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Matching statistic: St000034
St000034: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 1
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 1
[3,4,2,1] => 2
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 2
[4,3,1,2] => 2
[4,3,2,1] => 2
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 1
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 0
[1,3,5,4,2] => 1
[1,4,2,3,5] => 0
[1,4,2,5,3] => 0
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 1
Description
The maximum defect over any reduced expression for a permutation and any subexpression.
Matching statistic: St001964
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 67% ●values known / values provided: 83%●distinct values known / distinct values provided: 67%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 67% ●values known / values provided: 83%●distinct values known / distinct values provided: 67%
Values
[1] => [1,0]
=> [1,0]
=> ([],1)
=> 0
[1,2] => [1,0,1,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 0
[2,1] => [1,1,0,0]
=> [1,1,0,0]
=> ([],2)
=> 0
[1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(1,2)],3)
=> ? = 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 0
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> 0
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ([],3)
=> 0
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ([],3)
=> 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> ? ∊ {1,2,2}
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> ? ∊ {1,2,2}
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> ? ∊ {1,2,2}
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,5}
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,5}
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(1,4)],5)
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 0
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(2,4)],5)
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(2,4)],5)
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,5}
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,5}
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,5}
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,5}
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,5}
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,5}
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 1
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(2,3),(2,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,5}
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(2,3),(2,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,5}
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,5}
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,5}
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,5}
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,5}
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,5}
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,5}
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,5}
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,5}
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,5}
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,5}
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,5}
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4,4,4,5}
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Matching statistic: St000456
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 69%●distinct values known / distinct values provided: 67%
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 69%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => ([],1)
=> ? = 0 + 1
[1,2] => [1,2] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [1,2] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,3,2] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,1,3] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,3,1] => [1,2,3] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,1,2] => [1,3,2] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,1} + 1
[3,2,1] => [1,3,2] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,1} + 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,1,1,1,2,2,2,2} + 1
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,1,1,1,2,2,2,2} + 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,1,1,1,2,2,2,2} + 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,1,1,1,2,2,2,2} + 1
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,1,1,1,2,2,2,2} + 1
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,1,1,1,2,2,2,2} + 1
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,1,1,1,2,2,2,2} + 1
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,1,1,1,2,2,2,2} + 1
