Processing math: 100%

Your data matches 168 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000836
(load all 2 compositions to match this statistic)
Mapping
Mp00252: Permutations —restriction⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2] => 0
[1,3,2] => [1,2] => 0
[2,1,3] => [2,1] => 0
[2,3,1] => [2,1] => 0
[3,1,2] => [1,2] => 0
[3,2,1] => [2,1] => 0
[1,2,3,4] => [1,2,3] => 0
[1,2,4,3] => [1,2,3] => 0
[1,3,2,4] => [1,3,2] => 0
[1,3,4,2] => [1,3,2] => 0
[1,4,2,3] => [1,2,3] => 0
[1,4,3,2] => [1,3,2] => 0
[2,1,3,4] => [2,1,3] => 0
[2,1,4,3] => [2,1,3] => 0
[2,3,1,4] => [2,3,1] => 1
[2,3,4,1] => [2,3,1] => 1
[2,4,1,3] => [2,1,3] => 0
[2,4,3,1] => [2,3,1] => 1
[3,1,2,4] => [3,1,2] => 1
[3,1,4,2] => [3,1,2] => 1
[3,2,1,4] => [3,2,1] => 1
[3,2,4,1] => [3,2,1] => 1
[3,4,1,2] => [3,1,2] => 1
[3,4,2,1] => [3,2,1] => 1
[4,1,2,3] => [1,2,3] => 0
[4,1,3,2] => [1,3,2] => 0
[4,2,1,3] => [2,1,3] => 0
[4,2,3,1] => [2,3,1] => 1
[4,3,1,2] => [3,1,2] => 1
[4,3,2,1] => [3,2,1] => 1
[1,2,3,4,5] => [1,2,3,4] => 0
[1,2,3,5,4] => [1,2,3,4] => 0
[1,2,4,3,5] => [1,2,4,3] => 0
[1,2,4,5,3] => [1,2,4,3] => 0
[1,2,5,3,4] => [1,2,3,4] => 0
[1,2,5,4,3] => [1,2,4,3] => 0
[1,3,2,4,5] => [1,3,2,4] => 0
[1,3,2,5,4] => [1,3,2,4] => 0
[1,3,4,2,5] => [1,3,4,2] => 1
[1,3,4,5,2] => [1,3,4,2] => 1
[1,3,5,2,4] => [1,3,2,4] => 0
[1,3,5,4,2] => [1,3,4,2] => 1
[1,4,2,3,5] => [1,4,2,3] => 1
[1,4,2,5,3] => [1,4,2,3] => 1
[1,4,3,2,5] => [1,4,3,2] => 1
[1,4,3,5,2] => [1,4,3,2] => 1
[1,4,5,2,3] => [1,4,2,3] => 1
[1,4,5,3,2] => [1,4,3,2] => 1
[1,5,2,3,4] => [1,2,3,4] => 0
[1,5,2,4,3] => [1,2,4,3] => 0
Description
The number of descents of distance 2 of a permutation. This is, des2(π)=|{i:π(i)>π(i+2)}|.
