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Your data matches 357 different statistics following compositions of up to 3 maps.
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Matching statistic: St001550
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
St001550: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 0
[1,3,5,4,2] => 0
[1,4,2,3,5] => 0
[1,4,2,5,3] => 0
[1,4,3,2,5] => 0
[1,4,3,5,2] => 0
[1,4,5,2,3] => 0
Description
The number of inversions between exceedances where the greater exceedance is linked.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{ile}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(j) < \sigma(i) \wedge \sigma^{-1}(j) < j \}.$$
Matching statistic: St001551
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
St001551: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 0
[1,3,5,4,2] => 0
[1,4,2,3,5] => 0
[1,4,2,5,3] => 0
[1,4,3,2,5] => 0
[1,4,3,5,2] => 0
[1,4,5,2,3] => 0
Description
The number of restricted non-inversions between exceedances where the rightmost exceedance is linked.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \wedge \sigma^{-1}(j) < j \}.$$
Matching statistic: St000366
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [1,2,3] => 0
[3,2,1] => [1,3,2] => [1,3,2] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[3,1,4,2] => [1,3,4,2] => [1,4,3,2] => [1,2,4,3] => 0
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[3,2,4,1] => [1,3,4,2] => [1,4,3,2] => [1,2,4,3] => 0
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,4,3,2] => [1,4,2,3] => [1,2,4,3] => 0
[4,1,3,2] => [1,4,2,3] => [1,3,4,2] => [1,2,3,4] => 0
[4,2,1,3] => [1,4,3,2] => [1,4,2,3] => [1,2,4,3] => 0
[4,2,3,1] => [1,4,2,3] => [1,3,4,2] => [1,2,3,4] => 0
[4,3,1,2] => [1,4,2,3] => [1,3,4,2] => [1,2,3,4] => 0
[4,3,2,1] => [1,4,2,3] => [1,3,4,2] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4,5] => 0
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4,5] => 0
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4,5] => 0
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => 0
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,3,5,4] => 0
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => 0
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,3,5,4] => 0
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => 0
Description
The number of double descents of a permutation.
A double descent of a permutation $\pi$ is a position $i$ such that $\pi(i) > \pi(i+1) > \pi(i+2)$.
Matching statistic: St001549
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St001549: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St001549: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,1,2] => [3,2,1] => 0
[2,1,3] => [1,3,2] => [3,1,2] => [3,2,1] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => [4,2,3,1] => 0
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => [3,2,1,4] => 0
[1,3,4,2] => [1,3,4,2] => [3,4,1,2] => [4,3,2,1] => 0
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 0
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 0
[2,1,3,4] => [1,3,4,2] => [3,4,1,2] => [4,3,2,1] => 0
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 0
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [3,1,2,4] => [3,2,1,4] => 0
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [4,2,3,1] => 0
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => [4,2,3,1] => 0
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 0
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 0
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => [4,2,3,1] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [3,1,2,4] => [3,2,1,4] => 0
[4,2,1,3] => [1,3,2,4] => [3,1,2,4] => [3,2,1,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,2,3,4,1] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [4,2,3,1,5] => 0
[1,2,4,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => [5,4,3,2,1] => 0
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => [1,5,3,4,2] => 0
[1,2,5,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => [1,5,3,4,2] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => [4,5,3,1,2] => 1
[1,3,4,2,5] => [1,3,4,2,5] => [3,4,1,2,5] => [4,3,2,1,5] => 0
[1,3,4,5,2] => [1,3,4,5,2] => [3,4,5,1,2] => [5,4,3,2,1] => 0
