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Your data matches 16 different statistics following compositions of up to 3 maps.
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Matching statistic: St000731
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
St000731: 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] => 1
[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] => 1
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 1
[2,3,4,1] => 2
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 1
[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] => 1
[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] => 1
[1,3,4,5,2] => 2
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 0
[1,4,2,5,3] => 1
[1,4,3,2,5] => 0
[1,4,3,5,2] => 1
[1,4,5,2,3] => 0
Description
The number of double exceedences of a permutation.
A double exceedence is an index $\sigma(i)$ such that $i < \sigma(i) < \sigma(\sigma(i))$.
Matching statistic: St000366
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 90%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 90%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => 0
[2,3,1] => [3,1,2] => [3,2,1] => 1
[3,1,2] => [2,3,1] => [3,1,2] => 0
[3,2,1] => [3,2,1] => [2,3,1] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 1
[1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 0
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 0
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => [3,1,2,4] => [3,2,1,4] => 1
[2,3,4,1] => [4,1,2,3] => [4,3,2,1] => 2
[2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 1
[2,4,3,1] => [4,1,3,2] => [3,4,2,1] => 1
[3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 0
[3,1,4,2] => [2,4,1,3] => [4,3,1,2] => 1
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0
[3,2,4,1] => [4,2,1,3] => [2,4,3,1] => 1
[3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 0
[3,4,2,1] => [4,3,1,2] => [4,2,3,1] => 0
[4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 0
[4,1,3,2] => [2,4,3,1] => [3,4,1,2] => 0
[4,2,1,3] => [3,2,4,1] => [2,4,1,3] => 0
[4,2,3,1] => [4,2,3,1] => [2,3,4,1] => 0
[4,3,1,2] => [3,4,2,1] => [4,1,3,2] => 0
[4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 0
[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] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,4,3] => 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,3,4] => 0
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,3,2,5] => 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,4,3,2] => 2
[1,3,5,2,4] => [1,4,2,5,3] => [1,5,3,2,4] => 1
[1,3,5,4,2] => [1,5,2,4,3] => [1,4,5,3,2] => 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,2,3,5] => 0
[1,4,2,5,3] => [1,3,5,2,4] => [1,5,4,2,3] => 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 0
[1,4,3,5,2] => [1,5,3,2,4] => [1,3,5,4,2] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => 0
[1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => [1,5,4,3,2,6,7] => ? = 2
[1,3,4,5,2,7,6] => [1,5,2,3,4,7,6] => [1,5,4,3,2,7,6] => ? = 2
[1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => [1,6,5,4,3,2,7] => ? = 3
[1,3,4,5,7,2,6] => [1,6,2,3,4,7,5] => [1,7,5,4,3,2,6] => ? = 3
[1,3,4,6,2,5,7] => [1,5,2,3,6,4,7] => [1,6,4,3,2,5,7] => ? = 2
[1,3,4,6,2,7,5] => [1,5,2,3,7,4,6] => [1,7,6,4,3,2,5] => ? = 3
[1,3,4,6,5,2,7] => [1,6,2,3,5,4,7] => [1,5,6,4,3,2,7] => ? = 2
[1,3,4,6,7,2,5] => [1,6,2,3,7,4,5] => [1,6,4,3,2,7,5] => ? = 2
[1,3,4,6,7,5,2] => [1,7,2,3,6,4,5] => [1,7,5,6,4,3,2] => ? = 2
[1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => [1,7,4,3,2,5,6] => ? = 2
[1,3,4,7,2,6,5] => [1,5,2,3,7,6,4] => [1,6,7,4,3,2,5] => ? = 2
[1,3,4,7,5,2,6] => [1,6,2,3,5,7,4] => [1,5,7,4,3,2,6] => ? = 2
[1,3,4,7,6,2,5] => [1,6,2,3,7,5,4] => [1,7,4,3,2,6,5] => ? = 2
[1,3,5,2,4,6,7] => [1,4,2,5,3,6,7] => [1,5,3,2,4,6,7] => ? = 1
[1,3,5,2,4,7,6] => [1,4,2,5,3,7,6] => [1,5,3,2,4,7,6] => ? = 1
[1,3,5,2,6,4,7] => [1,4,2,6,3,5,7] => [1,6,5,3,2,4,7] => ? = 2
[1,3,5,2,6,7,4] => [1,4,2,7,3,5,6] => [1,7,6,5,3,2,4] => ? = 3
[1,3,5,2,7,4,6] => [1,4,2,6,3,7,5] => [1,7,5,3,2,4,6] => ? = 2
[1,3,5,2,7,6,4] => [1,4,2,7,3,6,5] => [1,6,7,5,3,2,4] => ? = 2
[1,3,5,4,7,2,6] => [1,6,2,4,3,7,5] => [1,4,7,5,3,2,6] => ? = 2
[1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => [1,5,3,2,6,4,7] => ? = 1
[1,3,5,6,2,7,4] => [1,5,2,7,3,4,6] => [1,5,3,2,7,6,4] => ? = 2
[1,3,5,6,4,2,7] => [1,6,2,5,3,4,7] => [1,6,4,5,3,2,7] => ? = 1
[1,3,5,6,7,2,4] => [1,6,2,7,3,4,5] => [1,7,5,3,2,6,4] => ? = 2
[1,3,5,6,7,4,2] => [1,7,2,6,3,4,5] => [1,6,4,7,5,3,2] => ? = 2
[1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => [1,5,3,2,7,4,6] => ? = 1
[1,3,5,7,2,6,4] => [1,5,2,7,3,6,4] => [1,5,3,2,6,7,4] => ? = 1
[1,3,5,7,4,2,6] => [1,6,2,5,3,7,4] => [1,7,4,5,3,2,6] => ? = 1
[1,3,5,7,4,6,2] => [1,7,2,5,3,6,4] => [1,6,7,4,5,3,2] => ? = 1
[1,3,5,7,6,2,4] => [1,6,2,7,3,5,4] => [1,6,5,3,2,7,4] => ? = 2
[1,3,6,2,4,5,7] => [1,4,2,5,6,3,7] => [1,6,3,2,4,5,7] => ? = 1
[1,3,6,2,4,7,5] => [1,4,2,5,7,3,6] => [1,7,6,3,2,4,5] => ? = 2
[1,3,6,2,5,4,7] => [1,4,2,6,5,3,7] => [1,5,6,3,2,4,7] => ? = 1
[1,3,6,2,5,7,4] => [1,4,2,7,5,3,6] => [1,5,7,6,3,2,4] => ? = 2
[1,3,6,2,7,4,5] => [1,4,2,6,7,3,5] => [1,6,3,2,4,7,5] => ? = 1
[1,3,6,2,7,5,4] => [1,4,2,7,6,3,5] => [1,7,5,6,3,2,4] => ? = 1
[1,3,6,4,2,7,5] => [1,5,2,4,7,3,6] => [1,4,7,6,3,2,5] => ? = 2
[1,3,6,4,7,5,2] => [1,7,2,4,6,3,5] => [1,4,7,5,6,3,2] => ? = 1
[1,3,6,5,2,4,7] => [1,5,2,6,4,3,7] => [1,6,3,2,5,4,7] => ? = 1
[1,3,6,5,2,7,4] => [1,5,2,7,4,3,6] => [1,7,6,3,2,5,4] => ? = 2
[1,3,6,5,4,2,7] => [1,6,2,5,4,3,7] => [1,5,4,6,3,2,7] => ? = 1
[1,3,6,5,7,2,4] => [1,6,2,7,4,3,5] => [1,6,3,2,7,5,4] => ? = 2
[1,3,6,7,2,4,5] => [1,5,2,6,7,3,4] => [1,7,4,6,3,2,5] => ? = 1
[1,3,6,7,2,5,4] => [1,5,2,7,6,3,4] => [1,6,3,2,5,7,4] => ? = 1
[1,3,6,7,4,2,5] => [1,6,2,5,7,3,4] => [1,6,3,2,7,4,5] => ? = 1
[1,3,6,7,5,2,4] => [1,6,2,7,5,3,4] => [1,5,6,3,2,7,4] => ? = 1
[1,3,6,7,5,4,2] => [1,7,2,6,5,3,4] => [1,5,7,4,6,3,2] => ? = 1
[1,3,7,2,4,5,6] => [1,4,2,5,6,7,3] => [1,7,3,2,4,5,6] => ? = 1
[1,3,7,2,4,6,5] => [1,4,2,5,7,6,3] => [1,6,7,3,2,4,5] => ? = 1
[1,3,7,2,5,4,6] => [1,4,2,6,5,7,3] => [1,5,7,3,2,4,6] => ? = 1
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: St001167
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 60%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [1] => [1,0]
=> 0
[1,2] => [1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [2,1] => [1,1] => [1,0,1,0]
=> 0
[1,2,3] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 0
[2,1,3] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 0
[2,3,1] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[3,1,2] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 0
[3,2,1] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[1,3,2,4] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,3,4,2] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,4,2,3] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,4,3,2] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[2,1,3,4] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[2,1,4,3] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[2,3,1,4] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[2,3,4,1] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2
[2,4,1,3] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[2,4,3,1] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[3,1,2,4] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[3,1,4,2] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[3,2,1,4] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,2,4,1] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[3,4,1,2] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[3,4,2,1] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[4,1,2,3] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[4,1,3,2] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[4,2,1,3] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[4,2,3,1] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[4,3,1,2] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[4,3,2,1] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,2,4,5,3] => [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,2,5,3,4] => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,2,5,4,3] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,3,4,2,5] => [1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,3,5,2,4] => [1,5,3,2,4] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,4,2,5,3] => [1,5,4,2,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,4,3,5,2] => [1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,4,5,2,3] => [1,5,2,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[8,7,6,5,4,3,2,1] => [5,4,6,3,7,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[7,6,8,5,4,3,2,1] => [5,4,8,3,6,2,7,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[7,8,5,6,4,3,2,1] => [6,4,5,3,8,2,7,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[7,8,6,4,5,3,2,1] => [4,5,6,3,8,2,7,1] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,6,7,4,5,3,2,1] => [4,5,7,3,6,2,8,1] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[7,6,5,4,8,3,2,1] => [4,8,5,3,6,2,7,1] => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
[6,5,7,4,8,3,2,1] => [4,8,3,7,5,2,6,1] => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 1
[7,8,6,5,3,4,2,1] => [5,3,6,4,8,2,7,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,6,7,5,3,4,2,1] => [5,3,7,4,6,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,7,5,6,3,4,2,1] => [6,3,5,4,7,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[6,7,8,3,4,5,2,1] => [8,3,4,5,7,2,6,1] => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 0
[7,8,6,5,4,2,3,1] => [5,4,6,2,8,3,7,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,6,7,5,4,2,3,1] => [5,4,7,2,6,3,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,7,5,6,4,2,3,1] => [6,4,5,2,7,3,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,7,6,4,5,2,3,1] => [4,5,6,2,7,3,8,1] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,6,5,7,3,2,4,1] => [7,3,5,2,6,4,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[6,7,8,5,2,3,4,1] => [5,2,8,3,7,4,6,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,5,6,7,2,3,4,1] => [7,2,6,3,5,4,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[7,8,3,2,4,5,6,1] => [3,8,2,4,5,6,7,1] => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[7,6,5,4,3,2,8,1] => [4,5,3,6,2,8,7,1] => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
