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Your data matches 315 different statistics following compositions of up to 3 maps.
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Matching statistic: St000007
(load all 63 compositions to match this statistic)
(load all 63 compositions to match this statistic)
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 1
[2,1] => 2
[1,2,3] => 1
[1,3,2] => 2
[2,1,3] => 1
[2,3,1] => 2
[3,1,2] => 2
[3,2,1] => 3
[1,2,3,4] => 1
[1,2,4,3] => 2
[1,3,2,4] => 1
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 3
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 1
[2,3,4,1] => 2
[2,4,1,3] => 2
[2,4,3,1] => 3
[3,1,2,4] => 1
[3,1,4,2] => 2
[3,2,1,4] => 1
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 3
[4,1,2,3] => 2
[4,1,3,2] => 3
[4,2,1,3] => 2
[4,2,3,1] => 3
[4,3,1,2] => 3
[4,3,2,1] => 4
[1,2,3,4,5] => 1
[1,2,3,5,4] => 2
[1,2,4,3,5] => 1
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 3
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 1
[1,3,4,5,2] => 2
[1,3,5,2,4] => 2
[1,3,5,4,2] => 3
[1,4,2,3,5] => 1
[1,4,2,5,3] => 2
[1,4,3,2,5] => 1
[1,4,3,5,2] => 2
[1,4,5,2,3] => 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: St000031
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 2
[2,1] => 1
[1,2,3] => 3
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 2
[1,2,3,4] => 4
[1,2,4,3] => 3
[1,3,2,4] => 3
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 3
[2,1,3,4] => 3
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 2
[3,1,2,4] => 2
[3,1,4,2] => 1
[3,2,1,4] => 3
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 3
[4,3,1,2] => 1
[4,3,2,1] => 2
[1,2,3,4,5] => 5
[1,2,3,5,4] => 4
[1,2,4,3,5] => 4
[1,2,4,5,3] => 3
[1,2,5,3,4] => 3
[1,2,5,4,3] => 4
[1,3,2,4,5] => 4
[1,3,2,5,4] => 3
[1,3,4,2,5] => 3
[1,3,4,5,2] => 2
[1,3,5,2,4] => 2
[1,3,5,4,2] => 3
[1,4,2,3,5] => 3
[1,4,2,5,3] => 2
[1,4,3,2,5] => 4
[1,4,3,5,2] => 3
[1,4,5,2,3] => 3
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St000314
(load all 48 compositions to match this statistic)
(load all 48 compositions to match this statistic)
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 2
[2,1] => 1
[1,2,3] => 3
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 2
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 4
[1,2,4,3] => 3
[1,3,2,4] => 3
[1,3,4,2] => 3
[1,4,2,3] => 2
[1,4,3,2] => 2
[2,1,3,4] => 3
[2,1,4,3] => 2
[2,3,1,4] => 3
[2,3,4,1] => 3
[2,4,1,3] => 2
[2,4,3,1] => 2
[3,1,2,4] => 2
[3,1,4,2] => 2
[3,2,1,4] => 2
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 2
[4,1,2,3] => 1
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 1
[1,2,3,4,5] => 5
[1,2,3,5,4] => 4
[1,2,4,3,5] => 4
[1,2,4,5,3] => 4
[1,2,5,3,4] => 3
[1,2,5,4,3] => 3
[1,3,2,4,5] => 4
[1,3,2,5,4] => 3
[1,3,4,2,5] => 4
[1,3,4,5,2] => 4
[1,3,5,2,4] => 3
[1,3,5,4,2] => 3
[1,4,2,3,5] => 3
[1,4,2,5,3] => 3
[1,4,3,2,5] => 3
[1,4,3,5,2] => 3
[1,4,5,2,3] => 3
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 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
St000542: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 1
