
[[1,3],[2]]
=> [2,1,3] => 1
[[1,3],[3]]
=> [2,1,3] => 1
[[2,2],[3]]
=> [3,1,2] => 2
[[2,3],[3]]
=> [2,1,3] => 1
[[1],[2],[3]]
=> [3,2,1] => 3
[[1,1,1,2]]
=> [1,2,3,4] => 1
[[1,1,2,2]]
=> [1,2,3,4] => 1
[[1,2,2,2]]
=> [1,2,3,4] => 1
[[2,2,2,2]]
=> [1,2,3,4] => 1
[[1,1,1],[2]]
=> [4,1,2,3] => 2
[[1,1,2],[2]]
=> [3,1,2,4] => 1
[[1,2,2],[2]]
=> [2,1,3,4] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => 2
[[1,5]]
=> [1,2] => 1
[[2,5]]
=> [1,2] => 1
[[3,5]]
=> [1,2] => 1
[[4,5]]
=> [1,2] => 1
[[5,5]]
=> [1,2] => 1
[[1],[5]]
=> [2,1] => 2
[[2],[5]]
=> [2,1] => 2
[[3],[5]]
=> [2,1] => 2
[[4],[5]]
=> [2,1] => 2
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000069
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => ([(0,1)],2)
=> 1
[[2,2]]
=> [1,2] => ([(0,1)],2)
=> 1
[[1],[2]]
=> [2,1] => ([],2)
=> 2
[[1,3]]
=> [1,2] => ([(0,1)],2)
=> 1
[[2,3]]
=> [1,2] => ([(0,1)],2)
=> 1
[[3,3]]
=> [1,2] => ([(0,1)],2)
=> 1
[[1],[3]]
=> [2,1] => ([],2)
=> 2
[[2],[3]]
=> [2,1] => ([],2)
=> 2
[[1,1,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[2,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,1],[2]]
=> [3,1,2] => ([(1,2)],3)
=> 2
[[1,2],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[[1,4]]
=> [1,2] => ([(0,1)],2)
=> 1
[[2,4]]
=> [1,2] => ([(0,1)],2)
=> 1
[[3,4]]
=> [1,2] => ([(0,1)],2)
=> 1
[[4,4]]
=> [1,2] => ([(0,1)],2)
=> 1
[[1],[4]]
=> [2,1] => ([],2)
=> 2
[[2],[4]]
=> [2,1] => ([],2)
=> 2
[[3],[4]]
=> [2,1] => ([],2)
=> 2
[[1,1,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[2,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[2,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[3,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[1,1],[3]]
=> [3,1,2] => ([(1,2)],3)
=> 2
[[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> 2
[[1,3],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[[1,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[[2,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> 2
[[2,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[[1],[2],[3]]
=> [3,2,1] => ([],3)
=> 3
[[1,1,1,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,1],[2]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[[1,1,2],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[[1,2,2],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[[1,5]]
=> [1,2] => ([(0,1)],2)
=> 1
[[2,5]]
=> [1,2] => ([(0,1)],2)
=> 1
[[3,5]]
=> [1,2] => ([(0,1)],2)
=> 1
[[4,5]]
=> [1,2] => ([(0,1)],2)
=> 1
[[5,5]]
=> [1,2] => ([(0,1)],2)
=> 1
[[1],[5]]
=> [2,1] => ([],2)
=> 2
[[2],[5]]
=> [2,1] => ([],2)
=> 2
[[3],[5]]
=> [2,1] => ([],2)
=> 2
[[4],[5]]
=> [2,1] => ([],2)
=> 2
Description
The number of maximal elements of a poset.