[6,5,7,4,3,2,8,1] => [4,8,7,3,5,2,6,1] => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
[6,5,4,3,7,2,8,1] => [4,3,8,7,5,2,6,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[5,4,6,3,7,2,8,1] => [8,7,3,4,2,6,5,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[3,4,5,2,6,7,8,1] => [8,7,6,5,2,4,3,1] => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 4
[5,2,3,4,6,7,8,1] => [2,3,4,8,7,6,5,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[4,3,2,5,6,7,8,1] => [3,2,8,7,6,5,4,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
[3,4,2,5,6,7,8,1] => [8,7,6,5,4,2,3,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 4
[4,2,3,5,6,7,8,1] => [2,3,8,7,6,5,4,1] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
[3,2,4,5,6,7,8,1] => [2,8,7,6,5,4,3,1] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[2,3,4,5,6,7,8,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[8,7,6,5,4,3,1,2] => [5,4,6,3,7,1,8,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[7,8,6,5,4,3,1,2] => [5,4,6,3,8,1,7,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,6,7,5,4,3,1,2] => [5,4,7,3,6,1,8,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,7,5,6,4,3,1,2] => [6,4,5,3,7,1,8,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,7,6,4,5,3,1,2] => [4,5,6,3,7,1,8,2] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,7,6,5,3,4,1,2] => [5,3,6,4,7,1,8,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,7,5,6,3,4,1,2] => [6,3,5,4,7,1,8,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[7,8,5,6,3,4,1,2] => [6,3,5,4,8,1,7,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[5,6,7,8,3,4,1,2] => [8,3,7,4,6,1,5,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[7,8,3,4,5,6,1,2] => [3,4,5,6,8,1,7,2] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0
[3,4,5,6,7,8,1,2] => [8,1,7,6,5,4,3,2] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
[8,7,6,5,4,2,1,3] => [5,4,6,2,7,1,8,3] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[7,6,8,5,4,2,1,3] => [5,4,8,2,6,1,7,3] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,7,6,5,4,1,2,3] => [5,4,6,1,7,2,8,3] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[6,7,8,5,4,1,2,3] => [5,4,8,1,7,2,6,3] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,5,6,7,4,1,2,3] => [7,4,6,1,5,2,8,3] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,7,6,4,5,1,2,3] => [4,5,6,1,7,2,8,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,6,7,4,5,1,2,3] => [4,5,7,1,6,2,8,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,7,4,5,6,1,2,3] => [6,5,4,1,7,2,8,3] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[7,8,4,5,6,1,2,3] => [6,5,4,1,8,2,7,3] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
Description
The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra.
The top of a module is the cokernel of the inclusion of the radical of the module into the module.
For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.
Matching statistic: St001253
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 60%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [1] => [1,0]
=> 0
[1,2] => [1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [2,1] => [1,1] => [1,0,1,0]
=> 0
[1,2,3] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 0
[2,1,3] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 0
[2,3,1] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[3,1,2] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 0
[3,2,1] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[1,3,2,4] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,3,4,2] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,4,2,3] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,4,3,2] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[2,1,3,4] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[2,1,4,3] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[2,3,1,4] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[2,3,4,1] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2
[2,4,1,3] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[2,4,3,1] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[3,1,2,4] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[3,1,4,2] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[3,2,1,4] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,2,4,1] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[3,4,1,2] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[3,4,2,1] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[4,1,2,3] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 0
[4,1,3,2] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[4,2,1,3] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[4,2,3,1] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[4,3,1,2] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[4,3,2,1] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,2,4,5,3] => [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,2,5,3,4] => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,2,5,4,3] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,3,4,2,5] => [1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,3,5,2,4] => [1,5,3,2,4] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,4,2,5,3] => [1,5,4,2,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,4,3,5,2] => [1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,4,5,2,3] => [1,5,2,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[8,7,6,5,4,3,2,1] => [5,4,6,3,7,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[7,6,8,5,4,3,2,1] => [5,4,8,3,6,2,7,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[7,8,5,6,4,3,2,1] => [6,4,5,3,8,2,7,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[7,8,6,4,5,3,2,1] => [4,5,6,3,8,2,7,1] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,6,7,4,5,3,2,1] => [4,5,7,3,6,2,8,1] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[7,6,5,4,8,3,2,1] => [4,8,5,3,6,2,7,1] => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
[6,5,7,4,8,3,2,1] => [4,8,3,7,5,2,6,1] => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 1
[7,8,6,5,3,4,2,1] => [5,3,6,4,8,2,7,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,6,7,5,3,4,2,1] => [5,3,7,4,6,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,7,5,6,3,4,2,1] => [6,3,5,4,7,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[6,7,8,3,4,5,2,1] => [8,3,4,5,7,2,6,1] => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 0
[7,8,6,5,4,2,3,1] => [5,4,6,2,8,3,7,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,6,7,5,4,2,3,1] => [5,4,7,2,6,3,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,7,5,6,4,2,3,1] => [6,4,5,2,7,3,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,7,6,4,5,2,3,1] => [4,5,6,2,7,3,8,1] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,6,5,7,3,2,4,1] => [7,3,5,2,6,4,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[6,7,8,5,2,3,4,1] => [5,2,8,3,7,4,6,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,5,6,7,2,3,4,1] => [7,2,6,3,5,4,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[7,8,3,2,4,5,6,1] => [3,8,2,4,5,6,7,1] => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[7,6,5,4,3,2,8,1] => [4,5,3,6,2,8,7,1] => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 1
[6,5,7,4,3,2,8,1] => [4,8,7,3,5,2,6,1] => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1
[6,5,4,3,7,2,8,1] => [4,3,8,7,5,2,6,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[5,4,6,3,7,2,8,1] => [8,7,3,4,2,6,5,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[3,4,5,2,6,7,8,1] => [8,7,6,5,2,4,3,1] => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 4
[5,2,3,4,6,7,8,1] => [2,3,4,8,7,6,5,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[4,3,2,5,6,7,8,1] => [3,2,8,7,6,5,4,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
[3,4,2,5,6,7,8,1] => [8,7,6,5,4,2,3,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 4
[4,2,3,5,6,7,8,1] => [2,3,8,7,6,5,4,1] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
[3,2,4,5,6,7,8,1] => [2,8,7,6,5,4,3,1] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5
[2,3,4,5,6,7,8,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
[8,7,6,5,4,3,1,2] => [5,4,6,3,7,1,8,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[7,8,6,5,4,3,1,2] => [5,4,6,3,8,1,7,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,6,7,5,4,3,1,2] => [5,4,7,3,6,1,8,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,7,5,6,4,3,1,2] => [6,4,5,3,7,1,8,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,7,6,4,5,3,1,2] => [4,5,6,3,7,1,8,2] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,7,6,5,3,4,1,2] => [5,3,6,4,7,1,8,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,7,5,6,3,4,1,2] => [6,3,5,4,7,1,8,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[7,8,5,6,3,4,1,2] => [6,3,5,4,8,1,7,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[5,6,7,8,3,4,1,2] => [8,3,7,4,6,1,5,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[7,8,3,4,5,6,1,2] => [3,4,5,6,8,1,7,2] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0
[3,4,5,6,7,8,1,2] => [8,1,7,6,5,4,3,2] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4
[8,7,6,5,4,2,1,3] => [5,4,6,2,7,1,8,3] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[7,6,8,5,4,2,1,3] => [5,4,8,2,6,1,7,3] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,7,6,5,4,1,2,3] => [5,4,6,1,7,2,8,3] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[6,7,8,5,4,1,2,3] => [5,4,8,1,7,2,6,3] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,5,6,7,4,1,2,3] => [7,4,6,1,5,2,8,3] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,7,6,4,5,1,2,3] => [4,5,6,1,7,2,8,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,6,7,4,5,1,2,3] => [4,5,7,1,6,2,8,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0
[8,7,4,5,6,1,2,3] => [6,5,4,1,7,2,8,3] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[7,8,4,5,6,1,2,3] => [6,5,4,1,8,2,7,3] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
Description
The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra.