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,2,4] => [5,4,3,2,1] => 0
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,2,4] => [5,4,3,2,1] => 0
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => 0
[1,4,2,5,3] => [1,4,2,5,3] => [1,4,5,2,3] => [1,5,4,3,2] => 0
[1,4,3,2,5] => [1,4,2,5,3] => [1,4,5,2,3] => [1,5,4,3,2] => 0
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => 0
[1,4,5,2,3] => [1,4,5,2,3] => [4,1,5,2,3] => [5,2,4,3,1] => 0
Description
The number of restricted non-inversions between exceedances.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \}.$$
Matching statistic: St001715
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St001715: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St001715: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 0
[2,1] => [2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,3,2] => [3,2,1] => [2,3,1] => 0
[1,3,2] => [1,3,2] => [3,2,1] => [2,3,1] => 0
[2,1,3] => [2,1,3] => [1,3,2] => [1,3,2] => 0
[2,3,1] => [2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[3,2,1] => [3,2,1] => [2,1,3] => [2,1,3] => 0
[1,2,3,4] => [1,4,3,2] => [4,3,2,1] => [2,3,4,1] => 0
[1,2,4,3] => [1,4,3,2] => [4,3,2,1] => [2,3,4,1] => 0
[1,3,2,4] => [1,4,3,2] => [4,3,2,1] => [2,3,4,1] => 0
[1,3,4,2] => [1,4,3,2] => [4,3,2,1] => [2,3,4,1] => 0
[1,4,2,3] => [1,4,3,2] => [4,3,2,1] => [2,3,4,1] => 0
[1,4,3,2] => [1,4,3,2] => [4,3,2,1] => [2,3,4,1] => 0
[2,1,3,4] => [2,1,4,3] => [1,4,3,2] => [1,3,4,2] => 0
[2,1,4,3] => [2,1,4,3] => [1,4,3,2] => [1,3,4,2] => 0
[2,3,1,4] => [2,4,1,3] => [1,4,2,3] => [1,4,2,3] => 0
[2,3,4,1] => [2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 0
[2,4,1,3] => [2,4,1,3] => [1,4,2,3] => [1,4,2,3] => 0
[2,4,3,1] => [2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 0
[3,1,2,4] => [3,1,4,2] => [4,1,3,2] => [3,4,1,2] => 0
[3,1,4,2] => [3,1,4,2] => [4,1,3,2] => [3,4,1,2] => 0
[3,2,1,4] => [3,2,1,4] => [2,1,4,3] => [2,1,4,3] => 0
[3,2,4,1] => [3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 0
[3,4,1,2] => [3,4,1,2] => [4,1,2,3] => [4,1,2,3] => 0
[3,4,2,1] => [3,4,2,1] => [3,1,2,4] => [3,1,2,4] => 0
[4,1,2,3] => [4,1,3,2] => [4,3,1,2] => [3,1,4,2] => 0
[4,1,3,2] => [4,1,3,2] => [4,3,1,2] => [3,1,4,2] => 0
[4,2,1,3] => [4,2,1,3] => [2,4,1,3] => [4,2,1,3] => 0
[4,2,3,1] => [4,2,3,1] => [2,3,1,4] => [3,2,1,4] => 0
[4,3,1,2] => [4,3,1,2] => [4,2,1,3] => [2,4,1,3] => 0
[4,3,2,1] => [4,3,2,1] => [3,2,1,4] => [2,3,1,4] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[1,2,3,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[1,2,4,3,5] => [1,5,4,3,2] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[1,2,4,5,3] => [1,5,4,3,2] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[1,2,5,3,4] => [1,5,4,3,2] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[1,2,5,4,3] => [1,5,4,3,2] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[1,3,2,4,5] => [1,5,4,3,2] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[1,3,2,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[1,3,4,2,5] => [1,5,4,3,2] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[1,3,4,5,2] => [1,5,4,3,2] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[1,3,5,2,4] => [1,5,4,3,2] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[1,3,5,4,2] => [1,5,4,3,2] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[1,4,2,3,5] => [1,5,4,3,2] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[1,4,2,5,3] => [1,5,4,3,2] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[1,4,3,2,5] => [1,5,4,3,2] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[1,4,3,5,2] => [1,5,4,3,2] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[1,4,5,2,3] => [1,5,4,3,2] => [5,4,3,2,1] => [2,3,4,5,1] => 0
Description
The number of non-records in a permutation.
A record in a permutation $\pi$ is a value $\pi(j)$ which is a left-to-right minimum, a left-to-right maximum, a right-to-left minimum, or a right-to-left maximum.
For example, in the permutation $\pi = [1, 4, 3, 2, 5]$, the values $1$ is a left-to-right minimum, $1, 4, 5$ are left-to-right maxima, $5, 2, 1$ are right-to-left minima and $5$ is a right-to-left maximum. Hence, $3$ is the unique non-record.
Permutations without non-records are called square [1].