[2,1] => 2
[1,2,3] => 1
[1,3,2] => 1
[2,1,3] => 2
[2,3,1] => 2
[3,1,2] => 2
[3,2,1] => 3
[1,2,3,4] => 1
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 2
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 2
[2,4,1,3] => 2
[2,4,3,1] => 2
[3,1,2,4] => 2
[3,1,4,2] => 2
[3,2,1,4] => 3
[3,2,4,1] => 3
[3,4,1,2] => 2
[3,4,2,1] => 3
[4,1,2,3] => 2
[4,1,3,2] => 2
[4,2,1,3] => 3
[4,2,3,1] => 3
[4,3,1,2] => 3
[4,3,2,1] => 4
[1,2,3,4,5] => 1
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 1
[1,3,2,4,5] => 1
[1,3,2,5,4] => 1
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 1
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: St000991
(load all 86 compositions to match this statistic)
(load all 86 compositions to match this statistic)
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 2
[2,1] => 1
[1,2,3] => 3
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 1
[3,1,2] => 2
[3,2,1] => 1
[1,2,3,4] => 4
[1,2,4,3] => 3
[1,3,2,4] => 3
[1,3,4,2] => 2
[1,4,2,3] => 3
[1,4,3,2] => 2
[2,1,3,4] => 3
[2,1,4,3] => 2
[2,3,1,4] => 2
[2,3,4,1] => 1
[2,4,1,3] => 2
[2,4,3,1] => 1
[3,1,2,4] => 3
[3,1,4,2] => 2
[3,2,1,4] => 2
[3,2,4,1] => 1
[3,4,1,2] => 2
[3,4,2,1] => 1
[4,1,2,3] => 3
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 1
[4,3,1,2] => 2
[4,3,2,1] => 1
[1,2,3,4,5] => 5
[1,2,3,5,4] => 4
[1,2,4,3,5] => 4
[1,2,4,5,3] => 3
[1,2,5,3,4] => 4
[1,2,5,4,3] => 3
[1,3,2,4,5] => 4
[1,3,2,5,4] => 3
[1,3,4,2,5] => 3
[1,3,4,5,2] => 2
[1,3,5,2,4] => 3
[1,3,5,4,2] => 2
[1,4,2,3,5] => 4
[1,4,2,5,3] => 3
[1,4,3,2,5] => 3
[1,4,3,5,2] => 2
[1,4,5,2,3] => 3
Description
The number of right-to-left minima of a permutation.
For the number of left-to-right maxima, see [[St000314]].
Matching statistic: St000010
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 1
[1,2] => [1,1]
=> 2
[2,1] => [2]
=> 1
[1,2,3] => [1,1,1]
=> 3
[1,3,2] => [2,1]
=> 2
[2,1,3] => [2,1]
=> 2
[2,3,1] => [3]
=> 1
[3,1,2] => [3]
=> 1
[3,2,1] => [2,1]
=> 2
[1,2,3,4] => [1,1,1,1]
=> 4
[1,2,4,3] => [2,1,1]
=> 3
[1,3,2,4] => [2,1,1]
=> 3
[1,3,4,2] => [3,1]
=> 2
[1,4,2,3] => [3,1]
=> 2
[1,4,3,2] => [2,1,1]
=> 3
[2,1,3,4] => [2,1,1]
=> 3
[2,1,4,3] => [2,2]
=> 2
[2,3,1,4] => [3,1]
=> 2
[2,3,4,1] => [4]
=> 1
[2,4,1,3] => [4]
=> 1
[2,4,3,1] => [3,1]
=> 2
[3,1,2,4] => [3,1]
=> 2
[3,1,4,2] => [4]
=> 1
[3,2,1,4] => [2,1,1]
=> 3
[3,2,4,1] => [3,1]
=> 2
[3,4,1,2] => [2,2]
=> 2
[3,4,2,1] => [4]
=> 1
[4,1,2,3] => [4]
=> 1
[4,1,3,2] => [3,1]
=> 2
[4,2,1,3] => [3,1]
=> 2
[4,2,3,1] => [2,1,1]
=> 3
[4,3,1,2] => [4]
=> 1
[4,3,2,1] => [2,2]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> 5
[1,2,3,5,4] => [2,1,1,1]
=> 4
[1,2,4,3,5] => [2,1,1,1]
=> 4
[1,2,4,5,3] => [3,1,1]
=> 3
[1,2,5,3,4] => [3,1,1]
=> 3
[1,2,5,4,3] => [2,1,1,1]
=> 4
[1,3,2,4,5] => [2,1,1,1]
=> 4
[1,3,2,5,4] => [2,2,1]
=> 3
[1,3,4,2,5] => [3,1,1]
=> 3
[1,3,4,5,2] => [4,1]
=> 2
[1,3,5,2,4] => [4,1]
=> 2
[1,3,5,4,2] => [3,1,1]
=> 3
[1,4,2,3,5] => [3,1,1]
=> 3
[1,4,2,5,3] => [4,1]
=> 2
[1,4,3,2,5] => [2,1,1,1]
=> 4
[1,4,3,5,2] => [3,1,1]
=> 3
[1,4,5,2,3] => [2,2,1]
=> 3
Description
The length of the partition.