For the first 196 values the statistic coincides also with the number of fixed points of $\tau \Omega^2$ composed with its inverse, see theorem 5.8. in the reference for more details.
The number of Dyck paths of length n where the statistics returns zero seems to be 2^(n-1).
Matching statistic: St001066
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 60%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [1] => [1,0]
=> 1 = 0 + 1
[1,2] => [1,2] => [2] => [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [2,1] => [1,1] => [1,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[2,1,3] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,3,1] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[3,1,2] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,1] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,4,2] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,3,2] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[2,1,3,4] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[2,3,1,4] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,4,1] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[2,4,1,3] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,1,2,4] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[3,1,4,2] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,4,1] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,4,1,2] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[3,4,2,1] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[4,1,2,3] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[4,1,3,2] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[4,2,1,3] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[4,2,3,1] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[4,3,1,2] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[4,3,2,1] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,5,4,3] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,5,2,4] => [1,5,3,2,4] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,2,5,3] => [1,5,4,2,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,3,5,2] => [1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,5,2,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[8,7,6,5,4,3,2,1] => [5,4,6,3,7,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[7,6,8,5,4,3,2,1] => [5,4,8,3,6,2,7,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[7,8,5,6,4,3,2,1] => [6,4,5,3,8,2,7,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[7,8,6,4,5,3,2,1] => [4,5,6,3,8,2,7,1] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,6,7,4,5,3,2,1] => [4,5,7,3,6,2,8,1] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[7,6,5,4,8,3,2,1] => [4,8,5,3,6,2,7,1] => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[6,5,7,4,8,3,2,1] => [4,8,3,7,5,2,6,1] => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 1 + 1
[7,8,6,5,3,4,2,1] => [5,3,6,4,8,2,7,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,6,7,5,3,4,2,1] => [5,3,7,4,6,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,7,5,6,3,4,2,1] => [6,3,5,4,7,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[6,7,8,3,4,5,2,1] => [8,3,4,5,7,2,6,1] => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[7,8,6,5,4,2,3,1] => [5,4,6,2,8,3,7,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,6,7,5,4,2,3,1] => [5,4,7,2,6,3,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,7,5,6,4,2,3,1] => [6,4,5,2,7,3,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,7,6,4,5,2,3,1] => [4,5,6,2,7,3,8,1] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,6,5,7,3,2,4,1] => [7,3,5,2,6,4,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[6,7,8,5,2,3,4,1] => [5,2,8,3,7,4,6,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,5,6,7,2,3,4,1] => [7,2,6,3,5,4,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[7,8,3,2,4,5,6,1] => [3,8,2,4,5,6,7,1] => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0 + 1
[7,6,5,4,3,2,8,1] => [4,5,3,6,2,8,7,1] => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[6,5,7,4,3,2,8,1] => [4,8,7,3,5,2,6,1] => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[6,5,4,3,7,2,8,1] => [4,3,8,7,5,2,6,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2 + 1
[5,4,6,3,7,2,8,1] => [8,7,3,4,2,6,5,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 + 1
[3,4,5,2,6,7,8,1] => [8,7,6,5,2,4,3,1] => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 4 + 1
[5,2,3,4,6,7,8,1] => [2,3,4,8,7,6,5,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3 + 1
[4,3,2,5,6,7,8,1] => [3,2,8,7,6,5,4,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[3,4,2,5,6,7,8,1] => [8,7,6,5,4,2,3,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[4,2,3,5,6,7,8,1] => [2,3,8,7,6,5,4,1] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[3,2,4,5,6,7,8,1] => [2,8,7,6,5,4,3,1] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5 + 1
[2,3,4,5,6,7,8,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 1
[8,7,6,5,4,3,1,2] => [5,4,6,3,7,1,8,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[7,8,6,5,4,3,1,2] => [5,4,6,3,8,1,7,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,6,7,5,4,3,1,2] => [5,4,7,3,6,1,8,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,7,5,6,4,3,1,2] => [6,4,5,3,7,1,8,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,7,6,4,5,3,1,2] => [4,5,6,3,7,1,8,2] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,7,6,5,3,4,1,2] => [5,3,6,4,7,1,8,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,7,5,6,3,4,1,2] => [6,3,5,4,7,1,8,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[7,8,5,6,3,4,1,2] => [6,3,5,4,8,1,7,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[5,6,7,8,3,4,1,2] => [8,3,7,4,6,1,5,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[7,8,3,4,5,6,1,2] => [3,4,5,6,8,1,7,2] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[3,4,5,6,7,8,1,2] => [8,1,7,6,5,4,3,2] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[8,7,6,5,4,2,1,3] => [5,4,6,2,7,1,8,3] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[7,6,8,5,4,2,1,3] => [5,4,8,2,6,1,7,3] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,7,6,5,4,1,2,3] => [5,4,6,1,7,2,8,3] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[6,7,8,5,4,1,2,3] => [5,4,8,1,7,2,6,3] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,5,6,7,4,1,2,3] => [7,4,6,1,5,2,8,3] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,7,6,4,5,1,2,3] => [4,5,6,1,7,2,8,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,6,7,4,5,1,2,3] => [4,5,7,1,6,2,8,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,7,4,5,6,1,2,3] => [6,5,4,1,7,2,8,3] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
[7,8,4,5,6,1,2,3] => [6,5,4,1,8,2,7,3] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
Description
The number of simple reflexive modules in the corresponding Nakayama algebra.