Matching statistic: St001906
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St001906: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St001906: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,1,2] => [3,2,1] => 0
[2,1,3] => [1,3,2] => [3,1,2] => [3,2,1] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [4,1,2,3] => [4,2,3,1] => 0
[1,3,2,4] => [1,3,2,4] => [3,1,2,4] => [3,2,1,4] => 0
[1,3,4,2] => [1,3,4,2] => [3,4,1,2] => [4,3,2,1] => 0
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 0
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 0
[2,1,3,4] => [1,3,4,2] => [3,4,1,2] => [4,3,2,1] => 0
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 0
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [3,1,2,4] => [3,2,1,4] => 0
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => [4,2,3,1] => 0
[3,1,2,4] => [1,2,4,3] => [4,1,2,3] => [4,2,3,1] => 0
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 0
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 0
[3,2,4,1] => [1,2,4,3] => [4,1,2,3] => [4,2,3,1] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [3,1,2,4] => [3,2,1,4] => 0
[4,2,1,3] => [1,3,2,4] => [3,1,2,4] => [3,2,1,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [5,2,3,4,1] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => [4,2,3,1,5] => 0
[1,2,4,5,3] => [1,2,4,5,3] => [4,5,1,2,3] => [5,4,3,2,1] => 0
[1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => [1,5,3,4,2] => 0
[1,2,5,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => [1,5,3,4,2] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [3,5,1,2,4] => [4,5,3,1,2] => 1
[1,3,4,2,5] => [1,3,4,2,5] => [3,4,1,2,5] => [4,3,2,1,5] => 0
[1,3,4,5,2] => [1,3,4,5,2] => [3,4,5,1,2] => [5,4,3,2,1] => 0
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,2,4] => [5,4,3,2,1] => 0
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,2,4] => [5,4,3,2,1] => 0
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => 0
[1,4,2,5,3] => [1,4,2,5,3] => [1,4,5,2,3] => [1,5,4,3,2] => 0
[1,4,3,2,5] => [1,4,2,5,3] => [1,4,5,2,3] => [1,5,4,3,2] => 0
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => 0
[1,4,5,2,3] => [1,4,5,2,3] => [4,1,5,2,3] => [5,2,4,3,1] => 0
Description
Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation.
Let $\pi$ be a permutation. Its total displacement [[St000830]] is $D(\pi) = \sum_i |\pi(i) - i|$, and its absolute length [[St000216]] is the minimal number $T(\pi)$ of transpositions whose product is $\pi$. Finally, let $I(\pi)$ be the number of inversions [[St000018]] of $\pi$.
This statistic equals $\left(D(\pi)-T(\pi)-I(\pi)\right)/2$.
Diaconis and Graham [1] proved that this statistic is always nonnegative.
Matching statistic: St000516
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000516: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000516: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => {{1}}
=> [1] => ? = 0
[1,2] => [1,2] => {{1},{2}}
=> [1,2] => 0
[2,1] => [1,2] => {{1},{2}}
=> [1,2] => 0
[1,2,3] => [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[1,3,2] => [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[2,1,3] => [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[2,3,1] => [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[3,1,2] => [1,3,2] => {{1},{2,3}}
=> [1,3,2] => 0
[3,2,1] => [1,3,2] => {{1},{2,3}}
=> [1,3,2] => 0
[1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 0
[1,4,3,2] => [1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 0
[2,1,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 0
[2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 0
[3,1,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 0
[3,1,4,2] => [1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => 0
[3,2,1,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 0
[3,2,4,1] => [1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => 0
[3,4,1,2] => [1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 0
[3,4,2,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 0
[4,1,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> [1,4,3,2] => 0
[4,1,3,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> [1,4,3,2] => 0
[4,2,1,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> [1,4,3,2] => 0
[4,2,3,1] => [1,4,2,3] => {{1},{2,4},{3}}
=> [1,4,3,2] => 0
[4,3,1,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> [1,4,3,2] => 0
[4,3,2,1] => [1,4,2,3] => {{1},{2,4},{3}}
=> [1,4,3,2] => 0
[1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 0
[1,2,5,4,3] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 0
[1,3,2,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 0
[1,3,5,4,2] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 0
[1,4,2,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 0
[1,4,2,5,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => 0
[1,4,3,2,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 0
[1,4,3,5,2] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => 0
[1,4,5,2,3] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 0
[1,4,5,3,2] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 0
Description
The number of stretching pairs of a permutation.
