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Your data matches 157 different statistics following compositions of up to 3 maps.
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Matching statistic: St000709
Mp00223: Permutations —runsort⟶ Permutations
St000709: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000709: 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] => 0
[1,3,5,4,2] => [1,3,5,2,4] => 0
[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] => 1
[1,4,5,3,2] => [1,4,5,2,3] => 1
Description
The number of occurrences of 14-2-3 or 14-3-2.
The number of permutations avoiding both of these patterns is the case $k=2$ of the third item in Corollary 34 of [1].
Matching statistic: St000566
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 52%●distinct values known / distinct values provided: 29%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 52%●distinct values known / distinct values provided: 29%
Values
[1,2] => [1] => [1]
=> []
=> ? ∊ {0,0}
[2,1] => [1] => [1]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[1,3,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[2,1,3] => [2,1] => [2]
=> []
=> ? ∊ {0,0,0,0,0,0}
[2,3,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0,0,0,0,0}
[3,1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[3,2,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0,0,0,0,0}
[1,2,3,4] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,2,4,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,3,2,4] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[1,3,4,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[1,4,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,4,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[2,1,3,4] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[2,1,4,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[2,3,1,4] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[2,3,4,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[2,4,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[2,4,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[3,1,2,4] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[3,1,4,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[3,2,1,4] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[3,2,4,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[3,4,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[3,4,2,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[4,1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[4,1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[4,2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[4,2,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[4,3,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[4,3,2,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,4,2,5] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,4,5,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,5,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,2,3,5] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,2,5,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,4,5,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,5,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,5,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,5,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,5,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,3,4,5] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,1,3,5,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,1,4,3,5] => [2,1,4,3] => [2,2]
=> [2]
=> 1
[2,1,4,5,3] => [2,1,4,3] => [2,2]
=> [2]
=> 1
[2,1,5,3,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,1,5,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 1
[2,3,1,4,5] => [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,1,5,4] => [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,4,1,5] => [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,4,5,1] => [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,5,1,4] => [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,5,4,1] => [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,1,3,5] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,1,5,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,3,1,5] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,3,5,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,5,1,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,5,3,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,1,3,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,5,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 1
[2,5,3,1,4] => [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,3,4,1] => [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,2,1,4,5] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,2,1,5,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,2,5,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,4,1,2,5] => [3,4,1,2] => [2,2]
=> [2]
=> 1
[3,4,1,5,2] => [3,4,1,2] => [2,2]
=> [2]
=> 1
[3,4,5,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 1
[3,5,2,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,5,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 1
[4,2,3,1,5] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,2,3,5,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,2,5,3,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,3,2,1,5] => [4,3,2,1] => [2,2]
=> [2]
=> 1
[4,3,2,5,1] => [4,3,2,1] => [2,2]
=> [2]
=> 1
[4,3,5,2,1] => [4,3,2,1] => [2,2]
=> [2]
=> 1
[4,5,2,3,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,5,3,2,1] => [4,3,2,1] => [2,2]
=> [2]
=> 1
[5,1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[5,1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[5,1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[5,1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[5,2,1,3,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[5,2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 1
Description
The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is
$$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
Matching statistic: St000621
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 52%●distinct values known / distinct values provided: 29%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 52%●distinct values known / distinct values provided: 29%
Values
[1,2] => [1] => [1]
=> []
=> ? ∊ {0,0}
[2,1] => [1] => [1]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[1,3,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[2,1,3] => [2,1] => [2]
=> []
=> ? ∊ {0,0,0,0,0,0}
[2,3,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0,0,0,0,0}
[3,1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[3,2,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0,0,0,0,0}
[1,2,3,4] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,2,4,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,3,2,4] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[1,3,4,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[1,4,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,4,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[2,1,3,4] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[2,1,4,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[2,3,1,4] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[2,3,4,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[2,4,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[2,4,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[3,1,2,4] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[3,1,4,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[3,2,1,4] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[3,2,4,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[3,4,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[3,4,2,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[4,1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[4,1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[4,2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[4,2,3,1] => [2,3,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[4,3,1,2] => [3,1,2] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[4,3,2,1] => [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,4,2,5] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,4,5,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,5,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,2,3,5] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,2,5,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,4,5,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[1,5,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,5,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,5,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,5,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,3,4,5] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,1,3,5,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,1,4,3,5] => [2,1,4,3] => [2,2]
=> [2]
=> 1
[2,1,4,5,3] => [2,1,4,3] => [2,2]
=> [2]
=> 1
[2,1,5,3,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,1,5,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 1
[2,3,1,4,5] => [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,1,5,4] => [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,4,1,5] => [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,4,5,1] => [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,5,1,4] => [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,5,4,1] => [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,1,3,5] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,1,5,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,3,1,5] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,3,5,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,5,1,3] => [2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,5,3,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,1,3,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,5,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 1
[2,5,3,1,4] => [2,3,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,3,4,1] => [2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,2,1,4,5] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,2,1,5,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,2,5,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,4,1,2,5] => [3,4,1,2] => [2,2]
=> [2]
=> 1
[3,4,1,5,2] => [3,4,1,2] => [2,2]
=> [2]
=> 1
[3,4,5,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 1
[3,5,2,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,5,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 1
[4,2,3,1,5] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,2,3,5,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,2,5,3,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,3,2,1,5] => [4,3,2,1] => [2,2]
=> [2]
=> 1
[4,3,2,5,1] => [4,3,2,1] => [2,2]
=> [2]
=> 1
[4,3,5,2,1] => [4,3,2,1] => [2,2]
=> [2]
=> 1
[4,5,2,3,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,5,3,2,1] => [4,3,2,1] => [2,2]
=> [2]
=> 1
[5,1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[5,1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[5,1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[5,1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[5,2,1,3,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[5,2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 1
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even.