Matching statistic: St000837
(load all 2 compositions to match this statistic)
Mapping
Mp00252: Permutations —restriction⟶ Permutations
St000837: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2] => 0
[1,3,2] => [1,2] => 0
[2,1,3] => [2,1] => 0
[2,3,1] => [2,1] => 0
[3,1,2] => [1,2] => 0
[3,2,1] => [2,1] => 0
[1,2,3,4] => [1,2,3] => 1
[1,2,4,3] => [1,2,3] => 1
[1,3,2,4] => [1,3,2] => 1
[1,3,4,2] => [1,3,2] => 1
[1,4,2,3] => [1,2,3] => 1
[1,4,3,2] => [1,3,2] => 1
[2,1,3,4] => [2,1,3] => 1
[2,1,4,3] => [2,1,3] => 1
[2,3,1,4] => [2,3,1] => 0
[2,3,4,1] => [2,3,1] => 0
[2,4,1,3] => [2,1,3] => 1
[2,4,3,1] => [2,3,1] => 0
[3,1,2,4] => [3,1,2] => 0
[3,1,4,2] => [3,1,2] => 0
[3,2,1,4] => [3,2,1] => 0
[3,2,4,1] => [3,2,1] => 0
[3,4,1,2] => [3,1,2] => 0
[3,4,2,1] => [3,2,1] => 0
[4,1,2,3] => [1,2,3] => 1
[4,1,3,2] => [1,3,2] => 1
[4,2,1,3] => [2,1,3] => 1
[4,2,3,1] => [2,3,1] => 0
[4,3,1,2] => [3,1,2] => 0
[4,3,2,1] => [3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4] => 2
[1,2,3,5,4] => [1,2,3,4] => 2
[1,2,4,3,5] => [1,2,4,3] => 2
[1,2,4,5,3] => [1,2,4,3] => 2
[1,2,5,3,4] => [1,2,3,4] => 2
[1,2,5,4,3] => [1,2,4,3] => 2
[1,3,2,4,5] => [1,3,2,4] => 2
[1,3,2,5,4] => [1,3,2,4] => 2
[1,3,4,2,5] => [1,3,4,2] => 1
[1,3,4,5,2] => [1,3,4,2] => 1
[1,3,5,2,4] => [1,3,2,4] => 2
[1,3,5,4,2] => [1,3,4,2] => 1
[1,4,2,3,5] => [1,4,2,3] => 1
[1,4,2,5,3] => [1,4,2,3] => 1
[1,4,3,2,5] => [1,4,3,2] => 1
[1,4,3,5,2] => [1,4,3,2] => 1
[1,4,5,2,3] => [1,4,2,3] => 1
[1,4,5,3,2] => [1,4,3,2] => 1
[1,5,2,3,4] => [1,2,3,4] => 2
[1,5,2,4,3] => [1,2,4,3] => 2
Description
The number of ascents of distance 2 of a permutation. This is, asc2(π)=|{i:π(i)<π(i+2)}|.
Matching statistic: St000815
(load all 2 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000815: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 74%●distinct values known / distinct values provided: 60%
Values
[1,2,3] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0}
[1,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0}
[2,1,3] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0}
[2,3,1] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0}
[3,1,2] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0}
[3,2,1] => [1,2,3] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0}
[1,2,3,4] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,2,4] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,3,2] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,3,4] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,4,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,1,4] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,4,1] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,1,3] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,3,1] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,2,4] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,4,2] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,1,4] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,4,1] => [1,2,4,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,1,2] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,2,1] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,2,3] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,3,2] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,1,3] => [1,3,2,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,3,1] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,1,2] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,2,1] => [1,2,3,4] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,3,4,5] => [1,2,3,4,5] => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [1,2,3,5,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,4,3,5] => [1,2,4,3,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [1,2,4,5,3] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [1,2,5,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [1,2,5,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,2,4,5] => [1,3,2,4,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [1,3,2,5,4] => [3,2]
 => [2]
 => 2
[1,3,4,2,5] => [1,3,4,2,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,3,4,5,2] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 2
[1,3,5,4,2] => [1,3,5,2,4] => [3,2]
 => [2]
 => 2
[1,4,2,3,5] => [1,4,2,3,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 2
[1,4,3,2,5] => [1,4,2,5,3] => [3,2]
 => [2]
 => 2
[1,4,3,5,2] => [1,4,2,3,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,5,2,3] => [1,4,5,2,3] => [3,2]
 => [2]
 => 2
[1,4,5,3,2] => [1,4,5,2,3] => [3,2]
 => [2]
 => 2
[1,5,2,3,4] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => 1
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => 1
[1,5,3,4,2] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,5,4,2,3] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,3,4,5] => [1,3,4,5,2] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,3,5,4] => [1,3,5,2,4] => [3,2]
 => [2]
 => 2
[2,1,4,3,5] => [1,4,2,3,5] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,4,5,3] => [1,4,5,2,3] => [3,2]
 => [2]
 => 2
[2,1,5,3,4] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,5,4,3] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,3,1,4,5] => [1,4,5,2,3] => [3,2]
 => [2]
 => 2
[2,3,1,5,4] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,4,1,3,5] => [1,3,5,2,4] => [3,2]
 => [2]
 => 2
[2,4,1,5,3] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => 1
[2,4,3,1,5] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => 1
[2,5,1,4,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => 2
[2,5,3,1,4] => [1,4,2,5,3] => [3,2]
 => [2]
 => 2
[2,5,4,1,3] => [1,3,2,5,4] => [3,2]
 => [2]
 => 2
[3,1,4,2,5] => [1,4,2,5,3] => [3,2]
 => [2]
 => 2
[3,1,4,5,2] => [1,4,5,2,3] => [3,2]
 => [2]
 => 2
[3,1,5,2,4] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => 1
[3,2,1,4,5] => [1,4,5,2,3] => [3,2]
 => [2]
 => 2
[3,2,4,1,5] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => 1
[3,2,5,1,4] => [1,4,2,5,3] => [3,2]
 => [2]
 => 2
[4,1,3,2,5] => [1,3,2,5,4] => [3,2]
 => [2]
 => 2
[4,1,3,5,2] => [1,3,5,2,4] => [3,2]
 => [2]
 => 2
[4,2,1,3,5] => [1,3,5,2,4] => [3,2]
 => [2]
 => 2
[4,2,5,1,3] => [1,3,2,5,4] => [3,2]
 => [2]
 => 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,2]
 => [2]
 => 2
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [4,2]
 => [2]
 => 2
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [4,2]
 => [2]
 => 2
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [4,2]
 => [2]
 => 2
