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Your data matches 169 different statistics following compositions of up to 3 maps.
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Matching statistic: St000732
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
St000732: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000732: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0
[2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 0
[2,1,3] => [1,3,2] => 0
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,2,3] => 0
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,3,4,2] => 0
[1,4,2,3] => [1,4,2,3] => 1
[1,4,3,2] => [1,4,2,3] => 1
[2,1,3,4] => [1,3,4,2] => 0
[2,1,4,3] => [1,4,2,3] => 1
[2,3,1,4] => [1,4,2,3] => 1
[2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => 0
[2,4,3,1] => [1,2,4,3] => 0
[3,1,2,4] => [1,2,4,3] => 0
[3,1,4,2] => [1,4,2,3] => 1
[3,2,1,4] => [1,4,2,3] => 1
[3,2,4,1] => [1,2,4,3] => 0
[3,4,1,2] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => 0
[4,2,1,3] => [1,3,2,4] => 0
[4,2,3,1] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,4,5,3] => 0
[1,2,5,3,4] => [1,2,5,3,4] => 1
[1,2,5,4,3] => [1,2,5,3,4] => 1
[1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,3,4,2,5] => 0
[1,3,4,5,2] => [1,3,4,5,2] => 0
[1,3,5,2,4] => [1,3,5,2,4] => 1
[1,3,5,4,2] => [1,3,5,2,4] => 1
[1,4,2,3,5] => [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => 1
[1,4,3,2,5] => [1,4,2,5,3] => 1
[1,4,3,5,2] => [1,4,2,3,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => 0
[1,4,5,3,2] => [1,4,5,2,3] => 0
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: St000366
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,3] => [1,3,2] => [1,3,2] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 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] => [1,3,4,2] => [1,4,2,3] => 0
[1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 1
[1,4,3,2] => [1,4,2,3] => [1,4,3,2] => 1
[2,1,3,4] => [1,3,4,2] => [1,4,2,3] => 0
[2,1,4,3] => [1,4,2,3] => [1,4,3,2] => 1
[2,3,1,4] => [1,4,2,3] => [1,4,3,2] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 0
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 0
[3,1,4,2] => [1,4,2,3] => [1,4,3,2] => 1
[3,2,1,4] => [1,4,2,3] => [1,4,3,2] => 1
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => 0
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 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] => [1,2,4,5,3] => [1,2,5,3,4] => 0
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,4,3] => 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => 1
[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] => [1,3,4,2,5] => [1,4,2,3,5] => 0
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => 0
[1,3,5,2,4] => [1,3,5,2,4] => [1,5,4,2,3] => 1
[1,3,5,4,2] => [1,3,5,2,4] => [1,5,4,2,3] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [1,5,3,2,4] => 1
[1,4,3,2,5] => [1,4,2,5,3] => [1,5,3,2,4] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,3,2,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => 0
[1,4,5,3,2] => [1,4,5,2,3] => [1,4,2,5,3] => 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: St000731
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,3] => [1,3,2] => [1,3,2] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 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] => [1,3,4,2] => [1,4,2,3] => 0
[1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 1
[1,4,3,2] => [1,4,2,3] => [1,3,4,2] => 1
[2,1,3,4] => [1,3,4,2] => [1,4,2,3] => 0
[2,1,4,3] => [1,4,2,3] => [1,3,4,2] => 1
[2,3,1,4] => [1,4,2,3] => [1,3,4,2] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 0
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 0
[3,1,4,2] => [1,4,2,3] => [1,3,4,2] => 1
[3,2,1,4] => [1,4,2,3] => [1,3,4,2] => 1
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => 0
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 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] => [1,2,4,5,3] => [1,2,5,3,4] => 0
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[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] => [1,3,4,2,5] => [1,4,2,3,5] => 0
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => 0
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => 1
[1,3,5,4,2] => [1,3,5,2,4] => [1,4,2,5,3] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [1,3,5,2,4] => 1
[1,4,3,2,5] => [1,4,2,5,3] => [1,3,5,2,4] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 0
[1,4,5,3,2] => [1,4,5,2,3] => [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: St000365
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000365: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000365: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [2,3,1] => 0
[2,1,3] => [1,3,2] => [1,3,2] => [2,3,1] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[1,3,4,2] => [1,3,4,2] => [1,4,2,3] => [3,2,4,1] => 0
[1,4,2,3] => [1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 1
[1,4,3,2] => [1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 1
[2,1,3,4] => [1,3,4,2] => [1,4,2,3] => [3,2,4,1] => 0
[2,1,4,3] => [1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 1
[2,3,1,4] => [1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 0
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 0
[3,1,4,2] => [1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 1
[3,2,1,4] => [1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 1
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => 0
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => [4,3,5,2,1] => 0
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => 0
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => [5,3,2,4,1] => 0
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => [4,3,2,5,1] => 0
[1,3,5,2,4] => [1,3,5,2,4] => [1,5,4,2,3] => [3,2,4,5,1] => 1
[1,3,5,4,2] => [1,3,5,2,4] => [1,5,4,2,3] => [3,2,4,5,1] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => [5,2,3,4,1] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [1,5,3,2,4] => [4,2,3,5,1] => 1
[1,4,3,2,5] => [1,4,2,5,3] => [1,5,3,2,4] => [4,2,3,5,1] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,3,2,5] => [5,2,3,4,1] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => [3,5,2,4,1] => 0
[1,4,5,3,2] => [1,4,5,2,3] => [1,4,2,5,3] => [3,5,2,4,1] => 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: St001384
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001384: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001384: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[2,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[1,2,3] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0}
[1,3,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0}
[2,1,3] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0}
[2,3,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0}
[3,1,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0}
[3,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,2,3,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1}
[1,2,4,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[1,3,2,4] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1}
[1,3,4,2] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1}
[1,4,2,3] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1}
[1,4,3,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[2,1,3,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1}
[2,1,4,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[2,3,1,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1}
[2,3,4,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1}
[2,4,1,3] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1}
[2,4,3,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[3,1,2,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1}
[3,1,4,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,2,1,4] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,2,4,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,4,1,2] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1}