This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1].
The case of an odd minimum is [[St000620]].
Matching statistic: St000506
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000506: Integer partitions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 86%
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000506: Integer partitions ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 86%
Values
[1,2] => [1,2] => [1,2] => []
=> ? = 0
[2,1] => [2,1] => [-2,1] => [2]
=> 0
[1,2,3] => [1,2,3] => [1,2,3] => []
=> ? ∊ {0,0,0}
[1,3,2] => [1,3,2] => [1,-3,2] => [2]
=> 0
[2,1,3] => [2,1,3] => [-2,1,3] => [2]
=> 0
[2,3,1] => [2,3,1] => [3,-2,1] => [1]
=> 0
[3,1,2] => [3,1,2] => [3,1,2] => []
=> ? ∊ {0,0,0}
[3,2,1] => [3,2,1] => [-2,-3,1] => []
=> ? ∊ {0,0,0}
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => []
=> ? ∊ {0,0,0,0,0,0}
[1,2,4,3] => [1,2,4,3] => [1,2,-4,3] => [2]
=> 0
[1,3,2,4] => [1,3,2,4] => [1,-3,2,4] => [2]
=> 0
[1,3,4,2] => [1,3,4,2] => [1,4,-3,2] => [1]
=> 0
[1,4,2,3] => [1,4,2,3] => [-4,1,2,3] => [4]
=> 0
[1,4,3,2] => [1,4,3,2] => [1,-3,-4,2] => []
=> ? ∊ {0,0,0,0,0,0}
[2,1,3,4] => [2,1,3,4] => [-2,1,3,4] => [2]
=> 0
[2,1,4,3] => [2,1,4,3] => [-2,1,-4,3] => [2,2]
=> 1
[2,3,1,4] => [2,3,1,4] => [3,-2,1,4] => [1]
=> 0
[2,3,4,1] => [2,3,4,1] => [3,4,-2,1] => [4]
=> 0
[2,4,1,3] => [2,4,1,3] => [-2,4,1,3] => [4]
=> 0
[2,4,3,1] => [2,4,3,1] => [3,-2,-4,1] => [3,1]
=> 1
[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => []
=> ? ∊ {0,0,0,0,0,0}
[3,1,4,2] => [3,1,4,2] => [-4,1,-3,2] => [3,1]
=> 1
[3,2,1,4] => [3,2,1,4] => [-2,-3,1,4] => []
=> ? ∊ {0,0,0,0,0,0}
[3,2,4,1] => [3,2,4,1] => [-2,4,-3,1] => [3,1]
=> 1
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => []
=> ? ∊ {0,0,0,0,0,0}
[3,4,2,1] => [3,4,2,1] => [2,-4,-3,1] => [3,1]
=> 1
[4,1,2,3] => [4,1,2,3] => [1,4,2,3] => []
=> ? ∊ {0,0,0,0,0,0}
[4,1,3,2] => [4,1,3,2] => [3,1,-4,2] => [4]
=> 0
[4,2,1,3] => [4,2,1,3] => [-4,-2,1,3] => [3,1]
=> 1
[4,2,3,1] => [4,2,3,1] => [2,3,-4,1] => [4]
=> 0
[4,3,1,2] => [4,3,1,2] => [-4,3,1,2] => [4]
=> 0
[4,3,2,1] => [4,3,2,1] => [-2,-3,-4,1] => [4]
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,-5,4] => [2]
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,-4,3,5] => [2]
=> 0
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,-4,3] => [1]
=> 0
[1,2,5,3,4] => [1,2,5,3,4] => [1,-5,2,3,4] => [4]
=> 0
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,-4,-5,3] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,2,4,5] => [1,3,2,4,5] => [1,-3,2,4,5] => [2]
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,-3,2,-5,4] => [2,2]
=> 1
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,-3,2,5] => [1]
=> 0
[1,3,4,5,2] => [1,3,4,5,2] => [1,4,5,-3,2] => [4]
=> 0
[1,3,5,2,4] => [1,3,5,2,4] => [1,-3,5,2,4] => [4]
=> 0
[1,3,5,4,2] => [1,3,5,4,2] => [1,4,-3,-5,2] => [3,1]
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [-4,1,2,3,5] => [4]
=> 0
[1,4,2,5,3] => [1,4,2,5,3] => [1,-5,2,-4,3] => [3,1]
=> 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,-3,-4,2,5] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,3,5,2] => [1,4,3,5,2] => [1,-3,5,-4,2] => [3,1]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [-4,1,5,2,3] => [3]
=> 0
[1,4,5,3,2] => [1,4,5,3,2] => [1,3,-5,-4,2] => [3,1]
=> 1
[1,5,2,3,4] => [1,5,2,3,4] => [5,1,2,3,4] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,2,4,3] => [1,5,2,4,3] => [-4,1,2,-5,3] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,3,2,4] => [1,5,3,2,4] => [1,-5,-3,2,4] => [3,1]
=> 1
[1,5,3,4,2] => [1,5,3,4,2] => [-3,1,4,-5,2] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,2,3] => [1,5,4,2,3] => [-4,-5,1,2,3] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,3,2] => [1,5,4,3,2] => [1,-3,-4,-5,2] => [4]