Matching statistic: St000314
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [2,1] => 1
[[2,2]]
=> [1,2] => [2,1] => 1
[[1],[2]]
=> [2,1] => [1,2] => 2
[[1,3]]
=> [1,2] => [2,1] => 1
[[2,3]]
=> [1,2] => [2,1] => 1
[[3,3]]
=> [1,2] => [2,1] => 1
[[1],[3]]
=> [2,1] => [1,2] => 2
[[2],[3]]
=> [2,1] => [1,2] => 2
[[1,1,2]]
=> [1,2,3] => [3,2,1] => 1
[[1,2,2]]
=> [1,2,3] => [3,2,1] => 1
[[2,2,2]]
=> [1,2,3] => [3,2,1] => 1
[[1,1],[2]]
=> [3,1,2] => [2,1,3] => 2
[[1,2],[2]]
=> [2,1,3] => [3,1,2] => 1
[[1,4]]
=> [1,2] => [2,1] => 1
[[2,4]]
=> [1,2] => [2,1] => 1
[[3,4]]
=> [1,2] => [2,1] => 1
[[4,4]]
=> [1,2] => [2,1] => 1
[[1],[4]]
=> [2,1] => [1,2] => 2
[[2],[4]]
=> [2,1] => [1,2] => 2
[[3],[4]]
=> [2,1] => [1,2] => 2
[[1,1,3]]
=> [1,2,3] => [3,2,1] => 1
[[1,2,3]]
=> [1,2,3] => [3,2,1] => 1
[[1,3,3]]
=> [1,2,3] => [3,2,1] => 1
[[2,2,3]]
=> [1,2,3] => [3,2,1] => 1
[[2,3,3]]
=> [1,2,3] => [3,2,1] => 1
[[3,3,3]]
=> [1,2,3] => [3,2,1] => 1
[[1,1],[3]]
=> [3,1,2] => [2,1,3] => 2
[[1,2],[3]]
=> [3,1,2] => [2,1,3] => 2
[[1,3],[2]]
=> [2,1,3] => [3,1,2] => 1
[[1,3],[3]]
=> [2,1,3] => [3,1,2] => 1
[[2,2],[3]]
=> [3,1,2] => [2,1,3] => 2
[[2,3],[3]]
=> [2,1,3] => [3,1,2] => 1
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => 3
[[1,1,1,2]]
=> [1,2,3,4] => [4,3,2,1] => 1
[[1,1,2,2]]
=> [1,2,3,4] => [4,3,2,1] => 1
[[1,2,2,2]]
=> [1,2,3,4] => [4,3,2,1] => 1
[[2,2,2,2]]
=> [1,2,3,4] => [4,3,2,1] => 1
[[1,1,1],[2]]
=> [4,1,2,3] => [3,2,1,4] => 2
[[1,1,2],[2]]
=> [3,1,2,4] => [4,2,1,3] => 1
[[1,2,2],[2]]
=> [2,1,3,4] => [4,3,1,2] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => [2,1,4,3] => 2
[[1,5]]
=> [1,2] => [2,1] => 1
[[2,5]]
=> [1,2] => [2,1] => 1
[[3,5]]
=> [1,2] => [2,1] => 1
[[4,5]]
=> [1,2] => [2,1] => 1
[[5,5]]
=> [1,2] => [2,1] => 1
[[1],[5]]
=> [2,1] => [1,2] => 2
[[2],[5]]
=> [2,1] => [1,2] => 2
[[3],[5]]
=> [2,1] => [1,2] => 2
[[4],[5]]
=> [2,1] => [1,2] => 2
Description
The number of left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a '''left-to-right-maximum''' if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
Matching statistic: St000542
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,2] => 1
[[2,2]]
=> [1,2] => [1,2] => 1
[[1],[2]]
=> [2,1] => [2,1] => 2
[[1,3]]
=> [1,2] => [1,2] => 1
[[2,3]]
=> [1,2] => [1,2] => 1
[[3,3]]
=> [1,2] => [1,2] => 1
[[1],[3]]
=> [2,1] => [2,1] => 2
[[2],[3]]
=> [2,1] => [2,1] => 2
[[1,1,2]]
=> [1,2,3] => [1,2,3] => 1
[[1,2,2]]
=> [1,2,3] => [1,2,3] => 1
[[2,2,2]]
=> [1,2,3] => [1,2,3] => 1
[[1,1],[2]]
=> [3,1,2] => [1,3,2] => 1
[[1,2],[2]]
=> [2,1,3] => [2,3,1] => 2
[[1,4]]