Matching statistic: St001483
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 60%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001483: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [1] => [1,0]
=> 1 = 0 + 1
[1,2] => [1,2] => [2] => [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [2,1] => [1,1] => [1,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[2,1,3] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,3,1] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[3,1,2] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,1] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,4,2] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,3,2] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[2,1,3,4] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[2,3,1,4] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,4,1] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[2,4,1,3] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,1,2,4] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[3,1,4,2] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,2,4,1] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,4,1,2] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[3,4,2,1] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[4,1,2,3] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[4,1,3,2] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[4,2,1,3] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[4,2,3,1] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[4,3,1,2] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[4,3,2,1] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,5,4,3] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,3,5,2,4] => [1,5,3,2,4] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,2,5,3] => [1,5,4,2,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,3,5,2] => [1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,5,2,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[8,7,6,5,4,3,2,1] => [5,4,6,3,7,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[7,6,8,5,4,3,2,1] => [5,4,8,3,6,2,7,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[7,8,5,6,4,3,2,1] => [6,4,5,3,8,2,7,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[7,8,6,4,5,3,2,1] => [4,5,6,3,8,2,7,1] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,6,7,4,5,3,2,1] => [4,5,7,3,6,2,8,1] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[7,6,5,4,8,3,2,1] => [4,8,5,3,6,2,7,1] => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[6,5,7,4,8,3,2,1] => [4,8,3,7,5,2,6,1] => [2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 1 + 1
[7,8,6,5,3,4,2,1] => [5,3,6,4,8,2,7,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,6,7,5,3,4,2,1] => [5,3,7,4,6,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,7,5,6,3,4,2,1] => [6,3,5,4,7,2,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[6,7,8,3,4,5,2,1] => [8,3,4,5,7,2,6,1] => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[7,8,6,5,4,2,3,1] => [5,4,6,2,8,3,7,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,6,7,5,4,2,3,1] => [5,4,7,2,6,3,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,7,5,6,4,2,3,1] => [6,4,5,2,7,3,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,7,6,4,5,2,3,1] => [4,5,6,2,7,3,8,1] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,6,5,7,3,2,4,1] => [7,3,5,2,6,4,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[6,7,8,5,2,3,4,1] => [5,2,8,3,7,4,6,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,5,6,7,2,3,4,1] => [7,2,6,3,5,4,8,1] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[7,8,3,2,4,5,6,1] => [3,8,2,4,5,6,7,1] => [2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0 + 1
[7,6,5,4,3,2,8,1] => [4,5,3,6,2,8,7,1] => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[6,5,7,4,3,2,8,1] => [4,8,7,3,5,2,6,1] => [2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 1 + 1
[6,5,4,3,7,2,8,1] => [4,3,8,7,5,2,6,1] => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2 + 1
[5,4,6,3,7,2,8,1] => [8,7,3,4,2,6,5,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2 + 1
[3,4,5,2,6,7,8,1] => [8,7,6,5,2,4,3,1] => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 4 + 1
[5,2,3,4,6,7,8,1] => [2,3,4,8,7,6,5,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3 + 1
[4,3,2,5,6,7,8,1] => [3,2,8,7,6,5,4,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[3,4,2,5,6,7,8,1] => [8,7,6,5,4,2,3,1] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 4 + 1
[4,2,3,5,6,7,8,1] => [2,3,8,7,6,5,4,1] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[3,2,4,5,6,7,8,1] => [2,8,7,6,5,4,3,1] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 5 + 1
[2,3,4,5,6,7,8,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6 + 1
[8,7,6,5,4,3,1,2] => [5,4,6,3,7,1,8,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[7,8,6,5,4,3,1,2] => [5,4,6,3,8,1,7,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,6,7,5,4,3,1,2] => [5,4,7,3,6,1,8,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,7,5,6,4,3,1,2] => [6,4,5,3,7,1,8,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,7,6,4,5,3,1,2] => [4,5,6,3,7,1,8,2] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,7,6,5,3,4,1,2] => [5,3,6,4,7,1,8,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,7,5,6,3,4,1,2] => [6,3,5,4,7,1,8,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[7,8,5,6,3,4,1,2] => [6,3,5,4,8,1,7,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[5,6,7,8,3,4,1,2] => [8,3,7,4,6,1,5,2] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[7,8,3,4,5,6,1,2] => [3,4,5,6,8,1,7,2] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[3,4,5,6,7,8,1,2] => [8,1,7,6,5,4,3,2] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 1
[8,7,6,5,4,2,1,3] => [5,4,6,2,7,1,8,3] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[7,6,8,5,4,2,1,3] => [5,4,8,2,6,1,7,3] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,7,6,5,4,1,2,3] => [5,4,6,1,7,2,8,3] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[6,7,8,5,4,1,2,3] => [5,4,8,1,7,2,6,3] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,5,6,7,4,1,2,3] => [7,4,6,1,5,2,8,3] => [1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,7,6,4,5,1,2,3] => [4,5,6,1,7,2,8,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,6,7,4,5,1,2,3] => [4,5,7,1,6,2,8,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 0 + 1
[8,7,4,5,6,1,2,3] => [6,5,4,1,7,2,8,3] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
[7,8,4,5,6,1,2,3] => [6,5,4,1,8,2,7,3] => [1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
Description
The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module.
Matching statistic: St000732
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000732: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 50%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000732: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[2,3,1] => [3,2,1] => [3,2,1] => [3,1,2] => 1
[3,1,2] => [3,1,2] => [3,1,2] => [2,3,1] => 0
[3,2,1] => [2,3,1] => [1,3,2] => [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] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => [1,4,2,3] => 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 0
[1,4,3,2] => [1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 0