=> 0
[2,1,3,4,5] => [2,1,3,4,5] => [-2,1,3,4,5] => [2]
=> 0
[2,1,3,5,4] => [2,1,3,5,4] => [-2,1,3,-5,4] => [2,2]
=> 1
[2,1,4,3,5] => [2,1,4,3,5] => [-2,1,-4,3,5] => [2,2]
=> 1
[2,1,4,5,3] => [2,1,4,5,3] => [-2,1,5,-4,3] => [2,1]
=> 1
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-5,1,3,4] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,5,4,3] => [2,1,5,4,3] => [-2,1,-4,-5,3] => [2]
=> 0
[2,3,1,4,5] => [2,3,1,4,5] => [3,-2,1,4,5] => [1]
=> 0
[2,3,1,5,4] => [2,3,1,5,4] => [3,-2,1,-5,4] => [2,1]
=> 1
[2,3,4,1,5] => [2,3,4,1,5] => [3,4,-2,1,5] => [4]
=> 0
[2,3,4,5,1] => [2,3,4,5,1] => [3,4,5,-2,1] => [2]
=> 0
[2,3,5,1,4] => [2,3,5,1,4] => [3,-2,5,1,4] => [1]
=> 0
[2,3,5,4,1] => [2,3,5,4,1] => [3,4,-2,-5,1] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,1,3,5] => [2,4,1,3,5] => [-2,4,1,3,5] => [4]
=> 0
[2,5,1,4,3] => [2,5,1,4,3] => [-2,4,1,-5,3] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,3,1,4] => [2,5,3,1,4] => [3,-5,-2,1,4] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,3,4,1] => [2,5,3,4,1] => [-2,3,4,-5,1] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,4,1,3] => [2,5,4,1,3] => [-2,-5,4,1,3] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,4,5,2] => [3,1,4,5,2] => [-4,1,5,-3,2] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,2,1,4,5] => [3,2,1,4,5] => [-2,-3,1,4,5] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,2,4,5,1] => [3,2,4,5,1] => [-2,4,5,-3,1] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,2,5,1,4] => [3,2,5,1,4] => [-2,-3,5,1,4] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,1,2,5] => [3,4,1,2,5] => [3,4,1,2,5] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,1,5,2] => [3,4,1,5,2] => [5,-4,1,-3,2] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,2,5,1] => [3,4,2,5,1] => [2,-4,5,-3,1] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,5,1,2] => [3,4,5,1,2] => [3,4,5,1,2] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,5,2,1] => [3,4,5,2,1] => [5,2,-4,-3,1] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,1,2,4] => [3,5,1,2,4] => [3,1,5,2,4] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,1,4,2] => [3,5,1,4,2] => [-5,4,1,-3,2] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,1,2,3,5] => [4,1,2,3,5] => [1,4,2,3,5] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,5,1,2,3] => [4,5,1,2,3] => [1,4,5,2,3] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,5,2,1,3] => [4,5,2,1,3] => [-5,-4,2,1,3] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[5,1,2,3,4] => [5,1,2,3,4] => [1,2,5,3,4] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[5,1,4,3,2] => [5,1,4,3,2] => [3,1,-4,-5,2] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[5,2,4,3,1] => [5,2,4,3,1] => [2,3,-4,-5,1] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[5,3,1,2,4] => [5,3,1,2,4] => [5,3,1,2,4] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[5,3,4,1,2] => [5,3,4,1,2] => [5,3,4,1,2] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[5,4,1,2,3] => [5,4,1,2,3] => [5,1,4,2,3] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[5,4,1,3,2] => [5,4,1,3,2] => [-4,3,1,-5,2] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[5,4,2,3,1] => [5,4,2,3,1] => [-4,2,3,-5,1] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[5,4,3,2,1] => [5,4,3,2,1] => [-2,-3,-4,-5,1] => []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,2,3,6,5,4] => [1,2,3,6,5,4] => [1,2,3,-5,-6,4] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,2,5,4,3,6] => [1,2,5,4,3,6] => [1,2,-4,-5,3,6] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
Description
The number of standard desarrangement tableaux of shape equal to the given partition.