=> [1,2] => [1,2] => 1
[[2,4]]
=> [1,2] => [1,2] => 1
[[3,4]]
=> [1,2] => [1,2] => 1
[[4,4]]
=> [1,2] => [1,2] => 1
[[1],[4]]
=> [2,1] => [2,1] => 2
[[2],[4]]
=> [2,1] => [2,1] => 2
[[3],[4]]
=> [2,1] => [2,1] => 2
[[1,1,3]]
=> [1,2,3] => [1,2,3] => 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => 1
[[1,3,3]]
=> [1,2,3] => [1,2,3] => 1
[[2,2,3]]
=> [1,2,3] => [1,2,3] => 1
[[2,3,3]]
=> [1,2,3] => [1,2,3] => 1
[[3,3,3]]
=> [1,2,3] => [1,2,3] => 1
[[1,1],[3]]
=> [3,1,2] => [1,3,2] => 1
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => 1
[[1,3],[2]]
=> [2,1,3] => [2,3,1] => 2
[[1,3],[3]]
=> [2,1,3] => [2,3,1] => 2
[[2,2],[3]]
=> [3,1,2] => [1,3,2] => 1
[[2,3],[3]]
=> [2,1,3] => [2,3,1] => 2
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 3
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => 1
[[1,1,1],[2]]
=> [4,1,2,3] => [1,2,4,3] => 1
[[1,1,2],[2]]
=> [3,1,2,4] => [1,3,4,2] => 1
[[1,2,2],[2]]
=> [2,1,3,4] => [2,3,4,1] => 2
[[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[1,5]]
=> [1,2] => [1,2] => 1
[[2,5]]
=> [1,2] => [1,2] => 1
[[3,5]]
=> [1,2] => [1,2] => 1
[[4,5]]
=> [1,2] => [1,2] => 1
[[5,5]]
=> [1,2] => [1,2] => 1
[[1],[5]]
=> [2,1] => [2,1] => 2
[[2],[5]]
=> [2,1] => [2,1] => 2
[[3],[5]]
=> [2,1] => [2,1] => 2
[[4],[5]]
=> [2,1] => [2,1] => 2
Description
The number of left-to-right-minima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a left-to-right-minimum if there does not exist a j < i such that $\sigma_j < \sigma_i$.
Matching statistic: St000740
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [2,1] => 1
[[2,2]]
=> [1,2] => [2,1] => 1
[[1],[2]]
=> [2,1] => [1,2] => 2
[[1,3]]
=> [1,2] => [2,1] => 1
[[2,3]]
=> [1,2] => [2,1] => 1
[[3,3]]
=> [1,2] => [2,1] => 1
[[1],[3]]
=> [2,1] => [1,2] => 2
[[2],[3]]
=> [2,1] => [1,2] => 2
[[1,1,2]]
=> [1,2,3] => [2,3,1] => 1
[[1,2,2]]
=> [1,2,3] => [2,3,1] => 1
[[2,2,2]]
=> [1,2,3] => [2,3,1] => 1
[[1,1],[2]]
=> [3,1,2] => [1,3,2] => 2
[[1,2],[2]]
=> [2,1,3] => [3,2,1] => 1
[[1,4]]
=> [1,2] => [2,1] => 1
[[2,4]]
=> [1,2] => [2,1] => 1
[[3,4]]
=> [1,2] => [2,1] => 1
[[4,4]]
=> [1,2] => [2,1] => 1
[[1],[4]]
=> [2,1] => [1,2] => 2
[[2],[4]]
=> [2,1] => [1,2] => 2
[[3],[4]]
=> [2,1] => [1,2] => 2
[[1,1,3]]
=> [1,2,3] => [2,3,1] => 1
[[1,2,3]]
=> [1,2,3] => [2,3,1] => 1
[[1,3,3]]
=> [1,2,3] => [2,3,1] => 1
[[2,2,3]]
=> [1,2,3] => [2,3,1] => 1
[[2,3,3]]
=> [1,2,3] => [2,3,1] => 1
[[3,3,3]]
=> [1,2,3] => [2,3,1] => 1
[[1,1],[3]]
=> [3,1,2] => [1,3,2] => 2
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => 2
[[1,3],[2]]
=> [2,1,3] => [3,2,1] => 1
[[1,3],[3]]