[3,4,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[4,1,2,3] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1}
[4,1,3,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[4,2,1,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[4,2,3,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[4,3,1,2] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1}
[4,3,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[1,2,3,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,2,3,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,2,4,3,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,2,4,5,3] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,2,5,3,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,2,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,3,2,4,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,3,2,5,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,3,4,2,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,3,4,5,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,3,5,2,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,3,5,4,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[1,4,2,3,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,4,2,5,3] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,4,3,2,5] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,4,3,5,2] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,4,5,2,3] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,4,5,3,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[1,5,2,3,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,5,2,4,3] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[1,5,3,2,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,5,3,4,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[1,5,4,2,3] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,5,4,3,2] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[2,1,3,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,1,3,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[2,1,4,3,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,1,4,5,3] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,1,5,3,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,1,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[2,3,1,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,3,1,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[2,3,4,1,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,3,4,5,1] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,3,5,1,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,3,5,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 2
[2,4,1,3,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,4,1,5,3] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[2,4,3,1,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[2,4,3,5,1] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[2,4,5,1,3] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,4,5,3,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 2
[2,5,1,3,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,5,1,4,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 2
[2,5,3,1,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[2,5,3,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 2
[2,5,4,1,3] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,5,4,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[3,1,2,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,1,2,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[3,1,4,2,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,4,5,2] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,5,2,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,5,4,2] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[3,2,1,4,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,4,1,2,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,4,5,1,2] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,5,1,2,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,1,2,3,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,3,1,2,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,3,5,1,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,5,1,2,3] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[5,1,2,3,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[5,3,1,2,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[5,3,4,1,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[5,4,1,2,3] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,2,3,4,5,6] => [6] => [[6],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,2,4,5,6] => [1,5] => [[5,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
Description
The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains.
Matching statistic: St000741
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => ([],2)
=> 1 = 0 + 1
[2,1] => [1,2] => [1,2] => ([],2)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> 1 = 0 + 1
[1,3,2] => [1,2,3] => [1,2,3] => ([],3)
=> 1 = 0 + 1
[2,1,3] => [1,2,3] => [1,2,3] => ([],3)
=> 1 = 0 + 1
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
=> 1 = 0 + 1
[3,1,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0} + 1
[3,2,1] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0} + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1} + 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1} + 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1} + 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1} + 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1} + 1
[3,1,4,2] => [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1} + 1
[3,2,4,1] => [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1} + 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1} + 1
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,1,3,2] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,2,3,1] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,3,1,2] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[1,5,2,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,5,2,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,5,3,2,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,5,3,4,2] => [1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,5,4,2,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[2,1,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[2,4,1,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[2,4,5,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[2,5,1,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,5,1,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[3,1,2,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[3,1,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[3,1,5,2,4] => [1,3,5,4,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[3,2,4,5,1] => [1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[3,2,5,1,4] => [1,3,5,4,2] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[3,4,1,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[3,4,1,5,2] => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[3,4,2,1,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[3,4,2,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[3,5,1,2,4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[3,5,1,4,2] => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[3,5,2,1,4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[3,5,2,4,1] => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[5,1,4,2,3] => [1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[5,2,4,1,3] => [1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2} + 1
[1,2,3,6,4,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,2,3,6,5,4] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,2,4,6,3,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,2,4,6,5,3] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,2,5,3,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
[1,2,5,4,3,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3} + 1
Description
The Colin de Verdière graph invariant.