A '''standard desarrangement tableau''' is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry $i$ such that $i+1$ appears to the right or above $i$ in the tableau (with respect to English tableau notation).
This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also:
* [[St000046]]: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition
* [[St000500]]: Eigenvalues of the random-to-random operator acting on the regular representation.
Matching statistic: St000941
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000941: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 48%●distinct values known / distinct values provided: 43%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000941: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 48%●distinct values known / distinct values provided: 43%
Values
[1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0}
[2,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0}
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0}
[2,3,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0}
[3,1,2] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0}
[3,2,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,0,0,0}
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,4] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,4,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,4,2,3] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,4,3,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,1,3,4] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 0
[2,3,1,4] => [3,2,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,3,4,1] => [4,2,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,4,1,3] => [3,4,1,2] => [2,1,1]
=> [1,1]
=> 0
[2,4,3,1] => [4,3,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,1,2,4] => [3,2,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,1,4,2] => [4,2,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,2,1,4] => [3,2,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,2,4,1] => [4,2,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,4,1,2] => [4,3,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,4,2,1] => [4,3,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,1,2,3] => [4,2,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,1,3,2] => [4,2,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,2,1,3] => [4,3,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,2,3,1] => [4,3,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,3,1,2] => [4,3,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,3,2,1] => [4,3,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,5,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[1,2,5,4,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,3,4,2,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0
[1,3,4,5,2] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 0
[1,3,5,2,4] => [1,4,5,2,3] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,2,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0
[1,4,2,5,3] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 0
[1,4,3,2,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0
[1,4,3,5,2] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 0
[1,4,5,2,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,5,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,2,3,4] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 0
[1,5,2,4,3] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 0
[1,5,3,2,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,3,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,2,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 1
[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] => [3,2,1,4,5] => [3,1,1]
=> [1,1]
=> 0
[2,3,1,5,4] => [3,2,1,5,4] => [3,2]
=> [2]
=> 0
[2,3,4,1,5] => [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> 0
[2,3,4,5,1] => [5,2,3,4,1] => [3,1,1]
=> [1,1]
=> 0
[2,3,5,1,4] => [4,2,5,1,3] => [3,1,1]
=> [1,1]
=> 0
[2,3,5,4,1] => [5,2,4,3,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,1,3,5] => [3,4,1,2,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[2,4,1,5,3] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 0
[2,4,3,1,5] => [4,3,2,1,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,3,5,1] => [5,3,2,4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,5,1,3] => [4,5,3,1,2] => [3,1,1]
=> [1,1]
=> 0
[2,4,5,3,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,1,3,4] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 0
[2,5,1,4,3] => [3,5,1,4,2] => [3,1,1]
=> [1,1]
=> 0
[2,5,3,1,4] => [4,5,3,1,2] => [3,1,1]
=> [1,1]
=> 0
[2,5,3,4,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,4,1,3] => [4,5,3,1,2] => [3,1,1]
=> [1,1]
=> 0
[2,5,4,3,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,2,4,5] => [3,2,1,4,5] => [3,1,1]
=> [1,1]
=> 0
[3,1,2,5,4] => [3,2,1,5,4] => [3,2]
=> [2]
=> 0
[3,1,4,2,5] => [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> 0
[3,1,4,5,2] => [5,2,3,4,1] => [3,1,1]
=> [1,1]
=> 0
[3,1,5,2,4] => [4,2,5,1,3] => [3,1,1]
=> [1,1]
=> 0
[3,1,5,4,2] => [5,2,4,3,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,2,1,4,5] => [3,2,1,4,5] => [3,1,1]
=> [1,1]
=> 0
[3,2,1,5,4] => [3,2,1,5,4] => [3,2]
=> [2]
=> 0
[3,2,4,1,5] => [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> 0
[3,2,5,4,1] => [5,2,4,3,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,1,2,5] => [4,3,2,1,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,1,5,2] => [5,3,2,4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,2,1,5] => [4,3,2,1,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,2,5,1] => [5,3,2,4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,5,1,2] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,5,2,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,1,4,2] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,2,4,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,4,1,2] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,4,2,1] => [5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
Description
The number of characters of the symmetric group whose value on the partition is even.