=> [2,1,3] => [3,2,1] => 1
[[2,2],[3]]
=> [3,1,2] => [1,3,2] => 2
[[2,3],[3]]
=> [2,1,3] => [3,2,1] => 1
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => 3
[[1,1,1,2]]
=> [1,2,3,4] => [2,3,4,1] => 1
[[1,1,2,2]]
=> [1,2,3,4] => [2,3,4,1] => 1
[[1,2,2,2]]
=> [1,2,3,4] => [2,3,4,1] => 1
[[2,2,2,2]]
=> [1,2,3,4] => [2,3,4,1] => 1
[[1,1,1],[2]]
=> [4,1,2,3] => [1,3,4,2] => 2
[[1,1,2],[2]]
=> [3,1,2,4] => [4,2,3,1] => 1
[[1,2,2],[2]]
=> [2,1,3,4] => [3,2,4,1] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => [4,1,3,2] => 2
[[1,5]]
=> [1,2] => [2,1] => 1
[[2,5]]
=> [1,2] => [2,1] => 1
[[3,5]]
=> [1,2] => [2,1] => 1
[[4,5]]
=> [1,2] => [2,1] => 1
[[5,5]]
=> [1,2] => [2,1] => 1
[[1],[5]]
=> [2,1] => [1,2] => 2
[[2],[5]]
=> [2,1] => [1,2] => 2
[[3],[5]]
=> [2,1] => [1,2] => 2
[[4],[5]]
=> [2,1] => [1,2] => 2
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St000991
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [2,1] => 1
[[2,2]]
=> [1,2] => [2,1] => 1
[[1],[2]]
=> [2,1] => [1,2] => 2
[[1,3]]
=> [1,2] => [2,1] => 1
[[2,3]]
=> [1,2] => [2,1] => 1
[[3,3]]
=> [1,2] => [2,1] => 1
[[1],[3]]
=> [2,1] => [1,2] => 2
[[2],[3]]
=> [2,1] => [1,2] => 2
[[1,1,2]]
=> [1,2,3] => [3,2,1] => 1
[[1,2,2]]
=> [1,2,3] => [3,2,1] => 1
[[2,2,2]]
=> [1,2,3] => [3,2,1] => 1
[[1,1],[2]]
=> [3,1,2] => [1,3,2] => 2
[[1,2],[2]]
=> [2,1,3] => [2,3,1] => 1
[[1,4]]
=> [1,2] => [2,1] => 1
[[2,4]]
=> [1,2] => [2,1] => 1
[[3,4]]
=> [1,2] => [2,1] => 1
[[4,4]]
=> [1,2] => [2,1] => 1
[[1],[4]]
=> [2,1] => [1,2] => 2
[[2],[4]]
=> [2,1] => [1,2] => 2
[[3],[4]]
=> [2,1] => [1,2] => 2
[[1,1,3]]
=> [1,2,3] => [3,2,1] => 1
[[1,2,3]]
=> [1,2,3] => [3,2,1] => 1
[[1,3,3]]
=> [1,2,3] => [3,2,1] => 1
[[2,2,3]]
=> [1,2,3] => [3,2,1] => 1
[[2,3,3]]
=> [1,2,3] => [3,2,1] => 1
[[3,3,3]]
=> [1,2,3] => [3,2,1] => 1
[[1,1],[3]]
=> [3,1,2] => [1,3,2] => 2
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => 2
[[1,3],[2]]
=> [2,1,3] => [2,3,1] => 1
[[1,3],[3]]
=> [2,1,3] => [2,3,1] => 1
[[2,2],[3]]
=> [3,1,2] => [1,3,2] => 2
[[2,3],[3]]
=> [2,1,3] => [2,3,1] => 1
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => 3
[[1,1,1,2]]
=> [1,2,3,4] => [4,3,2,1] => 1
[[1,1,2,2]]
=> [1,2,3,4] => [4,3,2,1] => 1
[[1,2,2,2]]
=> [1,2,3,4] => [4,3,2,1] => 1
[[2,2,2,2]]
=> [1,2,3,4] => [4,3,2,1] => 1
[[1,1,1],[2]]
=> [4,1,2,3] => [1,4,3,2] => 2
[[1,1,2],[2]]
=> [3,1,2,4] => [2,4,3,1] => 1
[[1,2,2],[2]]
=> [2,1,3,4] => [3,4,2,1] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => [2,1,4,3] => 2
[[1,5]]
=> [1,2] => [2,1] => 1
[[2,5]]
=> [1,2] => [2,1] => 1
[[3,5]]
=> [1,2] => [2,1] => 1
[[4,5]]