Matching statistic: St000015
(load all 52 compositions to match this statistic)
(load all 52 compositions to match this statistic)
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%
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1
[1,2] => [1,0,1,0]
=> 2
[2,1] => [1,1,0,0]
=> 1
[1,2,3] => [1,0,1,0,1,0]
=> 3
[1,3,2] => [1,0,1,1,0,0]
=> 2
[2,1,3] => [1,1,0,0,1,0]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> 2
[3,1,2] => [1,1,1,0,0,0]
=> 1
[3,2,1] => [1,1,1,0,0,0]
=> 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 4
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 4
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 4
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 3
Description
The number of peaks of a Dyck path.
Matching statistic: St000068
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00065: Permutations —permutation poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 1
[1,2] => ([(0,1)],2)
=> 1
[2,1] => ([],2)
=> 2
[1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,3,2] => ([(0,1),(0,2)],3)
=> 1
[2,1,3] => ([(0,2),(1,2)],3)
=> 2
[2,3,1] => ([(1,2)],3)
=> 2
[3,1,2] => ([(1,2)],3)
=> 2
[3,2,1] => ([],3)
=> 3
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 1
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 2
[2,4,3,1] => ([(1,2),(1,3)],4)
=> 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 2
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 2
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,2,4,1] => ([(1,3),(2,3)],4)
=> 3
[3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[3,4,2,1] => ([(2,3)],4)
=> 3
[4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[4,1,3,2] => ([(1,2),(1,3)],4)
=> 2
[4,2,1,3] => ([(1,3),(2,3)],4)
=> 3
[4,2,3,1] => ([(2,3)],4)
=> 3
[4,3,1,2] => ([(2,3)],4)
=> 3
[4,3,2,1] => ([],4)
=> 4
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 1
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> 1
Description
The number of minimal elements in a poset.
Matching statistic: St000069
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> 1
[1,2] => ([(0,1)],2)
=> 1
[2,1] => ([],2)
=> 2
[1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,3,2] => ([(0,1),(0,2)],3)
=> 2
[2,1,3] => ([(0,2),(1,2)],3)
=> 1
[2,3,1] => ([(1,2)],3)
=> 2
[3,1,2] => ([(1,2)],3)
=> 2
[3,2,1] => ([],3)
=> 3
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 3
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 2
[2,4,3,1] => ([(1,2),(1,3)],4)
=> 3
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 2
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[3,4,2,1] => ([(2,3)],4)
=> 3
[4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[4,1,3,2] => ([(1,2),(1,3)],4)
=> 3
[4,2,1,3] => ([(1,3),(2,3)],4)
=> 2
[4,2,3,1] => ([(2,3)],4)
=> 3
[4,3,1,2] => ([(2,3)],4)
=> 3
[4,3,2,1] => ([],4)
=> 4
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> 3
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> 3
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 2
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
Description
The number of maximal elements of a poset.