Matching statistic: St001604
(load all 13 compositions to match this statistic)
(load all 13 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
St001604: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 45%●distinct values known / distinct values provided: 29%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 45%●distinct values known / distinct values provided: 29%
Values
[1,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[2,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[1,2,3] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0}
[1,3,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0}
[2,1,3] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0}
[2,3,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0}
[3,1,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0}
[3,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[1,2,3,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[1,2,4,3] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[1,3,2,4] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[1,3,4,2] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[1,4,2,3] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[1,4,3,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[2,1,3,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[2,1,4,3] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[2,3,1,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[2,3,4,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[2,4,1,3] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[2,4,3,1] => [3,1] => [[3,3],[2]]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[3,1,2,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[3,1,4,2] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[3,2,1,4] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[3,2,4,1] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[3,4,1,2] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[3,4,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[4,1,2,3] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[4,1,3,2] => [3,1] => [[3,3],[2]]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[4,2,1,3] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[4,2,3,1] => [3,1] => [[3,3],[2]]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[4,3,1,2] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[4,3,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[1,2,3,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,3,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,4,3,5] => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,4,5,3] => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,5,3,4] => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,3,2,4,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,2,5,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,4,2,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,4,5,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,5,2,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,5,4,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,2,3,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,2,5,3] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,3,2,5] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,3,5,2] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,5,2,3] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,5,3,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,2,3,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[2,3,5,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[2,4,5,3,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[2,5,1,4,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[2,5,3,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[2,5,4,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[3,1,5,4,2] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[3,2,5,4,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[3,4,5,2,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[3,5,1,4,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[3,5,2,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[3,5,4,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[4,1,5,3,2] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[4,2,5,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[4,5,1,3,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[4,5,2,3,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[4,5,3,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
[5,1,2,4,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[5,1,3,4,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[5,1,4,3,2] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[5,2,1,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[5,2,3,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[5,2,4,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,2,3,4,6,5] => [4,2] => [[5,4],[3]]
=> [3]
=> 1
[1,2,3,6,5,4] => [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 1
[1,2,4,3,6,5] => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 0
[1,2,4,6,5,3] => [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 0
[1,2,5,3,6,4] => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 0
[1,2,5,4,3,6] => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 0
[1,2,5,4,6,3] => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 0
[1,2,5,6,4,3] => [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 0
[1,2,6,3,5,4] => [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 0
[1,2,6,4,3,5] => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 0
[1,2,6,4,5,3] => [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 0
[1,2,6,5,4,3] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 0
[1,3,2,6,5,4] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 0
[1,3,4,6,5,2] => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1
[1,3,5,6,4,2] => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1
[1,3,6,2,5,4] => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1
[1,3,6,4,5,2] => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1
[1,3,6,5,4,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 0
[1,4,2,6,5,3] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 0
[1,4,3,6,5,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 0
[1,4,5,6,3,2] => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1
[1,4,6,2,5,3] => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1
[1,4,6,3,5,2] => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1
[1,4,6,5,3,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 0
[1,5,2,6,4,3] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 0
[1,5,3,6,4,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 0
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001124
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 43%●distinct values known / distinct values provided: 29%
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 43%●distinct values known / distinct values provided: 29%
Values
[1,2] => [1,2] => [2,-1] => []
=> ? ∊ {0,0}
[2,1] => [2,1] => [1,-2] => [1]
=> ? ∊ {0,0}
[1,2,3] => [1,2,3] => [2,3,-1] => []
=> ? ∊ {0,0,0,0}
[1,3,2] => [1,3,2] => [3,2,-1] => [1]
=> ? ∊ {0,0,0,0}
[2,1,3] => [2,1,3] => [1,3,-2] => [1]
=> ? ∊ {0,0,0,0}
[2,3,1] => [2,3,1] => [1,2,-3] => [1,1]
=> 0
[3,1,2] => [3,1,2] => [3,1,-2] => []
=> ? ∊ {0,0,0,0}
[3,2,1] => [3,2,1] => [2,1,-3] => [2]
=> 0
[1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1}
[1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1}
[1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1}
[1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => [1,1]
=> 0
[1,4,2,3] => [1,4,2,3] => [3,4,2,-1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1}
[1,4,3,2] => [1,4,3,2] => [4,3,2,-1] => [2]
=> 0
[2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1}
[2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => [1,1]
=> 0