Matching statistic: St000454
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => ([],2)
=> 0
[2,1] => [1,2] => [1,2] => ([],2)
=> 0
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> 0
[1,3,2] => [1,3,2] => [1,2,3] => ([],3)
=> 0
[2,1,3] => [1,3,2] => [1,2,3] => ([],3)
=> 0
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
=> 0
[3,1,2] => [1,2,3] => [1,2,3] => ([],3)
=> 0
[3,2,1] => [1,2,3] => [1,2,3] => ([],3)
=> 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[1,2,4,3] => [1,2,4,3] => [1,2,3,4] => ([],4)
=> 0
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => ([],4)
=> 0
[1,3,4,2] => [1,3,4,2] => [1,2,3,4] => ([],4)
=> 0
[1,4,2,3] => [1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[1,4,3,2] => [1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[2,1,3,4] => [1,3,4,2] => [1,2,3,4] => ([],4)
=> 0
[2,1,4,3] => [1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[2,3,1,4] => [1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[2,4,1,3] => [1,3,2,4] => [1,2,3,4] => ([],4)
=> 0
[2,4,3,1] => [1,2,4,3] => [1,2,3,4] => ([],4)
=> 0
[3,1,2,4] => [1,2,4,3] => [1,2,3,4] => ([],4)
=> 0
[3,1,4,2] => [1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[3,2,1,4] => [1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[3,2,4,1] => [1,2,4,3] => [1,2,3,4] => ([],4)
=> 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[4,1,3,2] => [1,3,2,4] => [1,2,3,4] => ([],4)
=> 0
[4,2,1,3] => [1,3,2,4] => [1,2,3,4] => ([],4)
=> 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> 0
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> 0
[1,3,4,2,5] => [1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> 0
[1,3,4,5,2] => [1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> 0
[1,3,5,2,4] => [1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,3,5,4,2] => [1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,4,2,5,3] => [1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1}
[1,4,3,2,5] => [1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1}
[1,4,3,5,2] => [1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,4,5,3,2] => [1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,5,2,4,3] => [1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1}
[1,5,3,2,4] => [1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1}
[1,5,3,4,2] => [1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[2,4,1,5,3] => [1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1}
[2,4,3,1,5] => [1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1}
[2,5,1,4,3] => [1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1}
[2,5,3,1,4] => [1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1}
[3,1,4,2,5] => [1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1}
[3,1,5,2,4] => [1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1}
[3,2,4,1,5] => [1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1}
[3,2,5,1,4] => [1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1}
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,6,2,5,4] => [1,3,6,2,5,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,6,4,2,5] => [1,3,6,2,5,4] => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,6,3] => [1,4,2,5,6,3] => [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [1,2,4,6,5,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [1,2,4,6,5,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5,6] => [1,4,2,5,6,3] => [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [1,2,4,6,5,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [1,2,4,6,5,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,6,2,5] => [1,4,2,5,3,6] => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,6,2,3] => [1,4,5,6,2,3] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,6,3,2] => [1,4,5,6,2,3] => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,3,6,4] => [1,5,2,3,6,4] => [1,2,5,6,4,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,6,4,2] => [1,5,2,3,6,4] => [1,2,5,6,4,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,2,3,6] => [1,5,2,3,6,4] => [1,2,5,6,4,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,6,2] => [1,5,2,3,6,4] => [1,2,5,6,4,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,6,2,3,4] => [1,5,6,2,3,4] => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,6,3,4,2] => [1,5,6,2,3,4] => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,6,4,2,3] => [1,5,6,2,3,4] => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,6,4,3,2] => [1,5,6,2,3,4] => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,6,2,3,5,4] => [1,6,2,3,5,4] => [1,2,6,4,3,5] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [1,2,6,5,3,4] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,6,2,4,5,3] => [1,6,2,4,5,3] => [1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [1,2,6,4,5,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000567
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000567: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000567: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0}