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [4,2]
 => [2]
 => 2
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [4,2]
 => [2]
 => 2
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [4,2]
 => [2]
 => 2
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [4,1,1]
 => [1,1]
 => 1
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [4,1,1]
 => [1,1]
 => 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [4,2]
 => [2]
 => 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [4,2]
 => [2]
 => 2
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [4,2]
 => [2]
 => 2
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [4,2]
 => [2]
 => 2
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [4,2]
 => [2]
 => 2
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [4,2]
 => [2]
 => 2
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [4,2]
 => [2]
 => 2
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [4,2]
 => [2]
 => 2
[1,3,5,2,4,6] => [1,3,5,2,4,6] => [4,2]
 => [2]
 => 2
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [4,2]
 => [2]
 => 2
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [4,2]
 => [2]
 => 2
[1,3,5,4,6,2] => [1,3,5,2,4,6] => [4,2]
 => [2]
 => 2
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [4,2]
 => [2]
 => 2
Description
The number of semistandard Young tableaux of partition weight of given shape. The weight of a semistandard Young tableaux is the sequence (m1,m2,…), where mi is the number of occurrences of the number i in the tableau. This statistic counts those tableaux whose weight is a weakly decreasing sequence. Alternatively, this is the sum of the entries in the column specified by the partition of the change of basis matrix from Schur functions to monomial symmetric functions.
Matching statistic: St000668
(load all 2 compositions to match this statistic)
Mapping
Mp00252: Permutations —restriction⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 62%●distinct values known / distinct values provided: 60%
Values
[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]
 => 1
[1,2,4,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,3,2,4] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,3,4,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,4,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,4,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[2,1,3,4] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[2,1,4,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[2,3,1,4] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[2,3,4,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[2,4,1,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[2,4,3,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[3,1,2,4] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[3,1,4,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[3,2,1,4] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[3,2,4,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[3,4,1,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[3,4,2,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[4,1,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[4,1,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[4,2,1,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[4,2,3,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[4,3,1,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[4,3,2,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,4,2,5] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,5,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3,5] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,4,5,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,5,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,5,2,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,5,3,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,5,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,4,5] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,5,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,4,3,5] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,1,4,5,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,1,5,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,5,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,3,1,4,5] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,1,5,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,1,5] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,5,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,4,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,3,5] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,1,5] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,5,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,1,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,5,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,5,3,1,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,4,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,1,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,3,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,4,5] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,2,1,5,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,2,5,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,4,1,2,5] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,4,1,5,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,4,5,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,5,2,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,5,4,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[4,2,3,1,5] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,2,3,5,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,2,5,3,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,3,2,1,5] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,3,2,5,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,3,5,2,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,5,2,3,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,5,3,2,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[5,1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[5,1,2,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[5,1,3,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[5,1,4,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[5,2,1,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[5,2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
Description
The least common multiple of the parts of the partition.