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => [3,1,2,4] => 1
[2,3,4,1] => [4,3,2,1] => [4,3,2,1] => [4,1,2,3] => 2
[2,4,1,3] => [4,2,1,3] => [4,2,1,3] => [3,1,4,2] => 1
[2,4,3,1] => [3,4,2,1] => [1,4,3,2] => [1,4,2,3] => 1
[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => [2,3,1,4] => 0
[3,1,4,2] => [4,3,1,2] => [4,3,1,2] => [2,4,1,3] => 1
[3,2,1,4] => [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 0
[3,2,4,1] => [2,4,3,1] => [1,4,3,2] => [1,4,2,3] => 1
[3,4,1,2] => [4,1,3,2] => [4,1,3,2] => [3,4,2,1] => 0
[3,4,2,1] => [4,2,3,1] => [4,1,3,2] => [3,4,2,1] => 0
[4,1,2,3] => [4,1,2,3] => [4,1,2,3] => [2,3,4,1] => 0
[4,1,3,2] => [3,4,1,2] => [2,4,1,3] => [3,2,4,1] => 0
[4,2,1,3] => [2,4,1,3] => [2,4,1,3] => [3,2,4,1] => 0
[4,2,3,1] => [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[4,3,1,2] => [3,1,4,2] => [2,1,4,3] => [2,1,4,3] => 0
[4,3,2,1] => [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 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] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,3,4] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => 0
[1,2,5,4,3] => [1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,4,2,3,5] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,5,2,3,4] => 2
[1,3,5,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => [1,4,2,5,3] => 1
[1,3,5,4,2] => [1,4,5,3,2] => [1,2,5,4,3] => [1,2,5,3,4] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 0
[1,4,2,5,3] => [1,5,4,2,3] => [1,5,4,2,3] => [1,3,5,2,4] => 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,4,3,5,2] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,3,4] => 1
[1,4,5,2,3] => [1,5,2,4,3] => [1,5,2,4,3] => [1,4,5,3,2] => 0
[1,4,5,3,2] => [1,5,3,4,2] => [1,5,2,4,3] => [1,4,5,3,2] => 0
[1,3,4,5,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => [1,5,2,3,4,6,7] => ? = 2
[1,3,4,5,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => [1,5,2,3,4,7,6] => ? = 2
[1,3,4,5,6,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,2,3,4,5,7] => ? = 3
[1,3,4,5,6,7,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 4
[1,3,4,5,7,2,6] => [1,7,5,4,3,2,6] => [1,7,5,4,3,2,6] => [1,6,2,3,4,7,5] => ? = 3
[1,3,4,6,2,5,7] => [1,6,4,3,2,5,7] => [1,6,4,3,2,5,7] => [1,5,2,3,6,4,7] => ? = 2
[1,3,4,6,2,7,5] => [1,7,6,4,3,2,5] => [1,7,6,4,3,2,5] => [1,5,2,3,7,4,6] => ? = 3
[1,3,4,6,7,2,5] => [1,7,4,3,2,6,5] => [1,7,4,3,2,6,5] => [1,6,2,3,7,5,4] => ? = 2
[1,3,4,6,7,5,2] => [1,7,5,6,4,3,2] => [1,7,2,6,5,4,3] => [1,6,7,3,4,5,2] => ? = 2
[1,3,4,7,2,5,6] => [1,7,4,3,2,5,6] => [1,7,4,3,2,5,6] => [1,5,2,3,6,7,4] => ? = 2
[1,3,4,7,2,6,5] => [1,6,7,4,3,2,5] => [1,5,7,4,3,2,6] => [1,6,2,3,5,7,4] => ? = 2
[1,3,4,7,5,2,6] => [1,5,7,4,3,2,6] => [1,5,7,4,3,2,6] => [1,6,2,3,5,7,4] => ? = 2
[1,3,4,7,6,2,5] => [1,6,4,3,2,7,5] => [1,5,4,3,2,7,6] => [1,5,2,3,4,7,6] => ? = 2
[1,3,5,2,7,6,4] => [1,6,7,5,3,2,4] => [1,4,7,6,3,2,5] => [1,5,2,4,7,3,6] => ? = 2
[1,3,5,6,2,4,7] => [1,6,3,2,5,4,7] => [1,6,3,2,5,4,7] => [1,5,2,6,4,3,7] => ? = 1
[1,3,5,6,2,7,4] => [1,7,6,3,2,5,4] => [1,7,6,3,2,5,4] => [1,5,2,7,4,3,6] => ? = 2
[1,3,5,6,4,2,7] => [1,6,4,5,3,2,7] => [1,6,2,5,4,3,7] => [1,5,6,3,4,2,7] => ? = 1
[1,3,5,6,4,7,2] => [1,7,6,4,5,3,2] => [1,7,6,2,5,4,3] => [1,5,7,3,4,2,6] => ? = 2
[1,3,5,6,7,2,4] => [1,7,3,2,6,5,4] => [1,7,3,2,6,5,4] => [1,6,2,7,4,5,3] => ? = 2
[1,3,5,6,7,4,2] => [1,7,4,6,5,3,2] => [1,7,2,6,5,4,3] => [1,6,7,3,4,5,2] => ? = 2
[1,3,5,7,2,4,6] => [1,7,3,2,5,4,6] => [1,7,3,2,5,4,6] => [1,5,2,6,4,7,3] => ? = 1
[1,3,5,7,2,6,4] => [1,6,7,3,2,5,4] => [1,4,7,3,2,6,5] => [1,6,2,4,7,5,3] => ? = 1
[1,3,5,7,4,2,6] => [1,7,4,5,3,2,6] => [1,7,2,5,4,3,6] => [1,5,6,3,4,7,2] => ? = 1
[1,3,5,7,4,6,2] => [1,6,7,4,5,3,2] => [1,3,7,2,6,5,4] => [1,6,3,7,4,5,2] => ? = 1
[1,3,6,2,5,4,7] => [1,5,6,3,2,4,7] => [1,4,6,3,2,5,7] => [1,5,2,4,6,3,7] => ? = 1
[1,3,6,2,5,7,4] => [1,5,7,6,3,2,4] => [1,4,7,6,3,2,5] => [1,5,2,4,7,3,6] => ? = 2
[1,3,6,2,7,5,4] => [1,7,5,6,3,2,4] => [1,7,4,6,3,2,5] => [1,5,2,6,7,3,4] => ? = 1
[1,3,6,4,2,5,7] => [1,4,6,3,2,5,7] => [1,4,6,3,2,5,7] => [1,5,2,4,6,3,7] => ? = 1
[1,3,6,4,2,7,5] => [1,4,7,6,3,2,5] => [1,4,7,6,3,2,5] => [1,5,2,4,7,3,6] => ? = 2
[1,3,6,4,7,2,5] => [1,4,7,3,2,6,5] => [1,4,7,3,2,6,5] => [1,6,2,4,7,5,3] => ? = 1
[1,3,6,5,7,2,4] => [1,7,5,3,2,6,4] => [1,7,4,3,2,6,5] => [1,6,2,3,7,5,4] => ? = 2
[1,3,6,5,7,4,2] => [1,7,5,4,6,3,2] => [1,7,3,2,6,5,4] => [1,6,2,7,4,5,3] => ? = 2
[1,3,6,7,2,4,5] => [1,7,3,2,6,4,5] => [1,7,3,2,6,4,5] => [1,6,2,5,7,4,3] => ? = 1
[1,3,6,7,4,2,5] => [1,7,4,6,3,2,5] => [1,7,4,6,3,2,5] => [1,5,2,6,7,3,4] => ? = 1
[1,3,6,7,5,2,4] => [1,5,7,3,2,6,4] => [1,4,7,3,2,6,5] => [1,6,2,4,7,5,3] => ? = 1
[1,3,6,7,5,4,2] => [1,5,7,4,6,3,2] => [1,3,7,2,6,5,4] => [1,6,3,7,4,5,2] => ? = 1
[1,3,7,2,6,5,4] => [1,6,5,7,3,2,4] => [1,5,4,7,3,2,6] => [1,6,2,5,4,7,3] => ? = 1
[1,3,7,4,2,5,6] => [1,4,7,3,2,5,6] => [1,4,7,3,2,5,6] => [1,5,2,4,6,7,3] => ? = 1
[1,3,7,5,2,4,6] => [1,5,3,2,7,4,6] => [1,5,3,2,7,4,6] => [1,5,2,6,3,7,4] => ? = 1
[1,3,7,5,4,2,6] => [1,5,4,7,3,2,6] => [1,5,4,7,3,2,6] => [1,6,2,5,4,7,3] => ? = 1
[1,3,7,5,6,2,4] => [1,6,5,3,2,7,4] => [1,5,4,3,2,7,6] => [1,5,2,3,4,7,6] => ? = 2
[1,3,7,6,2,4,5] => [1,6,3,2,7,4,5] => [1,5,3,2,7,4,6] => [1,5,2,6,3,7,4] => ? = 1
[1,3,7,6,4,2,5] => [1,6,4,7,3,2,5] => [1,5,4,7,3,2,6] => [1,6,2,5,4,7,3] => ? = 1
[1,4,2,6,7,5,3] => [1,7,5,6,4,2,3] => [1,7,3,6,5,2,4] => [1,4,6,7,2,5,3] => ? = 1
[1,4,2,7,6,5,3] => [1,6,5,7,4,2,3] => [1,4,3,7,6,2,5] => [1,5,4,3,7,2,6] => ? = 1
[1,4,5,2,6,7,3] => [1,7,6,5,2,4,3] => [1,7,6,5,2,4,3] => [1,4,7,3,2,5,6] => ? = 2
[1,4,5,2,7,6,3] => [1,6,7,5,2,4,3] => [1,3,7,6,2,5,4] => [1,5,3,7,4,2,6] => ? = 1
[1,4,5,3,6,7,2] => [1,7,6,5,3,4,2] => [1,7,6,5,2,4,3] => [1,4,7,3,2,5,6] => ? = 2
[1,4,5,3,7,6,2] => [1,6,7,5,3,4,2] => [1,3,7,6,2,5,4] => [1,5,3,7,4,2,6] => ? = 1
Description
The number of double deficiencies of a permutation.
A double deficiency is an index $\sigma(i)$ such that $i > \sigma(i) > \sigma(\sigma(i))$.
Matching statistic: St001744
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001744: Permutations ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 50%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001744: Permutations ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 50%
Values
[1] => {{1}}
=> [1] => [1] => 0
[1,2] => {{1},{2}}
=> [1,2] => [1,2] => 0
[2,1] => {{1,2}}
=> [2,1] => [2,1] => 0
[1,2,3] => {{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
[1,3,2] => {{1},{2,3}}
=> [1,3,2] => [1,3,2] => 0
[2,1,3] => {{1,2},{3}}
=> [2,1,3] => [2,1,3] => 0
[2,3,1] => {{1,2,3}}
=> [2,3,1] => [3,1,2] => 1
[3,1,2] => {{1,3},{2}}
=> [3,2,1] => [2,3,1] => 0
[3,2,1] => {{1,3},{2}}
=> [3,2,1] => [2,3,1] => 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}}
=> [1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => 1
[1,4,2,3] => {{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => 0
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => 0
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => {{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => {{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => 1
[2,3,4,1] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 2
[2,4,1,3] => {{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => 1