This is the number of pairs $(i,j)$ with $\pi(i) < i < j < \pi(j)$.
Matching statistic: St000649
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000649: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000649: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => {{1}}
=> [1] => ? = 0
[1,2] => [1,2] => {{1},{2}}
=> [1,2] => 0
[2,1] => [1,2] => {{1},{2}}
=> [1,2] => 0
[1,2,3] => [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[1,3,2] => [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[2,1,3] => [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[2,3,1] => [1,2,3] => {{1},{2},{3}}
=> [1,2,3] => 0
[3,1,2] => [1,3,2] => {{1},{2,3}}
=> [1,3,2] => 0
[3,2,1] => [1,3,2] => {{1},{2,3}}
=> [1,3,2] => 0
[1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 0
[1,4,3,2] => [1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 0
[2,1,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 0
[2,4,3,1] => [1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => 0
[3,1,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 0
[3,1,4,2] => [1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => 0
[3,2,1,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 0
[3,2,4,1] => [1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => 0
[3,4,1,2] => [1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 0
[3,4,2,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => 0
[4,1,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> [1,4,3,2] => 0
[4,1,3,2] => [1,4,2,3] => {{1},{2,3,4}}
=> [1,3,4,2] => 0
[4,2,1,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> [1,4,3,2] => 0
[4,2,3,1] => [1,4,2,3] => {{1},{2,3,4}}
=> [1,3,4,2] => 0
[4,3,1,2] => [1,4,2,3] => {{1},{2,3,4}}
=> [1,3,4,2] => 0
[4,3,2,1] => [1,4,2,3] => {{1},{2,3,4}}
=> [1,3,4,2] => 0
[1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 0
[1,2,5,4,3] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 0
[1,3,2,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 0
[1,3,5,4,2] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => 0
[1,4,2,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 0
[1,4,2,5,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => 0
[1,4,3,2,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 0
[1,4,3,5,2] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => 0
[1,4,5,2,3] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 0
[1,4,5,3,2] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => 0
Description
The number of 3-excedences of a permutation.
This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+3$.
Matching statistic: St001552
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001552: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001552: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [1,2,3] => 0
[3,2,1] => [1,3,2] => [1,3,2] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [1,2,3,4] => 0
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[3,1,4,2] => [1,3,4,2] => [1,4,3,2] => [1,2,4,3] => 0
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[3,2,4,1] => [1,3,4,2] => [1,4,3,2] => [1,2,4,3] => 0
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,4,3,2] => [1,4,2,3] => [1,2,4,3] => 0
[4,1,3,2] => [1,4,2,3] => [1,3,4,2] => [1,2,3,4] => 0
[4,2,1,3] => [1,4,3,2] => [1,4,2,3] => [1,2,4,3] => 0
[4,2,3,1] => [1,4,2,3] => [1,3,4,2] => [1,2,3,4] => 0
[4,3,1,2] => [1,4,2,3] => [1,3,4,2] => [1,2,3,4] => 0
[4,3,2,1] => [1,4,2,3] => [1,3,4,2] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4,5] => 0
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4,5] => 0
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4,5] => 0
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4,5] => 0
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => 0
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,3,5,4] => 0
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => 0
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,3,5,4] => 0
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => 0
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => 0
Description
The number of inversions between excedances and fixed points of a permutation.