Matching statistic: St000708
(load all 2 compositions to match this statistic)
Mapping
Mp00252: Permutations —restriction⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 80%
Values
[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]
 => 1
[1,2,4,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,3,2,4] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,3,4,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,4,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,4,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[2,1,3,4] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[2,1,4,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[2,3,1,4] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[2,3,4,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[2,4,1,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[2,4,3,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[3,1,2,4] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[3,1,4,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[3,2,1,4] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[3,2,4,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[3,4,1,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[3,4,2,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[4,1,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[4,1,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[4,2,1,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[4,2,3,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[4,3,1,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[4,3,2,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,4,2,5] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,5,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3,5] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,4,5,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,5,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,5,2,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,5,3,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,5,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,4,5] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,5,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,4,3,5] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,1,4,5,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,1,5,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,5,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,3,1,4,5] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,1,5,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,1,5] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,5,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,4,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,3,5] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,1,5] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,5,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,1,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,5,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,5,3,1,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,4,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,1,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,3,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,4,5] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,2,1,5,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,2,5,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,4,1,2,5] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,4,1,5,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,4,5,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,5,2,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,5,4,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[4,2,3,1,5] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,2,3,5,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,2,5,3,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,3,2,1,5] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,3,2,5,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,3,5,2,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,5,2,3,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,5,3,2,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[5,1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[5,1,2,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[5,1,3,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[5,1,4,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[5,2,1,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[5,2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
Description
The product of the parts of an integer partition.
Matching statistic: St000933
(load all 2 compositions to match this statistic)
Mapping
Mp00252: Permutations —restriction⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 80%
Values
[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]
 => 1
[1,2,4,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,3,2,4] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,3,4,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,4,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[1,4,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[2,1,3,4] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[2,1,4,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[2,3,1,4] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[2,3,4,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[2,4,1,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[2,4,3,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[3,1,2,4] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[3,1,4,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[3,2,1,4] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[3,2,4,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[3,4,1,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[3,4,2,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[4,1,2,3] => [1,2,3] => [1,1,1]
 => [1,1]
 => 1
[4,1,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[4,2,1,3] => [2,1,3] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[4,2,3,1] => [2,3,1] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[4,3,1,2] => [3,1,2] => [3]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[4,3,2,1] => [3,2,1] => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,2,3,4,5] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,3,5,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,3,5] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,2,4,5,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,2,5,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,5,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,3,2,4,5] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,2,5,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,4,2,5] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,3,5,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3,5] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,4,3,5,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,4,5,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[1,5,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[1,5,2,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[1,5,3,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[1,5,3,4,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,4,5] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,3,5,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,4,3,5] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,1,4,5,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,1,5,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,1,5,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,3,1,4,5] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,1,5,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,1,5] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,5,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,4,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,3,5] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,1,5] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,5,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,1,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[2,5,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
[2,5,3,1,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,4,1] => [2,3,4,1] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,1,3] => [2,4,1,3] => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,3,1] => [2,4,3,1] => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,4,5] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,2,1,5,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,2,5,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,4,1,2,5] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,4,1,5,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,4,5,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[3,5,2,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 1
[3,5,4,1,2] => [3,4,1,2] => [2,2]
 => [2]
 => 2
[4,2,3,1,5] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,2,3,5,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,2,5,3,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,3,2,1,5] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,3,2,5,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,3,5,2,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[4,5,2,3,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 1
[4,5,3,2,1] => [4,3,2,1] => [2,2]
 => [2]
 => 2
[5,1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1
[5,1,2,4,3] => [1,2,4,3] => [2,1,1]
 => [1,1]
 => 1
[5,1,3,2,4] => [1,3,2,4] => [2,1,1]
 => [1,1]
 => 1
[5,1,4,3,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => 1
[5,2,1,3,4] => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 1
[5,2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => 2
Description
The number of multipartitions of sizes given by an integer partition. This is, for λ=(λ1,…,λn), this is the number of n-tuples (λ(1),…,λ(n)) of partitions λ(i) such that λ(i)⊢λi.
