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Your data matches 203 different statistics following compositions of up to 3 maps.
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Matching statistic: St000162
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
St000162: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 0
[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] => 1
[2,1,4,3] => 2
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 2
[3,4,2,1] => 1
[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] => 2
[1,2,3,4,5] => 0
[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] => 2
[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] => 2
Description
The number of nontrivial cycles in the cycle decomposition of a permutation.
This statistic is equal to the difference of the number of cycles of $\pi$ (see [[St000031]]) and the number of fixed points of $\pi$ (see [[St000022]]).
Matching statistic: St001665
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
St001665: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 0
[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] => 1
[2,1,4,3] => 2
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 1
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 2
[4,3,2,1] => 2
[1,2,3,4,5] => 0
[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] => 2
[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 pure excedances of a permutation.
A pure excedance of a permutation $\pi$ is a position $i < \pi_i$ such that there is no $j < i$ with $i\leq \pi_j < \pi_i$.
Matching statistic: St001737
(load all 61 compositions to match this statistic)
(load all 61 compositions to match this statistic)
St001737: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 0
[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] => 1
[2,1,4,3] => 2
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 2
[3,2,1,4] => 1
[3,2,4,1] => 2
[3,4,1,2] => 1
[3,4,2,1] => 1
[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] => 0
[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] => 2
[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] => 2
[1,4,3,2,5] => 1
[1,4,3,5,2] => 2
[1,4,5,2,3] => 1
Description
The number of descents of type 2 in a permutation.
A position $i\in[1,n-1]$ is a descent of type 2 of a permutation $\pi$ of $n$ letters, if it is a descent and if $\pi(j) < \pi(i)$ for all $j < i$.
Matching statistic: St000245
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 1
[2,1] => [2,1] => 0
[1,2,3] => [1,3,2] => 1
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [2,3,1] => 1
[3,1,2] => [3,1,2] => 1
[3,2,1] => [3,2,1] => 0
[1,2,3,4] => [1,4,3,2] => 1
[1,2,4,3] => [1,4,3,2] => 1
[1,3,2,4] => [1,4,3,2] => 1
[1,3,4,2] => [1,4,3,2] => 1
[1,4,2,3] => [1,4,3,2] => 1
[1,4,3,2] => [1,4,3,2] => 1
[2,1,3,4] => [2,1,4,3] => 1
[2,1,4,3] => [2,1,4,3] => 1
[2,3,1,4] => [2,4,1,3] => 2
[2,3,4,1] => [2,4,3,1] => 1
[2,4,1,3] => [2,4,1,3] => 2
[2,4,3,1] => [2,4,3,1] => 1
[3,1,2,4] => [3,1,4,2] => 1
[3,1,4,2] => [3,1,4,2] => 1
[3,2,1,4] => [3,2,1,4] => 1
[3,2,4,1] => [3,2,4,1] => 1
[3,4,1,2] => [3,4,1,2] => 2
[3,4,2,1] => [3,4,2,1] => 1
[4,1,2,3] => [4,1,3,2] => 1
[4,1,3,2] => [4,1,3,2] => 1
[4,2,1,3] => [4,2,1,3] => 1
[4,2,3,1] => [4,2,3,1] => 1
[4,3,1,2] => [4,3,1,2] => 1
[4,3,2,1] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,5,4,3,2] => 1
[1,2,3,5,4] => [1,5,4,3,2] => 1
[1,2,4,3,5] => [1,5,4,3,2] => 1
[1,2,4,5,3] => [1,5,4,3,2] => 1
[1,2,5,3,4] => [1,5,4,3,2] => 1
[1,2,5,4,3] => [1,5,4,3,2] => 1
[1,3,2,4,5] => [1,5,4,3,2] => 1
[1,3,2,5,4] => [1,5,4,3,2] => 1
[1,3,4,2,5] => [1,5,4,3,2] => 1
[1,3,4,5,2] => [1,5,4,3,2] => 1
[1,3,5,2,4] => [1,5,4,3,2] => 1
[1,3,5,4,2] => [1,5,4,3,2] => 1
[1,4,2,3,5] => [1,5,4,3,2] => 1
[1,4,2,5,3] => [1,5,4,3,2] => 1
[1,4,3,2,5] => [1,5,4,3,2] => 1
[1,4,3,5,2] => [1,5,4,3,2] => 1
[1,4,5,2,3] => [1,5,4,3,2] => 1
Description
The number of ascents of a permutation.