[2,4,3,1] => {{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => 1
[3,1,2,4] => {{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => 0
[3,1,4,2] => {{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => 1
[3,2,1,4] => {{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => 0
[3,2,4,1] => {{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => 1
[3,4,1,2] => {{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => 0
[3,4,2,1] => {{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => 0
[4,1,2,3] => {{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => 0
[4,1,3,2] => {{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => 0
[4,2,1,3] => {{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => 0
[4,2,3,1] => {{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => 0
[4,3,1,2] => {{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => 0
[4,3,2,1] => {{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => 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}}
=> [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,4,5,3] => 0
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,4,5,3] => 0
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => 2
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,4,5,2,3] => 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,4,5,2,3] => 1
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,3,4,2,5] => 0
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,3,5,2,4] => 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,3,4,2,5] => 0
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,3,5,2,4] => 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,2,5,3] => 0
[1,3,4,5,2,6,7] => {{1},{2,3,4,5},{6},{7}}
=> [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 2
[1,3,4,5,2,7,6] => {{1},{2,3,4,5},{6,7}}
=> [1,3,4,5,2,7,6] => [1,5,2,3,4,7,6] => ? = 2
[1,3,4,5,6,2,7] => {{1},{2,3,4,5,6},{7}}
=> [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 3
[1,3,4,5,6,7,2] => {{1},{2,3,4,5,6,7}}
=> [1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => ? = 4
[1,3,4,5,7,2,6] => {{1},{2,3,4,5,7},{6}}
=> [1,3,4,5,7,6,2] => [1,6,7,2,3,4,5] => ? = 3
[1,3,4,5,7,6,2] => {{1},{2,3,4,5,7},{6}}
=> [1,3,4,5,7,6,2] => [1,6,7,2,3,4,5] => ? = 3
[1,3,4,6,2,5,7] => {{1},{2,3,4,6},{5},{7}}
=> [1,3,4,6,5,2,7] => [1,5,6,2,3,4,7] => ? = 2
[1,3,4,6,2,7,5] => {{1},{2,3,4,6,7},{5}}
=> [1,3,4,6,5,7,2] => [1,5,7,2,3,4,6] => ? = 3
[1,3,4,6,5,2,7] => {{1},{2,3,4,6},{5},{7}}
=> [1,3,4,6,5,2,7] => [1,5,6,2,3,4,7] => ? = 2
[1,3,4,6,5,7,2] => {{1},{2,3,4,6,7},{5}}
=> [1,3,4,6,5,7,2] => [1,5,7,2,3,4,6] => ? = 3
[1,3,4,6,7,2,5] => {{1},{2,3,4,6},{5,7}}
=> [1,3,4,6,7,2,5] => [1,6,2,3,4,7,5] => ? = 2
[1,3,4,6,7,5,2] => {{1},{2,3,4,6},{5,7}}
=> [1,3,4,6,7,2,5] => [1,6,2,3,4,7,5] => ? = 2
[1,3,4,7,2,5,6] => {{1},{2,3,4,7},{5},{6}}
=> [1,3,4,7,5,6,2] => [1,5,6,7,2,3,4] => ? = 2
[1,3,4,7,2,6,5] => {{1},{2,3,4,7},{5},{6}}
=> [1,3,4,7,5,6,2] => [1,5,6,7,2,3,4] => ? = 2
[1,3,4,7,5,2,6] => {{1},{2,3,4,7},{5},{6}}
=> [1,3,4,7,5,6,2] => [1,5,6,7,2,3,4] => ? = 2
[1,3,4,7,5,6,2] => {{1},{2,3,4,7},{5},{6}}
=> [1,3,4,7,5,6,2] => [1,5,6,7,2,3,4] => ? = 2
[1,3,4,7,6,2,5] => {{1},{2,3,4,7},{5,6}}
=> [1,3,4,7,6,5,2] => [1,6,5,7,2,3,4] => ? = 2
[1,3,4,7,6,5,2] => {{1},{2,3,4,7},{5,6}}
=> [1,3,4,7,6,5,2] => [1,6,5,7,2,3,4] => ? = 2
[1,3,5,2,6,7,4] => {{1},{2,3,5,6,7},{4}}
=> [1,3,5,4,6,7,2] => [1,4,7,2,3,5,6] => ? = 3
[1,3,5,2,7,4,6] => {{1},{2,3,5,7},{4},{6}}
=> [1,3,5,4,7,6,2] => [1,4,6,7,2,3,5] => ? = 2
[1,3,5,2,7,6,4] => {{1},{2,3,5,7},{4},{6}}
=> [1,3,5,4,7,6,2] => [1,4,6,7,2,3,5] => ? = 2
[1,3,5,4,6,7,2] => {{1},{2,3,5,6,7},{4}}
=> [1,3,5,4,6,7,2] => [1,4,7,2,3,5,6] => ? = 3
[1,3,5,4,7,2,6] => {{1},{2,3,5,7},{4},{6}}
=> [1,3,5,4,7,6,2] => [1,4,6,7,2,3,5] => ? = 2
[1,3,5,4,7,6,2] => {{1},{2,3,5,7},{4},{6}}
=> [1,3,5,4,7,6,2] => [1,4,6,7,2,3,5] => ? = 2
[1,3,5,6,2,4,7] => {{1},{2,3,5},{4,6},{7}}
=> [1,3,5,6,2,4,7] => [1,5,2,3,6,4,7] => ? = 1
[1,3,5,6,2,7,4] => {{1},{2,3,5},{4,6,7}}
=> [1,3,5,6,2,7,4] => [1,5,2,3,7,4,6] => ? = 2
[1,3,5,6,4,2,7] => {{1},{2,3,5},{4,6},{7}}
=> [1,3,5,6,2,4,7] => [1,5,2,3,6,4,7] => ? = 1
[1,3,5,6,4,7,2] => {{1},{2,3,5},{4,6,7}}
=> [1,3,5,6,2,7,4] => [1,5,2,3,7,4,6] => ? = 2
[1,3,5,6,7,2,4] => {{1},{2,3,5,7},{4,6}}
=> [1,3,5,6,7,4,2] => [1,6,4,7,2,3,5] => ? = 2
[1,3,5,6,7,4,2] => {{1},{2,3,5,7},{4,6}}
=> [1,3,5,6,7,4,2] => [1,6,4,7,2,3,5] => ? = 2
[1,3,5,7,2,4,6] => {{1},{2,3,5},{4,7},{6}}
=> [1,3,5,7,2,6,4] => [1,5,2,3,6,7,4] => ? = 1
[1,3,5,7,2,6,4] => {{1},{2,3,5},{4,7},{6}}
=> [1,3,5,7,2,6,4] => [1,5,2,3,6,7,4] => ? = 1
[1,3,5,7,4,2,6] => {{1},{2,3,5},{4,7},{6}}
=> [1,3,5,7,2,6,4] => [1,5,2,3,6,7,4] => ? = 1
[1,3,5,7,4,6,2] => {{1},{2,3,5},{4,7},{6}}
=> [1,3,5,7,2,6,4] => [1,5,2,3,6,7,4] => ? = 1
[1,3,5,7,6,2,4] => {{1},{2,3,5,6},{4,7}}
=> [1,3,5,7,6,2,4] => [1,6,2,3,5,7,4] => ? = 2
[1,3,5,7,6,4,2] => {{1},{2,3,5,6},{4,7}}
=> [1,3,5,7,6,2,4] => [1,6,2,3,5,7,4] => ? = 2
[1,3,6,5,2,4,7] => {{1},{2,3,6},{4,5},{7}}
=> [1,3,6,5,4,2,7] => [1,5,4,6,2,3,7] => ? = 1
[1,3,6,5,2,7,4] => {{1},{2,3,6,7},{4,5}}
=> [1,3,6,5,4,7,2] => [1,5,4,7,2,3,6] => ? = 2
[1,3,6,5,4,2,7] => {{1},{2,3,6},{4,5},{7}}
=> [1,3,6,5,4,2,7] => [1,5,4,6,2,3,7] => ? = 1
[1,3,6,5,4,7,2] => {{1},{2,3,6,7},{4,5}}
=> [1,3,6,5,4,7,2] => [1,5,4,7,2,3,6] => ? = 2
[1,3,6,5,7,2,4] => {{1},{2,3,6},{4,5,7}}
=> [1,3,6,5,7,2,4] => [1,6,2,3,7,4,5] => ? = 2
[1,3,6,5,7,4,2] => {{1},{2,3,6},{4,5,7}}
=> [1,3,6,5,7,2,4] => [1,6,2,3,7,4,5] => ? = 2
[1,3,6,7,2,4,5] => {{1},{2,3,6},{4,7},{5}}
=> [1,3,6,7,5,2,4] => [1,5,6,2,3,7,4] => ? = 1
[1,3,6,7,2,5,4] => {{1},{2,3,6},{4,7},{5}}
=> [1,3,6,7,5,2,4] => [1,5,6,2,3,7,4] => ? = 1
[1,3,6,7,4,2,5] => {{1},{2,3,6},{4,7},{5}}
=> [1,3,6,7,5,2,4] => [1,5,6,2,3,7,4] => ? = 1
[1,3,6,7,4,5,2] => {{1},{2,3,6},{4,7},{5}}
=> [1,3,6,7,5,2,4] => [1,5,6,2,3,7,4] => ? = 1
[1,3,6,7,5,2,4] => {{1},{2,3,6},{4,7},{5}}
=> [1,3,6,7,5,2,4] => [1,5,6,2,3,7,4] => ? = 1
[1,3,6,7,5,4,2] => {{1},{2,3,6},{4,7},{5}}
=> [1,3,6,7,5,2,4] => [1,5,6,2,3,7,4] => ? = 1
[1,3,7,2,6,4,5] => {{1},{2,3,7},{4},{5,6}}
=> [1,3,7,4,6,5,2] => [1,4,6,5,7,2,3] => ? = 1
[1,3,7,2,6,5,4] => {{1},{2,3,7},{4},{5,6}}
=> [1,3,7,4,6,5,2] => [1,4,6,5,7,2,3] => ? = 1
Description
The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation.
Let $\nu$ be a (partial) permutation of $[k]$ with $m$ letters together with dashes between some of its letters. An occurrence of $\nu$ in a permutation $\tau$ is a subsequence $\tau_{a_1},\dots,\tau_{a_m}$
such that $a_i + 1 = a_{i+1}$ whenever there is a dash between the $i$-th and the $(i+1)$-st letter of $\nu$, which is order isomorphic to $\nu$.
Thus, $\nu$ is a vincular pattern, except that it is not required to be a permutation.
An arrow pattern of size $k$ consists of such a generalized vincular pattern $\nu$ and arrows $b_1\to c_1, b_2\to c_2,\dots$, such that precisely the numbers $1,\dots,k$ appear in the vincular pattern and the arrows.
Let $\Phi$ be the map [[Mp00087]]. Let $\tau$ be a permutation and $\sigma = \Phi(\tau)$. Then a subsequence $w = (x_{a_1},\dots,x_{a_m})$ of $\tau$ is an occurrence of the arrow pattern if $w$ is an occurrence of $\nu$, for each arrow $b\to c$ we have $\sigma(x_b) = x_c$ and $x_1 < x_2 < \dots < x_k$.