This is,
$$\operatorname{iefp}(\sigma) = \#\{1 \leq i,j \leq n \mid i < j = \sigma(j) < \sigma(i) \}.$$
Matching statistic: St000570
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000570: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000570: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => ? = 0 + 1
[1,2] => [1,2] => [2,1] => [2,1] => 1 = 0 + 1
[2,1] => [1,2] => [2,1] => [2,1] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [3,2,1] => [2,3,1] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 1 = 0 + 1
[2,1,3] => [1,3,2] => [3,1,2] => [3,1,2] => 1 = 0 + 1
[2,3,1] => [1,2,3] => [3,2,1] => [2,3,1] => 1 = 0 + 1
[3,1,2] => [1,2,3] => [3,2,1] => [2,3,1] => 1 = 0 + 1
[3,2,1] => [1,2,3] => [3,2,1] => [2,3,1] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [4,3,1,2] => [3,1,4,2] => 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [3,4,2,1] => 1 = 0 + 1
[1,3,4,2] => [1,3,4,2] => [4,2,1,3] => [2,4,1,3] => 1 = 0 + 1
[1,4,2,3] => [1,4,2,3] => [4,1,3,2] => [3,4,1,2] => 1 = 0 + 1
[1,4,3,2] => [1,4,2,3] => [4,1,3,2] => [3,4,1,2] => 1 = 0 + 1
[2,1,3,4] => [1,3,4,2] => [4,2,1,3] => [2,4,1,3] => 1 = 0 + 1
[2,1,4,3] => [1,4,2,3] => [4,1,3,2] => [3,4,1,2] => 1 = 0 + 1
[2,3,1,4] => [1,4,2,3] => [4,1,3,2] => [3,4,1,2] => 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => [3,4,2,1] => 1 = 0 + 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [3,1,4,2] => 1 = 0 + 1
[3,1,2,4] => [1,2,4,3] => [4,3,1,2] => [3,1,4,2] => 1 = 0 + 1
[3,1,4,2] => [1,4,2,3] => [4,1,3,2] => [3,4,1,2] => 1 = 0 + 1
[3,2,1,4] => [1,4,2,3] => [4,1,3,2] => [3,4,1,2] => 1 = 0 + 1
[3,2,4,1] => [1,2,4,3] => [4,3,1,2] => [3,1,4,2] => 1 = 0 + 1
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => [3,4,2,1] => 1 = 0 + 1
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => [3,4,2,1] => 1 = 0 + 1
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [2,3,4,1] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [2,3,4,5,1] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => [3,1,4,5,2] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [3,4,2,5,1] => 1 = 0 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [2,4,1,5,3] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [5,4,1,3,2] => [3,4,1,5,2] => 1 = 0 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [5,4,1,3,2] => [3,4,1,5,2] => 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => [2,4,5,3,1] => 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => [4,1,5,3,2] => 1 = 0 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [5,3,2,4,1] => [4,3,5,2,1] => 1 = 0 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [5,3,2,1,4] => [2,3,5,1,4] => 1 = 0 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [5,3,1,4,2] => [4,3,5,1,2] => 1 = 0 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [5,3,1,4,2] => [4,3,5,1,2] => 1 = 0 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [5,2,4,3,1] => [3,4,5,2,1] => 1 = 0 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [5,2,4,1,3] => [4,5,2,1,3] => 1 = 0 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [5,2,4,1,3] => [4,5,2,1,3] => 1 = 0 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [5,2,4,3,1] => [3,4,5,2,1] => 1 = 0 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [5,2,1,4,3] => [2,4,5,1,3] => 1 = 0 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [5,2,1,4,3] => [2,4,5,1,3] => 1 = 0 + 1
Description
The Edelman-Greene number of a permutation.
This is the sum of the coefficients of the expansion of the Stanley symmetric function $F_\omega$ in Schur functions. Equivalently, this is the number of semistandard tableaux whose column words - obtained by reading up columns starting with the leftmost - are reduced words for $\omega$.