[2,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[2,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[3,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[3,2,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0}
[1,2,3,4] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,2,4,3] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,3,2,4] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,3,4,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,4,2,3] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,4,3,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[2,1,3,4] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,3,1,4] => [2,4,1,3] => [2,1,1]
=> [1,1]
=> 1
[2,3,4,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[2,4,1,3] => [2,4,1,3] => [2,1,1]
=> [1,1]
=> 1
[2,4,3,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[3,1,2,4] => [3,1,4,2] => [2,2]
=> [2]
=> 0
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 0
[3,2,1,4] => [3,2,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[3,2,4,1] => [3,2,4,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[3,4,1,2] => [3,4,1,2] => [2,1,1]
=> [1,1]
=> 1
[3,4,2,1] => [3,4,2,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[4,1,2,3] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[4,1,3,2] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[4,2,1,3] => [4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[4,2,3,1] => [4,2,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[4,3,1,2] => [4,3,1,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[4,3,2,1] => [4,3,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,2,3,4,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,3,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,4,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,5,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,2,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,3,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,2,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,2,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,3,4,5] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,1,3,5,4] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,1,4,3,5] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,1,4,5,3] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,1,5,3,4] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,1,5,4,3] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,3,1,4,5] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,3,1,5,4] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,3,4,1,5] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,3,4,5,1] => [2,5,4,3,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,1,3,5] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,1,5,3] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,3,1,5] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,5,1,3] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,1,3,4] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,1,4,3] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,3,1,4] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,4,1,3] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[3,1,2,4,5] => [3,1,5,4,2] => [3,2]
=> [2]
=> 0
[3,1,2,5,4] => [3,1,5,4,2] => [3,2]
=> [2]
=> 0
[3,1,4,2,5] => [3,1,5,4,2] => [3,2]
=> [2]
=> 0
[3,1,4,5,2] => [3,1,5,4,2] => [3,2]
=> [2]
=> 0
[3,1,5,2,4] => [3,1,5,4,2] => [3,2]
=> [2]
=> 0
[3,1,5,4,2] => [3,1,5,4,2] => [3,2]
=> [2]
=> 0
[3,2,1,4,5] => [3,2,1,5,4] => [3,2]
=> [2]
=> 0
[3,2,1,5,4] => [3,2,1,5,4] => [3,2]
=> [2]
=> 0
[3,2,4,1,5] => [3,2,5,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,2,4,5,1] => [3,2,5,4,1] => [3,2]
=> [2]
=> 0
[3,2,5,1,4] => [3,2,5,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,2,5,4,1] => [3,2,5,4,1] => [3,2]
=> [2]
=> 0
[3,4,1,2,5] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,4,1,5,2] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,4,2,1,5] => [3,5,2,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,4,2,5,1] => [3,5,2,4,1] => [3,1,1]
=> [1,1]
=> 1
[3,4,5,1,2] => [3,5,4,1,2] => [3,1,1]
=> [1,1]
=> 1
[3,5,1,2,4] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,5,1,4,2] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,5,2,1,4] => [3,5,2,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,5,2,4,1] => [3,5,2,4,1] => [3,1,1]
=> [1,1]
=> 1
[3,5,4,1,2] => [3,5,4,1,2] => [3,1,1]
=> [1,1]
=> 1
[4,1,2,3,5] => [4,1,5,3,2] => [3,2]
=> [2]
=> 0
[4,1,2,5,3] => [4,1,5,3,2] => [3,2]
=> [2]
=> 0
[4,1,3,2,5] => [4,1,5,3,2] => [3,2]
=> [2]
=> 0
Description
The sum of the products of all pairs of parts.
This is the evaluation of the second elementary symmetric polynomial which is equal to
$$e_2(\lambda) = \binom{n+1}{2} - \sum_{i=1}^\ell\binom{\lambda_i+1}{2}$$
for a partition $\lambda = (\lambda_1,\dots,\lambda_\ell) \vdash n$, see [1].
This is the maximal number of inversions a permutation with the given shape can have, see [2, cor.2.4].