Matching statistic: St000365
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000365: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 60%
Mp00064: Permutations —reverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000365: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 60%
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] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [2,3,1] => [1,3,2] => 0
[2,1,3] => [2,1,3] => [3,1,2] => [3,1,2] => 0
[2,3,1] => [3,2,1] => [1,2,3] => [1,2,3] => 1
[3,1,2] => [3,1,2] => [2,1,3] => [2,1,3] => 0
[3,2,1] => [2,3,1] => [1,3,2] => [1,3,2] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [1,4,3,2] => 0
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [4,1,3,2] => 0
[1,3,4,2] => [1,4,3,2] => [2,3,4,1] => [1,2,4,3] => 1
[1,4,2,3] => [1,4,2,3] => [3,2,4,1] => [2,1,4,3] => 0
[1,4,3,2] => [1,3,4,2] => [2,4,3,1] => [1,4,3,2] => 0
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 0
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [2,4,1,3] => 0
[2,3,1,4] => [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 1
[2,3,4,1] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 2
[2,4,1,3] => [4,2,1,3] => [3,1,2,4] => [3,1,2,4] => 1
[2,4,3,1] => [3,4,2,1] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,2,4] => [3,1,2,4] => [4,2,1,3] => [4,2,1,3] => 0
[3,1,4,2] => [4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 1
[3,2,1,4] => [2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 0
[3,2,4,1] => [2,4,3,1] => [1,3,4,2] => [1,2,4,3] => 1
[3,4,1,2] => [4,1,3,2] => [2,3,1,4] => [1,3,2,4] => 0
[3,4,2,1] => [4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 0
[4,1,2,3] => [4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 0
[4,1,3,2] => [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 0
[4,2,1,3] => [2,4,1,3] => [3,1,4,2] => [2,1,4,3] => 0
[4,2,3,1] => [2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 0
[4,3,1,2] => [3,1,4,2] => [2,4,1,3] => [2,4,1,3] => 0
[4,3,2,1] => [3,2,4,1] => [1,4,2,3] => [1,4,2,3] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,5,4,3,2] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [5,1,4,3,2] => 0
[1,2,4,5,3] => [1,2,5,4,3] => [3,4,5,2,1] => [1,2,5,4,3] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [4,3,5,2,1] => [2,1,5,4,3] => 0
[1,2,5,4,3] => [1,2,4,5,3] => [3,5,4,2,1] => [1,5,4,3,2] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [5,4,1,3,2] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [2,5,1,4,3] => 0
[1,3,4,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => [5,1,2,4,3] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,5,4] => 2
[1,3,5,2,4] => [1,5,3,2,4] => [4,2,3,5,1] => [3,1,2,5,4] => 1
[1,3,5,4,2] => [1,4,5,3,2] => [2,3,5,4,1] => [1,2,5,4,3] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => [5,2,1,4,3] => 0
[1,4,2,5,3] => [1,5,4,2,3] => [3,2,4,5,1] => [2,1,3,5,4] => 1
[1,4,3,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => [5,1,4,3,2] => 0
[1,4,3,5,2] => [1,3,5,4,2] => [2,4,5,3,1] => [1,2,5,4,3] => 1
[1,4,5,2,3] => [1,5,2,4,3] => [3,4,2,5,1] => [1,3,2,5,4] => 0
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 0
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,1,6,5,4,3,2] => ? = 0
[1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [2,1,7,6,5,4,3] => ? = 0
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 0
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [7,6,1,5,4,3,2] => ? = 0
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,7,1,6,5,4,3] => ? = 0
[1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [7,1,2,6,5,4,3] => ? = 1
[1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [3,1,2,7,6,5,4] => ? = 1
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [7,2,1,6,5,4,3] => ? = 0
[1,2,3,6,4,7,5] => [1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [2,1,3,7,6,5,4] => ? = 1
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [7,1,6,5,4,3,2] => ? = 0
[1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [3,2,1,7,6,5,4] => ? = 0
[1,2,3,7,4,6,5] => [1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [2,1,7,6,5,4,3] => ? = 0
[1,2,3,7,5,4,6] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [2,1,7,6,5,4,3] => ? = 0
[1,2,3,7,5,6,4] => [1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [1,7,6,5,4,3,2] => ? = 0
[1,2,3,7,6,4,5] => [1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [2,7,1,6,5,4,3] => ? = 0
[1,2,3,7,6,5,4] => [1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [1,7,2,6,5,4,3] => ? = 0
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [7,6,5,1,4,3,2] => ? = 0
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [2,7,6,1,5,4,3] => ? = 0
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [7,2,6,1,5,4,3] => ? = 0
[1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [3,2,7,1,6,5,4] => ? = 0
[1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [2,7,6,1,5,4,3] => ? = 0
[1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => [7,6,1,2,5,4,3] => ? = 1
[1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => [3,7,1,2,6,5,4] => ? = 1
[1,2,4,5,6,3,7] => [1,2,6,5,4,3,7] => [7,3,4,5,6,2,1] => [7,1,2,3,6,5,4] => ? = 2
[1,2,4,5,7,3,6] => [1,2,7,5,4,3,6] => [6,3,4,5,7,2,1] => [4,1,2,3,7,6,5] => ? = 2
[1,2,4,6,3,5,7] => [1,2,6,4,3,5,7] => [7,5,3,4,6,2,1] => [7,3,1,2,6,5,4] => ? = 1
[1,2,4,6,3,7,5] => [1,2,7,6,4,3,5] => [5,3,4,6,7,2,1] => [3,1,2,4,7,6,5] => ? = 2
[1,2,4,6,5,3,7] => [1,2,5,6,4,3,7] => [7,3,4,6,5,2,1] => [7,1,2,6,5,4,3] => ? = 1
[1,2,4,6,7,3,5] => [1,2,7,4,3,6,5] => [5,6,3,4,7,2,1] => [2,4,1,3,7,6,5] => ? = 1
[1,2,4,7,3,5,6] => [1,2,7,4,3,5,6] => [6,5,3,4,7,2,1] => [4,3,1,2,7,6,5] => ? = 1
[1,2,4,7,3,6,5] => [1,2,6,7,4,3,5] => [5,3,4,7,6,2,1] => [3,1,2,7,6,5,4] => ? = 1
[1,2,4,7,5,3,6] => [1,2,5,7,4,3,6] => [6,3,4,7,5,2,1] => [3,1,2,7,6,5,4] => ? = 1
[1,2,4,7,6,3,5] => [1,2,6,4,3,7,5] => [5,7,3,4,6,2,1] => [3,7,1,2,6,5,4] => ? = 1
[1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [7,6,2,1,5,4,3] => ? = 0
[1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [3,7,2,1,6,5,4] => ? = 0
[1,2,5,3,6,4,7] => [1,2,6,5,3,4,7] => [7,4,3,5,6,2,1] => [7,2,1,3,6,5,4] => ? = 1
[1,2,5,3,6,7,4] => [1,2,7,6,5,3,4] => [4,3,5,6,7,2,1] => [2,1,3,4,7,6,5] => ? = 2
[1,2,5,3,7,4,6] => [1,2,7,5,3,4,6] => [6,4,3,5,7,2,1] => [4,2,1,3,7,6,5] => ? = 1
[1,2,5,3,7,6,4] => [1,2,6,7,5,3,4] => [4,3,5,7,6,2,1] => [2,1,3,7,6,5,4] => ? = 1
[1,2,5,4,3,6,7] => [1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [7,6,1,5,4,3,2] => ? = 0
[1,2,5,4,3,7,6] => [1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [2,7,1,6,5,4,3] => ? = 0
[1,2,5,4,6,3,7] => [1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => [7,1,2,6,5,4,3] => ? = 1
[1,2,5,4,7,3,6] => [1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [3,1,2,7,6,5,4] => ? = 1
[1,2,5,6,3,4,7] => [1,2,6,3,5,4,7] => [7,4,5,3,6,2,1] => [7,1,3,2,6,5,4] => ? = 0
[1,2,5,6,4,3,7] => [1,2,6,4,5,3,7] => [7,3,5,4,6,2,1] => [7,1,3,2,6,5,4] => ? = 0
[1,2,5,7,3,4,6] => [1,2,7,3,5,4,6] => [6,4,5,3,7,2,1] => [4,1,3,2,7,6,5] => ? = 0
[1,2,5,7,4,3,6] => [1,2,7,4,5,3,6] => [6,3,5,4,7,2,1] => [4,1,3,2,7,6,5] => ? = 0
[1,2,6,3,4,5,7] => [1,2,6,3,4,5,7] => [7,5,4,3,6,2,1] => [7,3,2,1,6,5,4] => ? = 0
Description
The number of double ascents of a permutation.
A double ascent of a permutation $\pi$ is a position $i$ such that $\pi(i) < \pi(i+1) < \pi(i+2)$.