The following 347 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001271The competition number of a graph. St001578The minimal number of edges to add or remove to make a graph a line graph. St000535The rank-width of a graph. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. 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 $eA$. St000068The number of minimal elements in a poset. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001175The size of a partition minus the hook length of the base cell. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000879The number of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000379The number of Hamiltonian cycles in a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001498The normalised height of a Nakayama algebra with magnitude 1. St000017The number of inversions of a standard tableau. St000142The number of even parts of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000256The number of parts from which one can substract 2 and still get an integer partition. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000296The length of the symmetric border of a binary word. St000347The inversion sum of a binary word. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000629The defect of a binary word. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001092The number of distinct even parts of a partition. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001214The aft of an integer partition. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001435The number of missing boxes in the first row. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001438The number of missing boxes of a skew partition. St001485The modular major index of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001586The number of odd parts smaller than the largest even part in an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001596The number of two-by-two squares inside a skew partition. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001730The number of times the path corresponding to a binary word crosses the base line. St000478Another weight of a partition according to Alladi. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000455The second largest eigenvalue of a graph if it is integral. St000878The number of ones minus the number of zeros of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000944The 3-degree of an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St000934The 2-degree of an integer partition. St000929The constant term of the character polynomial of an integer partition. St001651The Frankl number of a lattice. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000488The number of cycles of a permutation of length at most 2. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000787The number of flips required to make a perfect matching noncrossing. St001204Call 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. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St000056The decomposition (or block) number of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St001081The number of minimal length factorizations of a permutation into star transpositions. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001461The number of topologically connected components of the chord diagram of a permutation. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001095The number of non-isomorphic posets with precisely one further covering relation. St001964The interval resolution global dimension of a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001890The maximum magnitude of the Möbius function of a poset. St000661The number of rises of length 3 of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001593This is the number of standard Young tableaux of the given shifted shape. St000323The minimal crossing number of a graph. St000370The genus of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001793The difference between the clique number and the chromatic number of a graph. St001797The number of overfull subgraphs of a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001353The number of prime nodes in the modular decomposition of a graph. St001307The number of induced stars on four vertices in a graph. St000699The toughness times the least common multiple of 1,. St000567The sum of the products of all pairs of parts. St000936The number of even values of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000781The number of proper colouring schemes of a Ferrers diagram. St000322The skewness of a graph. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001871The number of triconnected components of a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001490The number of connected components of a skew partition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001330The hat guessing number of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000759The smallest missing part in an integer partition. St000897The number of different multiplicities of parts of an integer partition. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000297The number of leading ones in a binary word. St000475The number of parts equal to 1 in a partition. St000022The number of fixed points of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000069The number of maximal elements of a poset. St000667The greatest common divisor of the parts of the partition. St000407The number of occurrences of the pattern 2143 in a permutation. St000461The rix statistic of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001577The minimal number of edges to add or remove to make a graph a cograph. St001856The number of edges in the reduced word graph of a permutation. St001518The number of graphs with the same ordinary spectrum as the given graph. St001520The number of strict 3-descents. St000993The multiplicity of the largest part of an integer partition. St001555The order of a signed permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000889The number of alternating sign matrices with the same antidiagonal sums. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000181The number of connected components of the Hasse diagram for the poset. St001568The smallest positive integer that does not appear twice in the partition. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001429The number of negative entries in a signed permutation. St001621The number of atoms of a lattice. St001846The number of elements which do not have a complement in the lattice. St001820The size of the image of the pop stack sorting operator. St000895The number of ones on the main diagonal of an alternating sign matrix. St001434The number of negative sum pairs of a signed permutation. St001249Sum of the odd parts of a partition. St001383The BG-rank of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001889The size of the connectivity set of a signed permutation. St000731The number of double exceedences of a permutation. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001862The number of crossings of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001863The number of weak excedances of a signed permutation. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001624The breadth of a lattice. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000403The Szeged index minus the Wiener index of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001305The number of induced cycles on four vertices in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001395The number of strictly unfriendly partitions of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001316The domatic number of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St000273The domination number of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000916The packing number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001430The number of positive entries in a signed permutation. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St001281The normalized isoperimetric number of a graph. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001768The number of reduced words of a signed permutation. St000074The number of special entries. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000894The trace of an alternating sign matrix. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001335The cardinality of a minimal cycle-isolating set of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001524The degree of symmetry of a binary word. St001557The number of inversions of the second entry of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St000295The length of the border of a binary word. St000657The smallest part of an integer composition. St000942The number of critical left to right maxima of the parking functions. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001260The permanent of an alternating sign matrix. St001267The length of the Lyndon factorization of the binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001437The flex of a binary word. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001884The number of borders of a binary word. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000876The number of factors in the Catalan decomposition of a binary word. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001927Sparre Andersen's number of positives of a signed permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001857The number of edges in the reduced word graph of a signed permutation. St001052The length of the exterior of a permutation. St001410The minimal entry of a semistandard tableau. St000084The number of subtrees. St000328The maximum number of child nodes in a tree. St000264The girth of a graph, which is not a tree. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000095The number of triangles of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001765The number of connected components of the friends and strangers graph. St000782The indicator function of whether a given perfect matching is an L & P matching.
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