[2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => [1,1]
=> 0
[2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => [1,1,1]
=> 0
[2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1}
[2,4,3,1] => [2,4,3,1] => [1,3,2,-4] => [2,1]
=> 1
[3,1,2,4] => [3,1,2,4] => [3,1,4,-2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1}
[3,1,4,2] => [3,1,4,2] => [4,1,3,-2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1}
[3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => [2]
=> 0
[3,2,4,1] => [3,2,4,1] => [2,1,3,-4] => [2,1]
=> 1
[3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1}
[3,4,2,1] => [3,4,2,1] => [3,1,2,-4] => [3]
=> 0
[4,1,2,3] => [4,1,2,3] => [3,4,1,-2] => [2]
=> 0
[4,1,3,2] => [4,1,3,2] => [4,3,1,-2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1}
[4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1}
[4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => [3]
=> 0
[4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1}
[4,3,2,1] => [4,3,2,1] => [3,2,1,-4] => [2,1]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,-1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,-1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,-1] => [1,1]
=> 0
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,-1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,5,4,3] => [1,2,5,4,3] => [2,5,4,3,-1] => [2]
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,-1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,-1] => [1,1]
=> 0
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,-1] => [1,1]
=> 0
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,-1] => [1,1,1]
=> 0
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,-1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,5,4,2] => [1,3,5,4,2] => [5,2,4,3,-1] => [2,1]
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,-1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,-1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,2,5,-1] => [2]
=> 0
[1,4,3,5,2] => [1,4,3,5,2] => [5,3,2,4,-1] => [2,1]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,-1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,5,3,2] => [1,4,5,3,2] => [5,4,2,3,-1] => [3]
=> 0
[1,5,2,3,4] => [1,5,2,3,4] => [3,4,5,2,-1] => [2]
=> 0
[1,5,2,4,3] => [1,5,2,4,3] => [3,5,4,2,-1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,3,2,4] => [1,5,3,2,4] => [4,3,5,2,-1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,3,4,2] => [1,5,3,4,2] => [5,3,4,2,-1] => [3]
=> 0
[1,5,4,2,3] => [1,5,4,2,3] => [4,5,3,2,-1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,-1] => [2,1]
=> 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,-2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,-2] => [1,1]
=> 0
[2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,-2] => [1,1]
=> 0
[2,1,4,5,3] => [2,1,4,5,3] => [1,5,3,4,-2] => [1,1,1]
=> 0
[2,1,5,3,4] => [2,1,5,3,4] => [1,4,5,3,-2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,-2] => [2,1]
=> 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,2,4,5,-3] => [1,1]
=> 0
[2,3,1,5,4] => [2,3,1,5,4] => [1,2,5,4,-3] => [1,1,1]
=> 0
[2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,5,-4] => [1,1,1]
=> 0
[2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,-5] => [1,1,1,1]
=> 0
[2,3,5,1,4] => [2,3,5,1,4] => [1,2,5,3,-4] => [1,1]
=> 0
[2,3,5,4,1] => [2,3,5,4,1] => [1,2,4,3,-5] => [2,1,1]
=> 1
[2,4,1,3,5] => [2,4,1,3,5] => [1,4,2,5,-3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,1,5,3] => [2,4,1,5,3] => [1,5,2,4,-3] => [1,1]
=> 0
[2,4,3,1,5] => [2,4,3,1,5] => [1,3,2,5,-4] => [2,1]
=> 1
[2,4,3,5,1] => [2,4,3,5,1] => [1,3,2,4,-5] => [2,1,1]
=> 1
[2,4,5,1,3] => [2,4,5,1,3] => [1,5,2,3,-4] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,5,3,1] => [2,4,5,3,1] => [1,4,2,3,-5] => [3,1]
=> 1
[2,5,1,3,4] => [2,5,1,3,4] => [1,4,5,2,-3] => [2,1]
=> 1
[2,5,1,4,3] => [2,5,1,4,3] => [1,5,4,2,-3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,3,1,4] => [2,5,3,1,4] => [1,3,5,2,-4] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,3,4,1] => [2,5,3,4,1] => [1,3,4,2,-5] => [3,1]
=> 1
[2,5,4,1,3] => [2,5,4,1,3] => [1,5,3,2,-4] => [1,1]
=> 0
[2,5,4,3,1] => [2,5,4,3,1] => [1,4,3,2,-5] => [2,1,1]
=> 1
[3,1,2,4,5] => [3,1,2,4,5] => [3,1,4,5,-2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,2,5,4] => [3,1,2,5,4] => [3,1,5,4,-2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,4,2,5] => [3,1,4,2,5] => [4,1,3,5,-2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,4,5,2] => [3,1,4,5,2] => [5,1,3,4,-2] => [1,1]
=> 0
[3,1,5,2,4] => [3,1,5,2,4] => [4,1,5,3,-2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,5,4,2] => [3,1,5,4,2] => [5,1,4,3,-2] => [2]
=> 0
[3,2,1,4,5] => [3,2,1,4,5] => [2,1,4,5,-3] => [2]
=> 0
[3,2,1,5,4] => [3,2,1,5,4] => [2,1,5,4,-3] => [2,1]
=> 1
[3,2,4,1,5] => [3,2,4,1,5] => [2,1,3,5,-4] => [2,1]
=> 1
[3,2,4,5,1] => [3,2,4,5,1] => [2,1,3,4,-5] => [2,1,1]
=> 1
[3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,1,5,2] => [3,4,1,5,2] => [5,1,2,4,-3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,5,1,2] => [3,4,5,1,2] => [5,1,2,3,-4] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,1,4,2] => [3,5,1,4,2] => [5,1,4,2,-3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,2,1,4] => [3,5,2,1,4] => [3,1,5,2,-4] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,4,1,2] => [3,5,4,1,2] => [5,1,3,2,-4] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,1,3,2,5] => [4,1,3,2,5] => [4,3,1,5,-2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,1,3,5,2] => [4,1,3,5,2] => [5,3,1,4,-2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,1,5,3,2] => [4,1,5,3,2] => [5,4,1,3,-2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,2,1,3,5] => [4,2,1,3,5] => [2,4,1,5,-3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
Matching statistic: St001876
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 57%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 57%
Values
[1,2] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0}
[2,1] => [2,1] => ([],2)
=> ([],1)
=> ? ∊ {0,0}
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0}
[2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0}
[2,3,1] => [3,2,1] => ([],3)
=> ([],1)
=> ? ∊ {0,0,0,0,0}
[3,1,2] => [3,2,1] => ([],3)
=> ([],1)
=> ? ∊ {0,0,0,0,0}
[3,2,1] => [3,2,1] => ([],3)
=> ([],1)
=> ? ∊ {0,0,0,0,0}
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 0
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 0
[1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,4,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 0
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,3,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,3,4,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,4,1,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,4,3,1] => [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,1,2,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,1,4,2] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,2,4,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,4,1,2] => [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,4,2,1] => [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,1,2,3] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,1,3,2] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,2,1,3] => [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,2,3,1] => [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,3,1,2] => [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,3,2,1] => [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,5,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 0
[1,2,5,3,4] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 0
[1,2,5,4,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
[1,3,4,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
[1,3,4,5,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 0