Matching statistic: St000317
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000317: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 50%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000317: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 50%
Values
[1] => {{1}}
=> [1] => [1] => 0
[1,2] => {{1},{2}}
=> [1,2] => [1,2] => 0
[2,1] => {{1,2}}
=> [2,1] => [2,1] => 0
[1,2,3] => {{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
[1,3,2] => {{1},{2,3}}
=> [1,3,2] => [1,3,2] => 0
[2,1,3] => {{1,2},{3}}
=> [2,1,3] => [2,1,3] => 0
[2,3,1] => {{1,2,3}}
=> [2,3,1] => [3,1,2] => 1
[3,1,2] => {{1,3},{2}}
=> [3,2,1] => [3,2,1] => 0
[3,2,1] => {{1,3},{2}}
=> [3,2,1] => [3,2,1] => 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}}
=> [1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => 1
[1,4,2,3] => {{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => 0
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => 0
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => {{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => {{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => 1
[2,3,4,1] => {{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => 2
[2,4,1,3] => {{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => 1
[2,4,3,1] => {{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => 1
[3,1,2,4] => {{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => 0
[3,1,4,2] => {{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => 1
[3,2,1,4] => {{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => 0
[3,2,4,1] => {{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => 1
[3,4,1,2] => {{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => 0
[3,4,2,1] => {{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => 0
[4,1,2,3] => {{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => 0
[4,1,3,2] => {{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => 0
[4,2,1,3] => {{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => 0
[4,2,3,1] => {{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => 0
[4,3,1,2] => {{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => 0
[4,3,2,1] => {{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => 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}}
=> [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => 2
[1,3,5,2,4] => {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,5,2,4,3] => 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,5,2,4,3] => 1
[1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => 0
[1,4,2,5,3] => {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,5,3,2,4] => 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => 0
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,5,3,2,4] => 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,5,2,3] => 0
[1,2,3,4,5,6,7] => {{1},{2},{3},{4},{5},{6},{7}}
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,3,4,5,7,6] => {{1},{2},{3},{4},{5},{6,7}}
=> [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => ? = 0
[1,2,3,4,6,5,7] => {{1},{2},{3},{4},{5,6},{7}}
=> [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => ? = 0
[1,2,3,4,6,7,5] => {{1},{2},{3},{4},{5,6,7}}
=> [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => ? = 1
[1,2,3,4,7,5,6] => {{1},{2},{3},{4},{5,7},{6}}
=> [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => ? = 0
[1,2,3,4,7,6,5] => {{1},{2},{3},{4},{5,7},{6}}
=> [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => ? = 0
[1,2,3,5,4,6,7] => {{1},{2},{3},{4,5},{6},{7}}
=> [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 0
[1,2,3,5,4,7,6] => {{1},{2},{3},{4,5},{6,7}}
=> [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 0
[1,2,3,5,6,4,7] => {{1},{2},{3},{4,5,6},{7}}
=> [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => ? = 1
[1,2,3,5,6,7,4] => {{1},{2},{3},{4,5,6,7}}
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => ? = 2
[1,2,3,5,7,4,6] => {{1},{2},{3},{4,5,7},{6}}
=> [1,2,3,5,7,6,4] => [1,2,3,7,4,6,5] => ? = 1
[1,2,3,5,7,6,4] => {{1},{2},{3},{4,5,7},{6}}
=> [1,2,3,5,7,6,4] => [1,2,3,7,4,6,5] => ? = 1
[1,2,3,6,4,5,7] => {{1},{2},{3},{4,6},{5},{7}}
=> [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ? = 0
[1,2,3,6,4,7,5] => {{1},{2},{3},{4,6,7},{5}}
=> [1,2,3,6,5,7,4] => [1,2,3,7,5,4,6] => ? = 1
[1,2,3,6,5,4,7] => {{1},{2},{3},{4,6},{5},{7}}
=> [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ? = 0
[1,2,3,6,5,7,4] => {{1},{2},{3},{4,6,7},{5}}
=> [1,2,3,6,5,7,4] => [1,2,3,7,5,4,6] => ? = 1
[1,2,3,6,7,4,5] => {{1},{2},{3},{4,6},{5,7}}
=> [1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => ? = 0
[1,2,3,6,7,5,4] => {{1},{2},{3},{4,6},{5,7}}
=> [1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => ? = 0
[1,2,3,7,4,5,6] => {{1},{2},{3},{4,7},{5},{6}}
=> [1,2,3,7,5,6,4] => [1,2,3,7,5,6,4] => ? = 0
[1,2,3,7,4,6,5] => {{1},{2},{3},{4,7},{5},{6}}
=> [1,2,3,7,5,6,4] => [1,2,3,7,5,6,4] => ? = 0
[1,2,3,7,5,4,6] => {{1},{2},{3},{4,7},{5},{6}}
=> [1,2,3,7,5,6,4] => [1,2,3,7,5,6,4] => ? = 0
[1,2,3,7,5,6,4] => {{1},{2},{3},{4,7},{5},{6}}
=> [1,2,3,7,5,6,4] => [1,2,3,7,5,6,4] => ? = 0
[1,2,3,7,6,4,5] => {{1},{2},{3},{4,7},{5,6}}
=> [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 0
[1,2,3,7,6,5,4] => {{1},{2},{3},{4,7},{5,6}}
=> [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 0
[1,2,4,3,5,6,7] => {{1},{2},{3,4},{5},{6},{7}}
=> [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 0
[1,2,4,3,5,7,6] => {{1},{2},{3,4},{5},{6,7}}
=> [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ? = 0
[1,2,4,3,6,5,7] => {{1},{2},{3,4},{5,6},{7}}
=> [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ? = 0
[1,2,4,3,6,7,5] => {{1},{2},{3,4},{5,6,7}}
=> [1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => ? = 1
[1,2,4,3,7,5,6] => {{1},{2},{3,4},{5,7},{6}}
=> [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => ? = 0
[1,2,4,3,7,6,5] => {{1},{2},{3,4},{5,7},{6}}
=> [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => ? = 0
[1,2,4,5,3,6,7] => {{1},{2},{3,4,5},{6},{7}}
=> [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ? = 1
[1,2,4,5,3,7,6] => {{1},{2},{3,4,5},{6,7}}
=> [1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => ? = 1
[1,2,4,5,6,3,7] => {{1},{2},{3,4,5,6},{7}}
=> [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => ? = 2
[1,2,4,5,6,7,3] => {{1},{2},{3,4,5,6,7}}
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 3
[1,2,4,5,7,3,6] => {{1},{2},{3,4,5,7},{6}}
=> [1,2,4,5,7,6,3] => [1,2,7,3,4,6,5] => ? = 2
[1,2,4,5,7,6,3] => {{1},{2},{3,4,5,7},{6}}
=> [1,2,4,5,7,6,3] => [1,2,7,3,4,6,5] => ? = 2
[1,2,4,6,3,5,7] => {{1},{2},{3,4,6},{5},{7}}
=> [1,2,4,6,5,3,7] => [1,2,6,3,5,4,7] => ? = 1
[1,2,4,6,3,7,5] => {{1},{2},{3,4,6,7},{5}}
=> [1,2,4,6,5,7,3] => [1,2,7,3,5,4,6] => ? = 2
[1,2,4,6,5,3,7] => {{1},{2},{3,4,6},{5},{7}}
=> [1,2,4,6,5,3,7] => [1,2,6,3,5,4,7] => ? = 1
[1,2,4,6,5,7,3] => {{1},{2},{3,4,6,7},{5}}
=> [1,2,4,6,5,7,3] => [1,2,7,3,5,4,6] => ? = 2
[1,2,4,6,7,3,5] => {{1},{2},{3,4,6},{5,7}}
=> [1,2,4,6,7,3,5] => [1,2,6,3,7,4,5] => ? = 1
[1,2,4,6,7,5,3] => {{1},{2},{3,4,6},{5,7}}
=> [1,2,4,6,7,3,5] => [1,2,6,3,7,4,5] => ? = 1
[1,2,4,7,3,5,6] => {{1},{2},{3,4,7},{5},{6}}
=> [1,2,4,7,5,6,3] => [1,2,7,3,5,6,4] => ? = 1
[1,2,4,7,3,6,5] => {{1},{2},{3,4,7},{5},{6}}
=> [1,2,4,7,5,6,3] => [1,2,7,3,5,6,4] => ? = 1
[1,2,4,7,5,3,6] => {{1},{2},{3,4,7},{5},{6}}
=> [1,2,4,7,5,6,3] => [1,2,7,3,5,6,4] => ? = 1
[1,2,4,7,5,6,3] => {{1},{2},{3,4,7},{5},{6}}
=> [1,2,4,7,5,6,3] => [1,2,7,3,5,6,4] => ? = 1
[1,2,4,7,6,3,5] => {{1},{2},{3,4,7},{5,6}}
=> [1,2,4,7,6,5,3] => [1,2,7,3,6,5,4] => ? = 1
[1,2,4,7,6,5,3] => {{1},{2},{3,4,7},{5,6}}
=> [1,2,4,7,6,5,3] => [1,2,7,3,6,5,4] => ? = 1
[1,2,5,3,4,6,7] => {{1},{2},{3,5},{4},{6},{7}}
=> [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => ? = 0
[1,2,5,3,4,7,6] => {{1},{2},{3,5},{4},{6,7}}
=> [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => ? = 0
Description
The cycle descent number of a permutation.
Let $(i_1,\ldots,i_k)$ be a cycle of a permutation $\pi$ such that $i_1$ is its smallest element. A **cycle descent** of $(i_1,\ldots,i_k)$ is an $i_a$ for $1 \leq a < k$ such that $i_a > i_{a+1}$. The **cycle descent set** of $\pi$ is then the set of descents in all the cycles of $\pi$, and the **cycle descent number** is its cardinality.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001651The Frankl number of a lattice. St001845The number of join irreducibles minus the rank of a lattice. St001875The number of simple modules with projective dimension at most 1. St001862The number of crossings of a signed permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
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