=> [1,2] => [2,1] => 1
[[5,5]]
=> [1,2] => [2,1] => 1
[[1],[5]]
=> [2,1] => [1,2] => 2
[[2],[5]]
=> [2,1] => [1,2] => 2
[[3],[5]]
=> [2,1] => [1,2] => 2
[[4],[5]]
=> [2,1] => [1,2] => 2
Description
The number of right-to-left minima of a permutation.
For the number of left-to-right maxima, see [[St000314]].
Matching statistic: St000541
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [1,2] => 0 = 1 - 1
[[2,2]]
=> [1,2] => [1,2] => 0 = 1 - 1
[[1],[2]]
=> [2,1] => [2,1] => 1 = 2 - 1
[[1,3]]
=> [1,2] => [1,2] => 0 = 1 - 1
[[2,3]]
=> [1,2] => [1,2] => 0 = 1 - 1
[[3,3]]
=> [1,2] => [1,2] => 0 = 1 - 1
[[1],[3]]
=> [2,1] => [2,1] => 1 = 2 - 1
[[2],[3]]
=> [2,1] => [2,1] => 1 = 2 - 1
[[1,1,2]]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,2,2]]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,2,2]]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,1],[2]]
=> [3,1,2] => [1,3,2] => 0 = 1 - 1
[[1,2],[2]]
=> [2,1,3] => [2,3,1] => 1 = 2 - 1
[[1,4]]
=> [1,2] => [1,2] => 0 = 1 - 1
[[2,4]]
=> [1,2] => [1,2] => 0 = 1 - 1
[[3,4]]
=> [1,2] => [1,2] => 0 = 1 - 1
[[4,4]]
=> [1,2] => [1,2] => 0 = 1 - 1
[[1],[4]]
=> [2,1] => [2,1] => 1 = 2 - 1
[[2],[4]]
=> [2,1] => [2,1] => 1 = 2 - 1
[[3],[4]]
=> [2,1] => [2,1] => 1 = 2 - 1
[[1,1,3]]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,3,3]]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,2,3]]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[[2,3,3]]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[[3,3,3]]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[[1,1],[3]]
=> [3,1,2] => [1,3,2] => 0 = 1 - 1
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => 0 = 1 - 1
[[1,3],[2]]
=> [2,1,3] => [2,3,1] => 1 = 2 - 1
[[1,3],[3]]
=> [2,1,3] => [2,3,1] => 1 = 2 - 1
[[2,2],[3]]
=> [3,1,2] => [1,3,2] => 0 = 1 - 1
[[2,3],[3]]
=> [2,1,3] => [2,3,1] => 1 = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 2 = 3 - 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[1,1,1],[2]]
=> [4,1,2,3] => [1,2,4,3] => 0 = 1 - 1
[[1,1,2],[2]]
=> [3,1,2,4] => [1,3,4,2] => 0 = 1 - 1
[[1,2,2],[2]]
=> [2,1,3,4] => [2,3,4,1] => 1 = 2 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[1,5]]
=> [1,2] => [1,2] => 0 = 1 - 1
[[2,5]]
=> [1,2] => [1,2] => 0 = 1 - 1
[[3,5]]
=> [1,2] => [1,2] => 0 = 1 - 1
[[4,5]]
=> [1,2] => [1,2] => 0 = 1 - 1
[[5,5]]
=> [1,2] => [1,2] => 0 = 1 - 1
[[1],[5]]
=> [2,1] => [2,1] => 1 = 2 - 1
[[2],[5]]
=> [2,1] => [2,1] => 1 = 2 - 1
[[3],[5]]
=> [2,1] => [2,1] => 1 = 2 - 1
[[4],[5]]
=> [2,1] => [2,1] => 1 = 2 - 1
Description
The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right.