[1,3,5,2,4] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,2,3,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
[1,4,2,5,3] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 0
[1,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
[1,4,3,5,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 0
[1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,5,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,2,3,4] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 0
[1,5,2,4,3] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 0
[1,5,3,2,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,3,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,1,3,5,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 0
[2,1,4,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 0
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,5,3,4] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,5,4,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,1,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 0
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,4,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
[2,3,4,5,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
[2,3,5,1,4] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,3,5,4,1] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,1,3,5] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,4,3,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,3,5,1] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,5,1,3] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,4,5,3,1] => [5,4,3,2,1] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,1,3,4] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,5,1,4,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,5,3,1,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,5,3,4,1] => [5,4,3,2,1] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,4,1,3] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,5,4,3,1] => [5,4,3,2,1] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,2,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 0
[3,1,2,5,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,4,2,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
[3,1,4,5,2] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,1,5,4,2] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,2,1,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 0
[3,2,1,5,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,2,4,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
[3,2,4,5,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
[3,2,5,1,4] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,2,5,4,1] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,1,2,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,1,5,2] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,2,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,1,2,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,5,2,1,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,1,2,3,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
[4,1,2,5,3] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
[4,1,3,2,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St001877
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 43%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 43%
Values
[1,2] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0}
[2,1] => [2,1] => ([],2)
=> ([],1)
=> ? ∊ {0,0}
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0}
[2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0}
[2,3,1] => [3,2,1] => ([],3)
=> ([],1)
=> ? ∊ {0,0,0,0,0}
[3,1,2] => [3,2,1] => ([],3)
=> ([],1)
=> ? ∊ {0,0,0,0,0}
[3,2,1] => [3,2,1] => ([],3)
=> ([],1)
=> ? ∊ {0,0,0,0,0}
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 0
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 0
[1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,4,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 0
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,3,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,3,4,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,4,1,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,4,3,1] => [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,1,2,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,1,4,2] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,2,4,1] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,4,1,2] => [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,4,2,1] => [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,1,2,3] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,1,3,2] => [4,2,3,1] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,2,1,3] => [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,2,3,1] => [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,3,1,2] => [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,3,2,1] => [4,3,2,1] => ([],4)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,5,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 0
[1,2,5,3,4] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 0
[1,2,5,4,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
[1,3,4,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
[1,3,4,5,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 0
[1,3,5,2,4] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,2,3,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
[1,4,2,5,3] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 0
[1,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
[1,4,3,5,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 0
[1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,5,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,2,3,4] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 0
[1,5,2,4,3] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,2),(2,1)],3)
=> 0
[1,5,3,2,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,3,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,4,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,1,3,5,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 0
[2,1,4,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 0
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,5,3,4] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,5,4,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,1,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 0
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,4,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
[2,3,4,5,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
[2,3,5,1,4] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,3,5,4,1] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,1,3,5] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[2,4,1,5,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,4,3,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,3,5,1] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,5,1,3] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,4,5,3,1] => [5,4,3,2,1] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,1,3,4] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,5,1,4,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,5,3,1,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,5,3,4,1] => [5,4,3,2,1] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,4,1,3] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,5,4,3,1] => [5,4,3,2,1] => ([],5)
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,2,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 0
[3,1,2,5,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,4,2,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
[3,1,4,5,2] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
[3,1,5,2,4] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,1,5,4,2] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,2,1,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 0