Matching statistic: St000834
(load all 36 compositions to match this statistic)
(load all 36 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000834: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 1
[2,1] => [2,1] => 0
[1,2,3] => [1,3,2] => 1
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [2,3,1] => 1
[3,1,2] => [3,1,2] => 1
[3,2,1] => [3,2,1] => 0
[1,2,3,4] => [1,4,3,2] => 1
[1,2,4,3] => [1,4,3,2] => 1
[1,3,2,4] => [1,4,3,2] => 1
[1,3,4,2] => [1,4,3,2] => 1
[1,4,2,3] => [1,4,3,2] => 1
[1,4,3,2] => [1,4,3,2] => 1
[2,1,3,4] => [2,1,4,3] => 1
[2,1,4,3] => [2,1,4,3] => 1
[2,3,1,4] => [2,4,1,3] => 2
[2,3,4,1] => [2,4,3,1] => 1
[2,4,1,3] => [2,4,1,3] => 2
[2,4,3,1] => [2,4,3,1] => 1
[3,1,2,4] => [3,1,4,2] => 1
[3,1,4,2] => [3,1,4,2] => 1
[3,2,1,4] => [3,2,1,4] => 1
[3,2,4,1] => [3,2,4,1] => 1
[3,4,1,2] => [3,4,1,2] => 2
[3,4,2,1] => [3,4,2,1] => 1
[4,1,2,3] => [4,1,3,2] => 1
[4,1,3,2] => [4,1,3,2] => 1
[4,2,1,3] => [4,2,1,3] => 1
[4,2,3,1] => [4,2,3,1] => 1
[4,3,1,2] => [4,3,1,2] => 1
[4,3,2,1] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,5,4,3,2] => 1
[1,2,3,5,4] => [1,5,4,3,2] => 1
[1,2,4,3,5] => [1,5,4,3,2] => 1
[1,2,4,5,3] => [1,5,4,3,2] => 1
[1,2,5,3,4] => [1,5,4,3,2] => 1
[1,2,5,4,3] => [1,5,4,3,2] => 1
[1,3,2,4,5] => [1,5,4,3,2] => 1
[1,3,2,5,4] => [1,5,4,3,2] => 1
[1,3,4,2,5] => [1,5,4,3,2] => 1
[1,3,4,5,2] => [1,5,4,3,2] => 1
[1,3,5,2,4] => [1,5,4,3,2] => 1
[1,3,5,4,2] => [1,5,4,3,2] => 1
[1,4,2,3,5] => [1,5,4,3,2] => 1
[1,4,2,5,3] => [1,5,4,3,2] => 1
[1,4,3,2,5] => [1,5,4,3,2] => 1
[1,4,3,5,2] => [1,5,4,3,2] => 1
[1,4,5,2,3] => [1,5,4,3,2] => 1
Description
The number of right outer peaks of a permutation.
A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St001280
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 0
[1,2] => [1,1]
=> 0
[2,1] => [2]
=> 1
[1,2,3] => [1,1,1]
=> 0
[1,3,2] => [2,1]
=> 1
[2,1,3] => [2,1]
=> 1
[2,3,1] => [3]
=> 1
[3,1,2] => [3]
=> 1
[3,2,1] => [2,1]
=> 1
[1,2,3,4] => [1,1,1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> 1
[1,3,2,4] => [2,1,1]
=> 1
[1,3,4,2] => [3,1]
=> 1
[1,4,2,3] => [3,1]
=> 1
[1,4,3,2] => [2,1,1]
=> 1
[2,1,3,4] => [2,1,1]
=> 1
[2,1,4,3] => [2,2]
=> 2
[2,3,1,4] => [3,1]
=> 1
[2,3,4,1] => [4]
=> 1
[2,4,1,3] => [4]
=> 1
[2,4,3,1] => [3,1]
=> 1
[3,1,2,4] => [3,1]
=> 1
[3,1,4,2] => [4]
=> 1
[3,2,1,4] => [2,1,1]
=> 1
[3,2,4,1] => [3,1]
=> 1
[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]
=> 1
[4,2,1,3] => [3,1]
=> 1
[4,2,3,1] => [2,1,1]
=> 1
[4,3,1,2] => [4]
=> 1
[4,3,2,1] => [2,2]
=> 2
[1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> 1
[1,2,5,3,4] => [3,1,1]
=> 1
[1,2,5,4,3] => [2,1,1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> 2
[1,3,4,2,5] => [3,1,1]
=> 1
[1,3,4,5,2] => [4,1]
=> 1
[1,3,5,2,4] => [4,1]
=> 1
[1,3,5,4,2] => [3,1,1]
=> 1
[1,4,2,3,5] => [3,1,1]
=> 1
[1,4,2,5,3] => [4,1]
=> 1
[1,4,3,2,5] => [2,1,1,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> 1
[1,4,5,2,3] => [2,2,1]
=> 2
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000021
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 1
[2,1] => [2,1] => [1,2] => 0
[1,2,3] => [1,3,2] => [2,3,1] => 1
[1,3,2] => [1,3,2] => [2,3,1] => 1
[2,1,3] => [2,1,3] => [3,1,2] => 1
[2,3,1] => [2,3,1] => [2,1,3] => 1
[3,1,2] => [3,1,2] => [1,3,2] => 1
[3,2,1] => [3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,4,3,2] => [2,3,4,1] => 1
[1,2,4,3] => [1,4,3,2] => [2,3,4,1] => 1
[1,3,2,4] => [1,4,3,2] => [2,3,4,1] => 1
[1,3,4,2] => [1,4,3,2] => [2,3,4,1] => 1
[1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 1
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => 1
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 1
[2,3,1,4] => [2,4,1,3] => [2,1,4,3] => 2
[2,3,4,1] => [2,4,3,1] => [2,1,3,4] => 1
[2,4,1,3] => [2,4,1,3] => [2,1,4,3] => 2
[2,4,3,1] => [2,4,3,1] => [2,1,3,4] => 1
[3,1,2,4] => [3,1,4,2] => [1,3,2,4] => 1
[3,1,4,2] => [3,1,4,2] => [1,3,2,4] => 1
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => 1
[3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 1
[3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 2
[3,4,2,1] => [3,4,2,1] => [3,1,2,4] => 1
[4,1,2,3] => [4,1,3,2] => [1,4,2,3] => 1
[4,1,3,2] => [4,1,3,2] => [1,4,2,3] => 1
[4,2,1,3] => [4,2,1,3] => [1,2,4,3] => 1
[4,2,3,1] => [4,2,3,1] => [2,4,1,3] => 1
[4,3,1,2] => [4,3,1,2] => [1,3,4,2] => 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,3,4,5,1] => 1
Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000035
(load all 54 compositions to match this statistic)
(load all 54 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 0
[1,2] => [1,0,1,0]
=> [1,2] => 0
[2,1] => [1,1,0,0]
=> [2,1] => 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => 1
[2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => 1
[3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => 1
[3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 1
Description
The number of left outer peaks of a permutation.
A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$.
In other words, it is a peak in the word $[0,w_1,..., w_n]$.