Matching statistic: St000105
(load all 78 compositions to match this statistic)
(load all 78 compositions to match this statistic)
Mp00151: Permutations —to cycle type⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
=> 1
[1,2] => {{1},{2}}
=> 2
[2,1] => {{1,2}}
=> 1
[1,2,3] => {{1},{2},{3}}
=> 3
[1,3,2] => {{1},{2,3}}
=> 2
[2,1,3] => {{1,2},{3}}
=> 2
[2,3,1] => {{1,2,3}}
=> 1
[3,1,2] => {{1,2,3}}
=> 1
[3,2,1] => {{1,3},{2}}
=> 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> 4
[1,2,4,3] => {{1},{2},{3,4}}
=> 3
[1,3,2,4] => {{1},{2,3},{4}}
=> 3
[1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,4,2,3] => {{1},{2,3,4}}
=> 2
[1,4,3,2] => {{1},{2,4},{3}}
=> 3
[2,1,3,4] => {{1,2},{3},{4}}
=> 3
[2,1,4,3] => {{1,2},{3,4}}
=> 2
[2,3,1,4] => {{1,2,3},{4}}
=> 2
[2,3,4,1] => {{1,2,3,4}}
=> 1
[2,4,1,3] => {{1,2,3,4}}
=> 1
[2,4,3,1] => {{1,2,4},{3}}
=> 2
[3,1,2,4] => {{1,2,3},{4}}
=> 2
[3,1,4,2] => {{1,2,3,4}}
=> 1
[3,2,1,4] => {{1,3},{2},{4}}
=> 3
[3,2,4,1] => {{1,3,4},{2}}
=> 2
[3,4,1,2] => {{1,3},{2,4}}
=> 2
[3,4,2,1] => {{1,2,3,4}}
=> 1
[4,1,2,3] => {{1,2,3,4}}
=> 1
[4,1,3,2] => {{1,2,4},{3}}
=> 2
[4,2,1,3] => {{1,3,4},{2}}
=> 2
[4,2,3,1] => {{1,4},{2},{3}}
=> 3
[4,3,1,2] => {{1,2,3,4}}
=> 1
[4,3,2,1] => {{1,4},{2,3}}
=> 2
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 5
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 4
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 4
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 3
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 3
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 4
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 4
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 3
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 3
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> 2
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 3
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 3
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> 2
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 4
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 3
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 3
Description
The number of blocks in the set partition.
The generating function of this statistic yields the famous [[wiki:Stirling numbers of the second kind|Stirling numbers of the second kind]] $S_2(n,k)$ given by the number of [[SetPartitions|set partitions]] of $\{ 1,\ldots,n\}$ into $k$ blocks, see [1].
The following 305 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000011The number of touch points (or returns) of a Dyck path. St000025The number of initial rises of a Dyck path. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000084The number of subtrees. St000147The largest part of an integer partition. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000273The domination number of a graph. St000288The number of ones in a binary word. St000291The number of descents of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000325The width of the tree associated to a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000378The diagonal inversion number of an integer partition. St000390The number of runs of ones in a binary word. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000702The number of weak deficiencies of a permutation. St000703The number of deficiencies of a permutation. St000733The row containing the largest entry of a standard tableau. St000740The last entry of a permutation. St000912The number of maximal antichains in a poset. St000916The packing number of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one 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:
St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001829The common independence number of a graph. St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000133The "bounce" of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000159The number of distinct parts of the integer partition. St000234The number of global ascents of a permutation. St000245The number of ascents of a permutation. St000292The number of ascents of a binary word. St000306The bounce count of a Dyck path. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000340The number of non-final maximal constant sub-paths of length greater than one. St000439The position of the first down step of a Dyck path. St000519The largest length of a factor maximising the subword complexity. St000546The number of global descents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000996The number of exclusive left-to-right maxima 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. St001180Number of indecomposable injective modules with projective dimension at most 1. 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 $eA$. St001489The maximum of the number of descents and the number of inverse descents. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000006The dinv of a Dyck path. St000013The height of a Dyck path. St000035The number of left outer peaks of a permutation. St000054The first entry of the permutation. St000058The order of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000071The number of maximal chains in a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000153The number of adjacent cycles of a permutation. St000172The Grundy number of a graph. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000237The number of small exceedances. St000240The number of indices that are not small excedances. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000297The number of leading ones in a binary word. St000308The height of the tree associated to a permutation. St000382The first part of an integer composition. St000383The last part of an integer composition. St000389The number of runs of ones of odd length in a binary word. St000392The length of the longest run of ones in a binary word. St000507The number of ascents of a standard tableau. St000527The width of the poset. St000528The height of a poset. St000544The cop number of a graph. St000619The number of cyclic descents of a permutation. St000653The last descent of a permutation. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000734The last entry in the first row of a standard tableau. St000742The number of big ascents of a permutation after prepending zero. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000822The Hadwiger number of the graph. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000843The decomposition number of a perfect matching. St000908The length of the shortest maximal antichain in a poset. St000925The number of topologically connected components of a set partition. St000971The smallest closer of a set partition. St001029The size of the core of a graph. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001116The game chromatic number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call 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 special CNakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001372The length of a longest cyclic run of ones of a binary word. St001389The number of partitions of the same length below the given integer partition. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001480The number of simple summands of the module J^2/J^3. St001494The Alon-Tarsi number of a graph. St001530The depth of a Dyck path. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001963The tree-depth of a graph. St000028The number of stack-sorts needed to sort a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000080The rank of the poset. St000083The number of left oriented leafs of a binary tree except the first one. St000120The number of left tunnels of a Dyck path. St000141The maximum drop size of a permutation. St000168The number of internal nodes of an ordered tree. St000203The number of external nodes of a binary tree. St000211The rank of the set partition. St000236The number of cyclical small weak excedances. St000238The number of indices that are not small weak excedances. St000272The treewidth of a graph. St000316The number of non-left-to-right-maxima of a permutation. St000331The number of upper interactions of a Dyck path. St000332The positive inversions of an alternating sign matrix. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000354The number of recoils of a permutation. St000362The size of a minimal vertex cover of a graph. St000446The disorder of a permutation. St000536The pathwidth of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000632The jump number of the poset. St000651The maximal size of a rise in a permutation. St000662The staircase size of the code of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000738The first entry in the last row of a standard tableau. St000834The number of right outer peaks of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000871The number of very big ascents of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001152The number of pairs with even minimum in a perfect matching. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. 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. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001358The largest degree of a regular subgraph of a graph. St001427The number of descents of a signed permutation. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001726The number of visible inversions of a permutation. St001777The number of weak descents in an integer composition. St001812The biclique partition number of a graph. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000806The semiperimeter of the associated bargraph. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000061The number of nodes on the left branch of a binary tree. St000906The length of the shortest maximal chain in a poset. St000989The number of final rises of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000485The length of the longest cycle of a permutation. St000654The first descent of a permutation. St000668The least common multiple of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000708The product of the parts of an integer partition. St000717The number of ordinal summands of a poset. St000914The sum of the values of the Möbius function of a poset. St000990The first ascent of a permutation. St000216The absolute length of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St000741The Colin de Verdière graph invariant. St001118The acyclic chromatic index of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000444The length of the maximal rise of a Dyck path. St001330The hat guessing number of a graph. St001060The distinguishing index of a graph. St000939The number of characters of the symmetric group whose value on the partition is positive. St000260The radius of a connected graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001889The size of the connectivity set of a signed permutation. St001864The number of excedances of a signed permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000259The diameter of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000815The number of semistandard Young tableaux of partition weight of given shape. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001488The number of corners of a skew partition. St000454The largest eigenvalue of a graph if it is integral. St000420The number of Dyck paths that are weakly above a Dyck path. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. 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. St001500The global dimension of magnitude 1 Nakayama algebras. 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 $eA$. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001863The number of weak excedances of a signed permutation. St001896The number of right descents of a signed permutations. St000455The second largest eigenvalue of a graph if it is integral. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000456The monochromatic index of a connected graph. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000264The girth of a graph, which is not a tree. St001875The number of simple modules with projective dimension at most 1. St000942The number of critical left to right maxima of the parking functions. St001712The number of natural descents of a standard Young tableau. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. 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. St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St001128The exponens consonantiae of a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. 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. St000352The Elizalde-Pak rank of a permutation. St000366The number of double descents of a permutation. St001621The number of atoms of a lattice. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. 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. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. St000907The number of maximal antichains of minimal length in a poset.
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