Matching statistic: St001099
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001099: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001099: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0}
[2,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[2,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[3,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[3,2,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0}
[1,2,3,4] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,2,4,3] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,3,2,4] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,3,4,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,4,2,3] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,4,3,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[2,1,3,4] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,3,1,4] => [2,4,1,3] => [2,1,1]
=> [1,1]
=> 1
[2,3,4,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[2,4,1,3] => [2,4,1,3] => [2,1,1]
=> [1,1]
=> 1
[2,4,3,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[3,1,2,4] => [3,1,4,2] => [2,2]
=> [2]
=> 0
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 0
[3,2,1,4] => [3,2,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[3,2,4,1] => [3,2,4,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[3,4,1,2] => [3,4,1,2] => [2,1,1]
=> [1,1]
=> 1
[3,4,2,1] => [3,4,2,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[4,1,2,3] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[4,1,3,2] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[4,2,1,3] => [4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[4,2,3,1] => [4,2,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[4,3,1,2] => [4,3,1,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[4,3,2,1] => [4,3,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1}
[1,2,3,4,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,3,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,4,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,5,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,2,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,3,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,2,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,2,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,3,4,5] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,1,3,5,4] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,1,4,3,5] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,1,4,5,3] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,1,5,3,4] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,1,5,4,3] => [2,1,5,4,3] => [3,2]
=> [2]
=> 0
[2,3,1,4,5] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,3,1,5,4] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,3,4,1,5] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,3,4,5,1] => [2,5,4,3,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,1,3,5] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,1,5,3] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,3,1,5] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,5,1,3] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,1,3,4] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,1,4,3] => [2,5,1,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,3,1,4] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,4,1,3] => [2,5,4,1,3] => [3,1,1]
=> [1,1]
=> 1
[3,1,2,4,5] => [3,1,5,4,2] => [3,2]
=> [2]
=> 0
[3,1,2,5,4] => [3,1,5,4,2] => [3,2]
=> [2]
=> 0
[3,1,4,2,5] => [3,1,5,4,2] => [3,2]
=> [2]
=> 0
[3,1,4,5,2] => [3,1,5,4,2] => [3,2]
=> [2]
=> 0
[3,1,5,2,4] => [3,1,5,4,2] => [3,2]
=> [2]
=> 0
[3,1,5,4,2] => [3,1,5,4,2] => [3,2]
=> [2]
=> 0
[3,2,1,4,5] => [3,2,1,5,4] => [3,2]
=> [2]
=> 0
[3,2,1,5,4] => [3,2,1,5,4] => [3,2]
=> [2]
=> 0
[3,2,4,1,5] => [3,2,5,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,2,4,5,1] => [3,2,5,4,1] => [3,2]
=> [2]
=> 0
[3,2,5,1,4] => [3,2,5,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,2,5,4,1] => [3,2,5,4,1] => [3,2]
=> [2]
=> 0
[3,4,1,2,5] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,4,1,5,2] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,4,2,1,5] => [3,5,2,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,4,2,5,1] => [3,5,2,4,1] => [3,1,1]
=> [1,1]
=> 1
[3,4,5,1,2] => [3,5,4,1,2] => [3,1,1]
=> [1,1]
=> 1
[3,5,1,2,4] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,5,1,4,2] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,5,2,1,4] => [3,5,2,1,4] => [3,1,1]
=> [1,1]
=> 1
[3,5,2,4,1] => [3,5,2,4,1] => [3,1,1]
=> [1,1]
=> 1
[3,5,4,1,2] => [3,5,4,1,2] => [3,1,1]
=> [1,1]
=> 1
[4,1,2,3,5] => [4,1,5,3,2] => [3,2]
=> [2]
=> 0
[4,1,2,5,3] => [4,1,5,3,2] => [3,2]
=> [2]
=> 0
[4,1,3,2,5] => [4,1,5,3,2] => [3,2]
=> [2]
=> 0
Description
The 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.
For a generating function $f$ the associated formal group law is the symmetric function $f(f^{(-1)}(x_1) + f^{(-1)}(x_2), \dots)$, see [1].
This statistic records the coefficient of the monomial symmetric function $m_\lambda$ times the product of the factorials of the parts of $\lambda$ in the formal group law for leaf labelled binary trees, with generating function $f(x) = 1-\sqrt{1-2x}$, see [1, sec. 3.2]
Fix a set of distinguishable vertices and a coloring of the vertices so that $\lambda_i$ are colored $i$. This statistic gives the number of rooted binary trees with leaves labeled with this set of vertices and internal vertices unlabeled so that no pair of 'twin' leaves have the same color.