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,4,5,3,2] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,5,2,3,4] => [1,2,5,4,3] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,5,2,4,3] => [1,2,5,3,4] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[1,5,3,2,4] => [1,2,5,4,3] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,5,3,4,2] => [1,2,5,3,4] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[1,5,4,2,3] => [1,2,5,3,4] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[1,5,4,3,2] => [1,2,5,3,4] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[2,1,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,1,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,1,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,1,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,1,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[2,1,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[2,3,1,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,3,1,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,3,4,1,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,3,4,5,1] => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,3,5,1,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[2,3,5,4,1] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[2,4,1,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,4,1,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[2,4,3,1,5] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,4,3,5,1] => [1,2,4,5,3] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[2,4,5,1,3] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,4,5,3,1] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,5,1,3,4] => [1,2,5,4,3] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[2,5,3,1,4] => [1,2,5,4,3] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[3,1,4,5,2] => [1,3,4,5,2] => [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[3,1,5,2,4] => [1,3,5,4,2] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[3,2,4,5,1] => [1,3,4,5,2] => [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[3,2,5,1,4] => [1,3,5,4,2] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[3,5,1,2,4] => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[3,5,1,4,2] => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[3,5,2,1,4] => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[3,5,2,4,1] => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[4,1,2,5,3] => [1,4,5,3,2] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[4,1,5,3,2] => [1,4,3,5,2] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[4,2,1,5,3] => [1,4,5,3,2] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[4,2,5,3,1] => [1,4,3,5,2] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[4,5,1,2,3] => [1,4,2,5,3] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[4,5,2,1,3] => [1,4,2,5,3] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[4,5,3,1,2] => [1,4,2,5,3] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[4,5,3,2,1] => [1,4,2,5,3] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[5,1,2,3,4] => [1,5,4,3,2] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[5,1,4,2,3] => [1,5,3,4,2] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[5,2,1,3,4] => [1,5,4,3,2] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[5,2,4,1,3] => [1,5,3,4,2] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St000940
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000940: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 66%●distinct values known / distinct values provided: 50%
Mp00190: Signed permutations —Foata-Han⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000940: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 66%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => [1]
=> ? = 0
[1,2] => [1,2] => [1,2] => [1,1]
=> 0
[2,1] => [2,1] => [-2,1] => []
=> ? = 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[1,3,2] => [1,3,2] => [1,-3,2] => [1]
=> ? ∊ {0,1}
[2,1,3] => [2,1,3] => [-2,1,3] => [1]
=> ? ∊ {0,1}
[2,3,1] => [2,3,1] => [3,-2,1] => [2]
=> 0
[3,1,2] => [3,1,2] => [3,1,2] => [3]
=> 0
[3,2,1] => [3,2,1] => [-2,-3,1] => [3]
=> 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,2,4,3] => [1,2,4,3] => [1,2,-4,3] => [1,1]
=> 0
[1,3,2,4] => [1,3,2,4] => [1,-3,2,4] => [1,1]
=> 0
[1,3,4,2] => [1,3,4,2] => [1,4,-3,2] => [2,1]
=> 1
[1,4,2,3] => [1,4,2,3] => [-4,1,2,3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,4,3,2] => [1,4,3,2] => [1,-3,-4,2] => [3,1]
=> 2
[2,1,3,4] => [2,1,3,4] => [-2,1,3,4] => [1,1]
=> 0
[2,1,4,3] => [2,1,4,3] => [-2,1,-4,3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,3,1,4] => [2,3,1,4] => [3,-2,1,4] => [2,1]
=> 1
[2,3,4,1] => [2,3,4,1] => [3,4,-2,1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,4,1,3] => [2,4,1,3] => [-2,4,1,3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1}
[2,4,3,1] => [2,4,3,1] => [3,-2,-4,1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => [3,1]
=> 2
[3,1,4,2] => [3,1,4,2] => [-4,1,-3,2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,2,1,4] => [3,2,1,4] => [-2,-3,1,4] => [3,1]
=> 2
[3,2,4,1] => [3,2,4,1] => [-2,4,-3,1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1}