For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.
Matching statistic: St000015
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1
[[2,2]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1
[[1],[2]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 2
[[1,3]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1
[[2,3]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1
[[3,3]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1
[[1],[3]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 2
[[2],[3]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 2
[[1,1,2]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[[1,2,2]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[[2,2,2]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[[1,1],[2]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[1,2],[2]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[[1,4]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1
[[2,4]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1
[[3,4]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1
[[4,4]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1
[[1],[4]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 2
[[2],[4]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 2
[[3],[4]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 2
[[1,1,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[[1,2,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[[1,3,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[[2,2,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[[2,3,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[[3,3,3]]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[[1,1],[3]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[1,2],[3]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[1,3],[2]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[[1,3],[3]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[[2,2],[3]]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[[2,3],[3]]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[[1,1,1,2]]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[[1,1,2,2]]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[[1,2,2,2]]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[[2,2,2,2]]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[[1,1,1],[2]]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[[1,1,2],[2]]
=> [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
[[1,2,2],[2]]
=> [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[[1,5]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1
[[2,5]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1
[[3,5]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1
[[4,5]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1
[[5,5]]
=> [1,2] => [2,1] => [1,1,0,0]
=> 1
[[1],[5]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 2
[[2],[5]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 2
[[3],[5]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 2
[[4],[5]]
=> [2,1] => [1,2] => [1,0,1,0]
=> 2
Description
The number of peaks of a Dyck path.
Matching statistic: St000031
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [2,1] => [2,1] => 1
[[2,2]]
=> [1,2] => [2,1] => [2,1] => 1
[[1],[2]]
=> [2,1] => [1,2] => [1,2] => 2
[[1,3]]
=> [1,2] => [2,1] => [2,1] => 1
[[2,3]]
=> [1,2] => [2,1] => [2,1] => 1
[[3,3]]
=> [1,2] => [2,1] => [2,1] => 1
[[1],[3]]
=> [2,1] => [1,2] => [1,2] => 2
[[2],[3]]
=> [2,1] => [1,2] => [1,2] => 2
[[1,1,2]]
=> [1,2,3] => [3,2,1] => [2,3,1] => 1
[[1,2,2]]
=> [1,2,3] => [3,2,1] => [2,3,1] => 1
[[2,2,2]]