Matching statistic: St000934
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000934: Integer partitions ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 75%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000934: Integer partitions ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 75%
Values
[1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0}
[2,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[2,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0}
[3,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0}
[3,2,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[1,2,3,4] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,2,4,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,4] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,4,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,3,4] => [2,1,4,3] => [2,2]
=> [2]
=> 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 1
[2,3,1,4] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1}
[2,3,4,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1}
[2,4,1,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1}
[2,4,3,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1}
[3,1,2,4] => [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1}
[3,1,4,2] => [3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1}
[3,2,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,2,4,1] => [3,2,4,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1}
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 1
[3,4,2,1] => [3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1}
[4,1,2,3] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1}
[4,1,3,2] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1}
[4,2,1,3] => [4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1}
[4,2,3,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,3,1,2] => [4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1}
[4,3,2,1] => [4,3,2,1] => [2,2]
=> [2]
=> 1
[1,2,3,4,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,2,3,5,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,2,4,3,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,2,4,5,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,2,5,3,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,2,5,4,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,3,2,4,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,3,2,5,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,3,4,2,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,3,4,5,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,3,5,2,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,3,5,4,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,4,2,3,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,4,2,5,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,4,3,2,5] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,4,3,5,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,4,5,2,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,4,5,3,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,5,2,3,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,5,2,4,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,5,3,2,4] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,5,3,4,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,5,4,2,3] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[1,5,4,3,2] => [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 0
[2,1,3,4,5] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,1,3,5,4] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,1,4,3,5] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,1,4,5,3] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,1,5,3,4] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,1,5,4,3] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,3,1,4,5] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,1,5,4] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,1,5] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,5,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 1
[2,3,5,1,4] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,4,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 1
[2,4,1,3,5] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,1,5] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,5,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 1
[2,4,5,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 1
[2,5,1,3,4] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,4,3] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,1,4] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,4,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 1
[2,5,4,1,3] => [2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,3,1] => [2,5,4,3,1] => [3,2]
=> [2]
=> 1
[3,1,2,4,5] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,2,5,4] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,2,5] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,5,2] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,5,2,4] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,5,4,2] => [3,1,5,4,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,4,5] => [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 0
[3,2,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 0
[3,2,4,1,5] => [3,2,5,1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,5,1,4] => [3,2,5,1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,2,1,5] => [3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,2,5,1] => [3,5,2,4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,4,5,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,2,1,4] => [3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,2,4,1] => [3,5,2,4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,5,4,2,1] => [3,5,4,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,2,3,5] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,2,5,3] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,3,2,5] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[4,1,3,5,2] => [4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The 2-degree of an integer partition.
For an integer partition $\lambda$, this is given by the exponent of 2 in the Gram determinant of the integal Specht module of the symmetric group indexed by $\lambda$.
The following 159 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001280The number of parts of an integer partition that are at least two. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. 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. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. 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. St000941The number of characters of the symmetric group whose value on the partition is even. St001964The interval resolution global dimension of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000929The constant term of the character polynomial of an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001651The Frankl number of a lattice. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000455The second largest eigenvalue of a graph if it is integral. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000938The number of zeros of the symmetric group character corresponding to the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. 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. St000284The Plancherel distribution on integer partitions. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001128The exponens consonantiae of a partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000225Difference between largest and smallest parts in a partition. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. 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 $K[x]/(x^n)$. 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. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001866The nesting alignments of a signed permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000260The radius of a connected graph. St000936The number of even values of the symmetric group character corresponding to the partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001846The number of elements which do not have a complement in the lattice. St001820The size of the image of the pop stack sorting operator. St001862The number of crossings of a signed permutation. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001561The value of the elementary symmetric function evaluated at 1. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000068The number of minimal elements in a poset. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001625The Möbius invariant of a lattice. St001621The number of atoms of a lattice. St001845The number of join irreducibles minus the rank of a lattice. St001095The number of non-isomorphic posets with precisely one further covering relation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001864The number of excedances of a signed permutation. St000022The number of fixed points of a permutation. St000259The diameter 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. St001330The hat guessing number of a graph. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000632The jump number of the poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. 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 $K[x]/(x^n)$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001821The sorting index of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001768The number of reduced words of a signed permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000679The pruning number of an ordered tree. St001624The breadth of a lattice. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. 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. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St000449The number of pairs of vertices of a graph with distance 4. 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. St001578The minimal number of edges to add or remove to make a graph a line graph. 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. St001871The number of triconnected components of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given 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. St001271The competition number of a graph.
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