[3,2,1,5,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,2,4,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
[3,2,4,5,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
[3,2,5,1,4] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,2,5,4,1] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,1,2,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,1,5,2] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,2,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,5,1,2,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,5,2,1,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,1,2,3,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
[4,1,2,5,3] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
[4,1,3,2,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 0
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St001964
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 29% ●values known / values provided: 38%●distinct values known / distinct values provided: 29%
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 29% ●values known / values provided: 38%●distinct values known / distinct values provided: 29%
Values
[1,2] => [1,2] => [2,1] => ([],2)
=> 0
[2,1] => [1,2] => [2,1] => ([],2)
=> 0
[1,2,3] => [1,2,3] => [3,2,1] => ([],3)
=> 0
[1,3,2] => [1,2,3] => [3,2,1] => ([],3)
=> 0
[2,1,3] => [1,2,3] => [3,2,1] => ([],3)
=> 0
[2,3,1] => [1,2,3] => [3,2,1] => ([],3)
=> 0
[3,1,2] => [1,3,2] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {0,0}
[3,2,1] => [1,3,2] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {0,0}
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[1,4,2,3] => [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,4,3,2] => [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[2,4,1,3] => [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[3,1,4,2] => [1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[3,2,4,1] => [1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1}
[4,1,2,3] => [1,4,3,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 0
[4,1,3,2] => [1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 0
[4,2,1,3] => [1,4,3,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 0
[4,2,3,1] => [1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 0
[4,3,1,2] => [1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 0
[4,3,2,1] => [1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,5,4,2] => [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,5,2,3] => [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,4,5,3,2] => [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[1,5,2,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> 0
[1,5,2,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 0
[1,5,3,2,4] => [1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> 0
[1,5,3,4,2] => [1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 0
[1,5,4,2,3] => [1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 0
[1,5,4,3,2] => [1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 0
[2,1,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,1,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,5,4,1] => [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,1,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,1,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,3,1,5] => [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,3,5,1] => [1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,5,1,3] => [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,4,5,3,1] => [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5,1,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> 0
[2,5,1,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 0
[2,5,3,1,4] => [1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> 0
[2,5,3,4,1] => [1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 0
[2,5,4,1,3] => [1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 0
[2,5,4,3,1] => [1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 0
[3,1,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,2,5,4] => [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,5,2,4] => [1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,5,4,2] => [1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,2,1,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,2,1,5,4] => [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,2,4,1,5] => [1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,2,4,5,1] => [1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,2,5,1,4] => [1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,2,5,4,1] => [1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,1,2,5] => [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,1,5,2] => [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,2,1,5] => [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,2,5,1] => [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,5,1,2] => [1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4,5,2,1] => [1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[4,1,2,3,5] => [1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 0
[4,1,3,2,5] => [1,4,2,3,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 0
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
The following 147 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001280The number of parts of an integer partition that are at least two. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. 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. St000940The number of characters of the symmetric group whose value on the partition is zero. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000741The Colin de Verdière graph invariant. St000567The sum of the products of all pairs of parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001651The Frankl number of a lattice. 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. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000934The 2-degree of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000260The radius of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. 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. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. 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. St000936The number of even values of the symmetric group character corresponding to the partition. St001128The exponens consonantiae of a partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001846The number of elements which do not have a complement in the lattice. St001820The size of the image of the pop stack sorting operator. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001862The number of crossings of a signed permutation. 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. 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. St001625The Möbius invariant of a lattice. St001845The number of join irreducibles minus the rank of a lattice. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. 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. St001875The number of simple modules with projective dimension at most 1. St000022The number of fixed points of a permutation. St000731The number of double exceedences 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. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in 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. St001866The nesting alignments of a signed permutation. 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. St001867The number of alignments of type EN of a signed permutation. 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 $K[x]/(x^n)$. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. 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. St001868The number of alignments of type NE of 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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