=> [1,2,3] => [3,2,1] => [2,3,1] => 1
[[1,1],[2]]
=> [3,1,2] => [2,1,3] => [2,1,3] => 2
[[1,2],[2]]
=> [2,1,3] => [3,1,2] => [3,1,2] => 1
[[1,4]]
=> [1,2] => [2,1] => [2,1] => 1
[[2,4]]
=> [1,2] => [2,1] => [2,1] => 1
[[3,4]]
=> [1,2] => [2,1] => [2,1] => 1
[[4,4]]
=> [1,2] => [2,1] => [2,1] => 1
[[1],[4]]
=> [2,1] => [1,2] => [1,2] => 2
[[2],[4]]
=> [2,1] => [1,2] => [1,2] => 2
[[3],[4]]
=> [2,1] => [1,2] => [1,2] => 2
[[1,1,3]]
=> [1,2,3] => [3,2,1] => [2,3,1] => 1
[[1,2,3]]
=> [1,2,3] => [3,2,1] => [2,3,1] => 1
[[1,3,3]]
=> [1,2,3] => [3,2,1] => [2,3,1] => 1
[[2,2,3]]
=> [1,2,3] => [3,2,1] => [2,3,1] => 1
[[2,3,3]]
=> [1,2,3] => [3,2,1] => [2,3,1] => 1
[[3,3,3]]
=> [1,2,3] => [3,2,1] => [2,3,1] => 1
[[1,1],[3]]
=> [3,1,2] => [2,1,3] => [2,1,3] => 2
[[1,2],[3]]
=> [3,1,2] => [2,1,3] => [2,1,3] => 2
[[1,3],[2]]
=> [2,1,3] => [3,1,2] => [3,1,2] => 1
[[1,3],[3]]
=> [2,1,3] => [3,1,2] => [3,1,2] => 1
[[2,2],[3]]
=> [3,1,2] => [2,1,3] => [2,1,3] => 2
[[2,3],[3]]
=> [2,1,3] => [3,1,2] => [3,1,2] => 1
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 3
[[1,1,1,2]]
=> [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 1
[[1,1,2,2]]
=> [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 1
[[1,2,2,2]]
=> [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 1
[[2,2,2,2]]
=> [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 1
[[1,1,1],[2]]
=> [4,1,2,3] => [3,2,1,4] => [2,3,1,4] => 2
[[1,1,2],[2]]
=> [3,1,2,4] => [4,2,1,3] => [2,4,1,3] => 1
[[1,2,2],[2]]
=> [2,1,3,4] => [4,3,1,2] => [3,1,4,2] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2
[[1,5]]
=> [1,2] => [2,1] => [2,1] => 1
[[2,5]]
=> [1,2] => [2,1] => [2,1] => 1
[[3,5]]
=> [1,2] => [2,1] => [2,1] => 1
[[4,5]]
=> [1,2] => [2,1] => [2,1] => 1
[[5,5]]
=> [1,2] => [2,1] => [2,1] => 1
[[1],[5]]
=> [2,1] => [1,2] => [1,2] => 2
[[2],[5]]
=> [2,1] => [1,2] => [1,2] => 2
[[3],[5]]
=> [2,1] => [1,2] => [1,2] => 2
[[4],[5]]
=> [2,1] => [1,2] => [1,2] => 2
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St000054
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,2] => [2,1] => [1,2] => 1
[[2,2]]
=> [1,2] => [2,1] => [1,2] => 1
[[1],[2]]
=> [2,1] => [1,2] => [2,1] => 2
[[1,3]]
=> [1,2] => [2,1] => [1,2] => 1
[[2,3]]
=> [1,2] => [2,1] => [1,2] => 1
[[3,3]]
=> [1,2] => [2,1] => [1,2] => 1
[[1],[3]]
=> [2,1] => [1,2] => [2,1] => 2
[[2],[3]]
=> [2,1] => [1,2] => [2,1] => 2
[[1,1,2]]
=> [1,2,3] => [2,3,1] => [1,3,2] => 1
[[1,2,2]]
=> [1,2,3] => [2,3,1] => [1,3,2] => 1
[[2,2,2]]
=> [1,2,3] => [2,3,1] => [1,3,2] => 1
[[1,1],[2]]
=> [3,1,2] => [1,3,2] => [2,3,1] => 2
[[1,2],[2]]
=> [2,1,3] => [3,2,1] => [1,2,3] => 1
[[1,4]]
=> [1,2] => [2,1] => [1,2] => 1
[[2,4]]
=> [1,2] => [2,1] => [1,2] => 1
[[3,4]]
=> [1,2] => [2,1] => [1,2] => 1
[[4,4]]
=> [1,2] => [2,1] => [1,2] => 1
[[1],[4]]
=> [2,1] => [1,2] => [2,1] => 2
[[2],[4]]
=> [2,1] => [1,2] => [2,1] => 2
[[3],[4]]
=> [2,1] => [1,2] => [2,1] => 2
[[1,1,3]]
=> [1,2,3] => [2,3,1] => [1,3,2] => 1
[[1,2,3]]
=> [1,2,3] => [2,3,1] => [1,3,2] => 1
[[1,3,3]]
=> [1,2,3] => [2,3,1] => [1,3,2] => 1
[[2,2,3]]
=> [1,2,3] => [2,3,1] => [1,3,2] => 1
[[2,3,3]]
=> [1,2,3] => [2,3,1] => [1,3,2] => 1
[[3,3,3]]
=> [1,2,3] => [2,3,1] => [1,3,2] => 1
[[1,1],[3]]
=> [3,1,2] => [1,3,2] => [2,3,1] => 2
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => [2,3,1] => 2
[[1,3],[2]]
=> [2,1,3] => [3,2,1] => [1,2,3] => 1
[[1,3],[3]]
=> [2,1,3] => [3,2,1] => [1,2,3] => 1
[[2,2],[3]]
=> [3,1,2] => [1,3,2] => [2,3,1] => 2
[[2,3],[3]]
=> [2,1,3] => [3,2,1] => [1,2,3] => 1
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [3,2,1] => 3
[[1,1,1,2]]