Matching statistic: St001630
(load all 4 compositions to match this statistic)
Mapping
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 40%
Values
[1,2,3] => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0}
[1,3,2] => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0}
[2,1,3] => [2,1] => ([],2)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0}
[2,3,1] => [2,1] => ([],2)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0}
[3,1,2] => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0}
[3,2,1] => [2,1] => ([],2)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0}
[1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[1,3,2,4] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,3,4,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,4,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[1,4,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[2,1,4,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[2,3,1,4] => [2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[2,3,4,1] => [2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[2,4,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[2,4,3,1] => [2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[3,1,2,4] => [3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[3,1,4,2] => [3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[3,2,1,4] => [3,2,1] => ([],3)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[3,2,4,1] => [3,2,1] => ([],3)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[3,4,1,2] => [3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[3,4,2,1] => [3,2,1] => ([],3)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 1
[4,1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[4,2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[4,2,3,1] => [2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[4,3,1,2] => [3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[4,3,2,1] => [3,2,1] => ([],3)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1}
[1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[1,2,4,3,5] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[1,2,5,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,4,5] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3,5] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,5,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[1,5,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,3,4,5] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,3,5,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,3,5] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,5,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,1,5] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[2,3,4,5,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[2,3,5,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[2,4,1,3,5] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[2,4,1,5,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[2,4,5,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[2,5,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[2,5,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[3,1,4,2,5] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[3,1,4,5,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[3,1,5,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[3,4,1,2,5] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,4,1,5,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,4,5,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,5,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[3,5,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,1,2,3,5] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[4,1,2,5,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[4,1,5,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[4,5,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[5,2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[5,2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[5,3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[5,3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[5,4,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 1
[1,2,3,4,5,6] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,3,4,6,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,3,6,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,6,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,4,5,2,6] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,3,4,5,6,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,3,4,6,5,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,3,5,2,4,6] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,3,5,2,6,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,3,5,6,2,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,3,6,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,3,6,5,2,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,4,2,5,3,6] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,4,2,5,6,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,4,2,6,5,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 1
[1,4,5,2,3,6] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St000454
(load all 19 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 80%
Values
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
 => 0
[1,3,2] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => ? ∊ {0,0}
[2,1,3] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => ? ∊ {0,0}
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
 => 0
[3,1,2] => [1,2,3] => [1,2,3] => ([],3)
 => 0
[3,2,1] => [1,2,3] => [1,2,3] => ([],3)
 => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[1,2,4,3] => [1,2,4,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,3,2,4] => [1,3,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,3,4,2] => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1}
[1,4,2,3] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[1,4,3,2] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[2,1,3,4] => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1}
[2,1,4,3] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[2,3,1,4] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[2,4,1,3] => [1,3,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1}
[2,4,3,1] => [1,2,4,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1}
[3,1,2,4] => [1,2,4,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1}
[3,1,4,2] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[3,2,1,4] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[3,2,4,1] => [1,2,4,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1}
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[4,1,3,2] => [1,3,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1}
[4,2,1,3] => [1,3,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,1,1,1,1,1,1}
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,4,5] => [1,3,2,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [1,3,2,5,4] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [1,3,4,2,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,3,5,2,4] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [1,3,5,2,4] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3,5] => [1,4,2,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,5,2] => [1,4,2,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 1
[1,4,5,2,3] => [1,4,5,2,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [1,4,5,2,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,3,4] => [1,5,2,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 1
[1,5,2,4,3] => [1,5,2,4,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,2,4] => [1,5,2,4,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,4,2] => [1,5,2,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 1
[1,5,4,2,3] => [1,5,2,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 1
[1,5,4,3,2] => [1,5,2,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 1
[2,1,3,4,5] => [1,3,4,5,2] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,3,5,4] => [1,3,5,2,4] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,3,5] => [1,4,2,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 1
[2,1,4,5,3] => [1,4,5,2,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,5,3,4] => [1,5,2,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 1
[2,1,5,4,3] => [1,5,2,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 1