This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.
Matching statistic: St000374
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00080: Set partitions —to permutation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => {{1}}
=> [1] => 0
[1,2] => {{1},{2}}
=> [1,2] => 0
[2,1] => {{1,2}}
=> [2,1] => 1
[1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[1,3,2] => {{1},{2,3}}
=> [1,3,2] => 1
[2,1,3] => {{1,2},{3}}
=> [2,1,3] => 1
[2,3,1] => {{1,2,3}}
=> [2,3,1] => 1
[3,1,2] => {{1,3},{2}}
=> [3,2,1] => 1
[3,2,1] => {{1,3},{2}}
=> [3,2,1] => 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 1
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 1
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => 1
[1,4,2,3] => {{1},{2,4},{3}}
=> [1,4,3,2] => 1
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,4,3,2] => 1
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,3,4] => 1
[2,1,4,3] => {{1,2},{3,4}}
=> [2,1,4,3] => 2
[2,3,1,4] => {{1,2,3},{4}}
=> [2,3,1,4] => 1
[2,3,4,1] => {{1,2,3,4}}
=> [2,3,4,1] => 1
[2,4,1,3] => {{1,2,4},{3}}
=> [2,4,3,1] => 1
[2,4,3,1] => {{1,2,4},{3}}
=> [2,4,3,1] => 1
[3,1,2,4] => {{1,3},{2},{4}}
=> [3,2,1,4] => 1
[3,1,4,2] => {{1,3,4},{2}}
=> [3,2,4,1] => 1
[3,2,1,4] => {{1,3},{2},{4}}
=> [3,2,1,4] => 1
[3,2,4,1] => {{1,3,4},{2}}
=> [3,2,4,1] => 1
[3,4,1,2] => {{1,3},{2,4}}
=> [3,4,1,2] => 2
[3,4,2,1] => {{1,3},{2,4}}
=> [3,4,1,2] => 2
[4,1,2,3] => {{1,4},{2},{3}}
=> [4,2,3,1] => 1
[4,1,3,2] => {{1,4},{2},{3}}
=> [4,2,3,1] => 1
[4,2,1,3] => {{1,4},{2},{3}}
=> [4,2,3,1] => 1
[4,2,3,1] => {{1,4},{2},{3}}
=> [4,2,3,1] => 1
[4,3,1,2] => {{1,4},{2,3}}
=> [4,3,2,1] => 1
[4,3,2,1] => {{1,4},{2,3}}
=> [4,3,2,1] => 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}}
=> [1,2,3,5,4] => 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => 1
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => 2
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => 1
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => 1
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => 1
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,4,5,2,3] => 2
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000672
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 1
[2,1] => [2,1] => [2,1] => 0
[1,2,3] => [1,3,2] => [1,3,2] => 1
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [2,3,1] => [1,3,2] => 1
[3,1,2] => [3,1,2] => [3,1,2] => 1
[3,2,1] => [3,2,1] => [3,2,1] => 0
[1,2,3,4] => [1,4,3,2] => [1,4,3,2] => 1
[1,2,4,3] => [1,4,3,2] => [1,4,3,2] => 1
[1,3,2,4] => [1,4,3,2] => [1,4,3,2] => 1
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 1
[2,1,3,4] => [2,1,4,3] => [2,1,4,3] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
[2,3,1,4] => [2,4,1,3] => [2,4,1,3] => 2
[2,3,4,1] => [2,4,3,1] => [1,4,3,2] => 1
[2,4,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[2,4,3,1] => [2,4,3,1] => [1,4,3,2] => 1
[3,1,2,4] => [3,1,4,2] => [2,1,4,3] => 1
[3,1,4,2] => [3,1,4,2] => [2,1,4,3] => 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[3,2,4,1] => [3,2,4,1] => [2,1,4,3] => 1
[3,4,1,2] => [3,4,1,2] => [2,4,1,3] => 2
[3,4,2,1] => [3,4,2,1] => [1,4,3,2] => 1
[4,1,2,3] => [4,1,3,2] => [4,1,3,2] => 1
[4,1,3,2] => [4,1,3,2] => [4,1,3,2] => 1
[4,2,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[4,2,3,1] => [4,2,3,1] => [4,1,3,2] => 1
[4,3,1,2] => [4,3,1,2] => [4,3,1,2] => 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1
Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
The following 193 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000703The number of deficiencies of a permutation. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a 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. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000010The length of the partition. St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000507The number of ascents of a standard tableau. St000012The area of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000142The number of even parts of a partition. St000147The largest part of an integer partition. St000155The number of exceedances (also excedences) of a permutation. St000157The number of descents of a standard tableau. St000168The number of internal nodes of an ordered tree. St000238The number of indices that are not small weak excedances. St000291The number of descents of a binary word. St000316The number of non-left-to-right-maxima of a permutation. 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. 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. St000386The number of factors DDU in a Dyck path. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000519The largest length of a factor maximising the subword complexity. St000659The number of rises of length at least 2 of a Dyck path. St000662The staircase size of the code of a permutation. St000665The number of rafts of a permutation. St000670The reversal length of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000996The number of exclusive left-to-right maxima of a permutation. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with 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. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001298The number of repeated entries in the Lehmer code of a permutation. St001333The cardinality of a minimal edge-isolating set of a graph. St001427The number of descents of a signed permutation. St001489The maximum of the number of descents and the number of inverse descents. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001657The number of twos in an integer partition. St001726The number of visible inversions of a permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001874Lusztig's a-function for the symmetric group. St000015The number of peaks of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000167The number of leaves of an ordered tree. St000288The number of ones in a binary word. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000378The diagonal inversion number of an integer partition. St000443The number of long tunnels of a Dyck path. St000542The number of left-to-right-minima of a permutation. St000676The number of odd rises of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000935The number of ordered refinements of an integer partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001389The number of partitions of the same length below the given integer partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. 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. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000251The number of nonsingleton blocks of a set partition. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St000919The number of maximal left branches of a binary tree. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000702The number of weak deficiencies of a permutation. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000658The number of rises of length 2 of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000829The Ulam distance of a permutation to the identity permutation. St000984The number of boxes below precisely one peak. St001139The number of occurrences of hills of size 2 in a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. 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. St000668The least common multiple of the parts of the partition. St001432The order dimension of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000183The side length of the Durfee square of an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001128The exponens consonantiae of a partition. St000897The number of different multiplicities of parts of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000260The radius of a connected graph. St000259The diameter of a connected graph. 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. St001568The smallest positive integer that does not appear twice in the partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000640The rank of the largest boolean interval in a poset. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000456The monochromatic index of a connected graph. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000544The cop number of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001339The irredundance number of a graph. St001933The largest multiplicity of a part in an integer partition. St000706The product of the factorials of the multiplicities of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001330The hat guessing number of a graph. St000284The Plancherel distribution on integer partitions. 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. 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. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the 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. 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. 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. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001877Number of indecomposable injective modules with projective dimension 2. St000455The second largest eigenvalue of a graph if it is integral. St000454The largest eigenvalue of a graph if it is integral. St000256The number of parts from which one can substract 2 and still get an integer partition. St000353The number of inner valleys of a permutation. St000711The number of big exceedences of a permutation. St000092The number of outer peaks of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001720The minimal length of a chain of small intervals in a lattice. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001896The number of right descents of a signed permutations. St001863The number of weak excedances of a signed permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000633The size of the automorphism group of a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001935The number of ascents in a parking function. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001597The Frobenius rank of a skew partition. St000862The number of parts of the shifted shape of a permutation. St001624The breadth of a lattice.
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