=> [1,2,3,4] => [2,3,4,1] => [1,4,3,2] => 1
[[1,1,2,2]]
=> [1,2,3,4] => [2,3,4,1] => [1,4,3,2] => 1
[[1,2,2,2]]
=> [1,2,3,4] => [2,3,4,1] => [1,4,3,2] => 1
[[2,2,2,2]]
=> [1,2,3,4] => [2,3,4,1] => [1,4,3,2] => 1
[[1,1,1],[2]]
=> [4,1,2,3] => [1,3,4,2] => [2,4,3,1] => 2
[[1,1,2],[2]]
=> [3,1,2,4] => [4,2,3,1] => [1,3,2,4] => 1
[[1,2,2],[2]]
=> [2,1,3,4] => [3,2,4,1] => [1,4,2,3] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => [4,1,3,2] => [2,3,1,4] => 2
[[1,5]]
=> [1,2] => [2,1] => [1,2] => 1
[[2,5]]
=> [1,2] => [2,1] => [1,2] => 1
[[3,5]]
=> [1,2] => [2,1] => [1,2] => 1
[[4,5]]
=> [1,2] => [2,1] => [1,2] => 1
[[5,5]]
=> [1,2] => [2,1] => [1,2] => 1
[[1],[5]]
=> [2,1] => [1,2] => [2,1] => 2
[[2],[5]]
=> [2,1] => [1,2] => [2,1] => 2
[[3],[5]]
=> [2,1] => [1,2] => [2,1] => 2
[[4],[5]]
=> [2,1] => [1,2] => [2,1] => 2
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
The following 130 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000061The number of nodes on the left branch of a binary tree. St000654The first descent of a permutation. St000990The first ascent of a permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001829The common independence number of a graph. St000053The number of valleys of the Dyck path. St000133The "bounce" of a permutation. St000306The bounce count of a Dyck path. St000989The number of final rises of a permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000068The number of minimal elements in a poset. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000260The radius of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000444The length of the maximal rise of a Dyck path. St000668The least common multiple of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000744The length of the path to the largest entry in a standard Young tableau. St000937The number of positive values of the symmetric group character corresponding to the partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000259The diameter of a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000942The number of critical left to right maxima of the parking functions. St001937The size of the center of a parking function. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. 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. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001733The number of weak left to right maxima of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St000981The length of the longest zigzag subpath. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. 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. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St001330The hat guessing number of a graph. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001323The independence gap of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001118The acyclic chromatic index of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001624The breadth of a lattice. St000454The largest eigenvalue of a graph if it is integral. St000456The monochromatic index of a connected graph. St000464The Schultz index of a connected graph. St001281The normalized isoperimetric number of a graph. St001545The second Elser number of a connected graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000736The last entry in the first row of a semistandard tableau. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000103The sum of the entries of a semistandard tableau. St000264The girth of a graph, which is not a tree. 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
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