[2,3,1,4,5] => [1,4,5,2,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,1,5,4] => [1,5,2,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 1
[2,3,4,1,5] => [1,5,2,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[2,3,5,1,4] => [1,4,2,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 1
[2,3,5,4,1] => [1,2,3,5,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[2,4,1,3,5] => [1,3,5,2,4] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,1,5,3] => [1,5,2,4,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,1,5] => [1,5,2,4,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,3,5,1] => [1,2,4,3,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,1,3] => [1,3,2,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,4,5,3,1] => [1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [1,3,4,2,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,4,3] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,1,4] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,3,4,1] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,1,3] => [1,3,2,5,4] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,4,3,1] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,2,4,5] => [1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,2,5,4] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,2,5] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,4,5,2] => [1,4,5,2,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,5,2,4] => [1,5,2,4,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,1,5,4,2] => [1,5,2,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 1
[3,2,1,4,5] => [1,4,5,2,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,5,4] => [1,5,2,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 1
[3,4,1,5,2] => [1,5,2,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 1
[3,4,2,1,5] => [1,5,2,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 1
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[3,5,1,4,2] => [1,4,2,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 1
[3,5,2,1,4] => [1,4,2,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 1
[3,5,4,1,2] => [1,2,3,5,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[3,5,4,2,1] => [1,2,3,5,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[4,1,2,3,5] => [1,2,3,5,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[4,1,5,2,3] => [1,5,2,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 1
[4,1,5,3,2] => [1,5,2,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 1
[4,2,1,5,3] => [1,5,2,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 1
[4,2,3,1,5] => [1,5,2,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => 1
[4,2,3,5,1] => [1,2,3,5,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
Description
The largest eigenvalue of a graph if it is integral. If a graph is d-regular, then its largest eigenvalue equals d. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000284
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000284: Integer partitions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 40%
Values
[1,2,3] => [1,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0}
[1,3,2] => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0}
[2,1,3] => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0}
[2,3,1] => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0}
[3,1,2] => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0}
[3,2,1] => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0}
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,4,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[1,3,2,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[1,3,4,2] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[1,4,2,3] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[1,4,3,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[2,1,3,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[2,1,4,3] => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[2,3,1,4] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[2,3,4,1] => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[2,4,1,3] => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[2,4,3,1] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[3,1,2,4] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[3,1,4,2] => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[3,2,1,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[3,2,4,1] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[3,4,1,2] => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[3,4,2,1] => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[4,1,2,3] => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[4,1,3,2] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[4,2,1,3] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[4,2,3,1] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[4,3,1,2] => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[4,3,2,1] => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,4,5,3] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3,5] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,2,3] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,3,4] => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,2,4] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,4,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,5,4,2,3] => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[2,1,3,5,4] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,3,5] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,5,3] => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,5,3,4] => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[4,2,3,1,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[5,2,3,4,1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,2,3,4,6,5] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,3,5,4,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,3,5,6,4] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,3,6,4,5] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,3,6,5,4] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,3,5,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,3,6,5] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,2,4,5,3,6] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,4,6,5,3] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,5,3,4,6] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,5,4,3,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,5,4,6,3] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,5,6,3,4] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,2,6,3,5,4] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,6,4,3,5] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,6,4,5,3] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,6,5,4,3] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,3,2,4,5,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[1,3,2,4,6,5] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,3,2,5,4,6] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,3,2,6,5,4] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,3,4,2,5,6] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,3,5,4,2,6] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,3,6,4,5,2] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,4,2,3,5,6] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,4,3,2,5,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[1,4,3,2,6,5] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,4,3,5,2,6] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,4,3,6,5,2] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,4,5,2,3,6] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,4,6,2,5,3] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,5,2,4,3,6] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,5,3,2,4,6] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,5,3,4,2,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[1,5,3,4,6,2] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,5,3,6,2,4] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,5,4,3,2,6] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
Description
The Plancherel distribution on integer partitions. This is defined as the distribution induced by the RSK shape of the uniform distribution on permutations. In other words, this is the size of the preimage of the map 'Robinson-Schensted tableau shape' from permutations to integer partitions. Equivalently, this is given by the square of the number of standard Young tableaux of the given shape.
Matching statistic: St000681
Mapping
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0}
[1,3,2] => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0}
[2,1,3] => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0}
[2,3,1] => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0}
[3,1,2] => [3]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0}
[3,2,1] => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0}
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,4,3] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[1,3,2,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[1,3,4,2] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[1,4,2,3] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[1,4,3,2] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[2,1,3,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[2,1,4,3] => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[2,3,1,4] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[2,3,4,1] => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[2,4,1,3] => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[2,4,3,1] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[3,1,2,4] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[3,1,4,2] => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[3,2,1,4] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[3,2,4,1] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[3,4,1,2] => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[3,4,2,1] => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[4,1,2,3] => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[4,1,3,2] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[4,2,1,3] => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[4,2,3,1] => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[4,3,1,2] => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[4,3,2,1] => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1}
[1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[1,2,3,5,4] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,4,3,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,4,5,3] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,3,2,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,3,2,5,4] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3,5] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,4,3,5,2] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,2,3] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,3,4] => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,2,4] => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,4,2] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,5,4,2,3] => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[2,1,3,5,4] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,3,5] => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,5,3] => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,5,3,4] => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[4,2,3,1,5] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[5,2,3,4,1] => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 3
[1,2,3,4,6,5] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[1,2,3,5,4,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[1,2,3,5,6,4] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,3,6,4,5] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,3,6,5,4] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[1,2,4,3,5,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[1,2,4,3,6,5] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,2,4,5,3,6] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,4,6,5,3] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,5,3,4,6] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,5,4,3,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[1,2,5,4,6,3] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,5,6,3,4] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,2,6,3,5,4] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,6,4,3,5] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,2,6,4,5,3] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[1,2,6,5,4,3] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,3,2,4,5,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[1,3,2,4,6,5] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,3,2,5,4,6] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,3,2,6,5,4] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,3,4,2,5,6] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,3,5,4,2,6] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,3,6,4,5,2] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,4,2,3,5,6] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,4,3,2,5,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[1,4,3,2,6,5] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,4,3,5,2,6] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,4,3,6,5,2] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,4,5,2,3,6] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,4,6,2,5,3] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,5,2,4,3,6] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,5,3,2,4,6] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,5,3,4,2,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 2
[1,5,3,4,6,2] => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[1,5,3,6,2,4] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[1,5,4,3,2,6] => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
Description
The Grundy value of Chomp on Ferrers diagrams. Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1]. This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
The following 158 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000770The major index of an integer partition when read from bottom to top. St000937The number of positive values of the symmetric group character corresponding to the partition. St001128The exponens consonantiae of a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000260The radius 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. St000455The second largest eigenvalue of a graph if it is integral. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001520The number of strict 3-descents. St000538The number of even inversions of a permutation. St001556The number of inversions of the third entry of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. 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. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001623The number of doubly irreducible elements of a lattice. St001964The interval resolution global dimension of a poset. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000941The number of characters of the symmetric group whose value on the partition is even. St000944The 3-degree of an integer partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001644The dimension of a graph. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. 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. St000717The number of ordinal summands of a poset. St000080The rank of the poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. St000245The number of ascents of a permutation. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St001863The number of weak excedances of a signed permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000097The order of the largest clique of the graph. St000632The jump number of the poset. St001301The first Betti number of the order complex associated with 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". St001845The number of join irreducibles minus the rank of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. 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. St001637The number of (upper) dissectors of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001668The number of points of the poset minus the width of the poset. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001638The book thickness of a graph. St001689The number of celebrities in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001736The total number of cycles in a graph. St001272The number of graphs with the same degree sequence. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001656The monophonic position number of a graph. St000370The genus of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001309The number of four-cliques in a graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St000068The number of minimal elements in a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000098The chromatic number of a graph. St001946The number of descents in a parking function. St001171The vector space dimension of Ext1A(Io,A) when Io is the tilting module corresponding to the permutation o in the Auslander algebra A of K[x]/(xn). 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]/(xn). 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. St001857The number of edges in the reduced word graph of a signed permutation. St000905The number of different multiplicities of parts of an integer composition. St001896The number of right descents of a signed permutations. St001905The number of preferred parking spots in a parking function less than the index of the car. St001581The achromatic number of a graph. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001624The breadth of a lattice. St001621The number of atoms of a lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001875The number of simple modules with projective dimension at most 1. St001490The number of connected components of a skew partition. St000322The skewness of a graph. St000528The height of a poset. 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. St000447The number of pairs of vertices of a graph with distance 3. 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. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. 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. St001827The number of two-component spanning forests of a graph. 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. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001518The number of graphs with the same ordinary spectrum as the given graph. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001642The Prague dimension of a graph. St001734The lettericity of a graph. St000906The length of the shortest maximal chain in a poset. St001391The disjunction number of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001828The Euler characteristic of a graph. St000643The size of the largest orbit of antichains under Panyushev complementation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length 3. St001271The competition number of a graph. St001765The number of connected components of the friends and strangers graph. St000822The Hadwiger number of the graph. St000299The number of nonisomorphic vertex-induced subtrees. St001625The Möbius invariant of a lattice. St000741The Colin de Verdière graph invariant.