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => [2,2]
=> 0
[3,4,2,1] => [3,4,2,1] => [2,-4,-3,1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,1,2,3] => [4,1,2,3] => [1,4,2,3] => [3,1]
=> 2
[4,1,3,2] => [4,1,3,2] => [3,1,-4,2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,2,1,3] => [4,2,1,3] => [-4,-2,1,3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,2,3,1] => [4,2,3,1] => [2,3,-4,1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,3,1,2] => [4,3,1,2] => [-4,3,1,2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1}
[4,3,2,1] => [4,3,2,1] => [-2,-3,-4,1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1}
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,-5,4] => [1,1,1]
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,-4,3,5] => [1,1,1]
=> 0
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,-4,3] => [2,1,1]
=> 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,-5,2,3,4] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,-4,-5,3] => [3,1,1]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,-3,2,4,5] => [1,1,1]
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,-3,2,-5,4] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,-3,2,5] => [2,1,1]
=> 1
[1,3,4,5,2] => [1,3,4,5,2] => [1,4,5,-3,2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,5,2,4] => [1,3,5,2,4] => [1,-3,5,2,4] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,5,4,2] => [1,3,5,4,2] => [1,4,-3,-5,2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,2,3,5] => [1,4,2,3,5] => [-4,1,2,3,5] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,2,5,3] => [1,4,2,5,3] => [1,-5,2,-4,3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,3,2,5] => [1,4,3,2,5] => [1,-3,-4,2,5] => [3,1,1]
=> 1
[1,4,3,5,2] => [1,4,3,5,2] => [1,-3,5,-4,2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,5,2,3] => [1,4,5,2,3] => [-4,1,5,2,3] => [2]
=> 0
[1,4,5,3,2] => [1,4,5,3,2] => [1,3,-5,-4,2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,2,3,4] => [1,5,2,3,4] => [5,1,2,3,4] => [5]
=> 2
[1,5,2,4,3] => [1,5,2,4,3] => [-4,1,2,-5,3] => [5]
=> 2
[1,5,3,2,4] => [1,5,3,2,4] => [1,-5,-3,2,4] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,3,4,2] => [1,5,3,4,2] => [-3,1,4,-5,2] => [5]
=> 2
[1,5,4,2,3] => [1,5,4,2,3] => [-4,-5,1,2,3] => [5]
=> 2
[1,5,4,3,2] => [1,5,4,3,2] => [1,-3,-4,-5,2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,1,3,4,5] => [2,1,3,4,5] => [-2,1,3,4,5] => [1,1,1]
=> 0
[2,1,3,5,4] => [2,1,3,5,4] => [-2,1,3,-5,4] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,1,4,3,5] => [2,1,4,3,5] => [-2,1,-4,3,5] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,1,4,5,3] => [2,1,4,5,3] => [-2,1,5,-4,3] => [2]
=> 0
[2,1,5,3,4] => [2,1,5,3,4] => [-2,-5,1,3,4] => [5]
=> 2
[2,1,5,4,3] => [2,1,5,4,3] => [-2,1,-4,-5,3] => [3]
=> 0
[2,3,1,4,5] => [2,3,1,4,5] => [3,-2,1,4,5] => [2,1,1]
=> 1
[2,3,1,5,4] => [2,3,1,5,4] => [3,-2,1,-5,4] => [2]
=> 0
[2,3,4,1,5] => [2,3,4,1,5] => [3,4,-2,1,5] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,3,4,5,1] => [2,3,4,5,1] => [3,4,5,-2,1] => [3]
=> 0
[2,3,5,1,4] => [2,3,5,1,4] => [3,-2,5,1,4] => [4]
=> 1
[2,3,5,4,1] => [2,3,5,4,1] => [3,4,-2,-5,1] => [5]
=> 2
[2,4,1,3,5] => [2,4,1,3,5] => [-2,4,1,3,5] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,4,1,5,3] => [2,4,1,5,3] => [5,-2,1,-4,3] => [3]
=> 0
[2,4,3,1,5] => [2,4,3,1,5] => [3,-2,-4,1,5] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[2,4,3,5,1] => [2,4,3,5,1] => [3,-2,5,-4,1] => [3]
=> 0
[2,4,5,1,3] => [2,4,5,1,3] => [-2,4,5,1,3] => [2]
=> 0
[2,4,5,3,1] => [2,4,5,3,1] => [5,3,-2,-4,1] => [2]
=> 0
[2,5,1,3,4] => [2,5,1,3,4] => [-2,1,5,3,4] => [3]
=> 0
[2,5,1,4,3] => [2,5,1,4,3] => [-2,4,1,-5,3] => [5]
=> 2
[2,5,3,1,4] => [2,5,3,1,4] => [3,-5,-2,1,4] => [5]
=> 2
[2,5,3,4,1] => [2,5,3,4,1] => [-2,3,4,-5,1] => [5]
=> 2
[2,5,4,1,3] => [2,5,4,1,3] => [-2,-5,4,1,3] => [5]
=> 2
[2,5,4,3,1] => [2,5,4,3,1] => [3,-2,-4,-5,1] => [4]
=> 1
[3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => [3,1,1]
=> 1
[3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,-5,4] => [3]
=> 0
[3,1,4,2,5] => [3,1,4,2,5] => [-4,1,-3,2,5] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[3,2,4,1,5] => [3,2,4,1,5] => [-2,4,-3,1,5] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[3,4,2,1,5] => [3,4,2,1,5] => [2,-4,-3,1,5] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[3,5,2,1,4] => [3,5,2,1,4] => [-5,2,-3,1,4] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[3,5,2,4,1] => [3,5,2,4,1] => [-5,2,4,-3,1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,1,3,2,5] => [4,1,3,2,5] => [3,1,-4,2,5] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,2,1,3,5] => [4,2,1,3,5] => [-4,-2,1,3,5] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,2,3,1,5] => [4,2,3,1,5] => [2,3,-4,1,5] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,3,1,2,5] => [4,3,1,2,5] => [-4,3,1,2,5] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,3,2,1,5] => [4,3,2,1,5] => [-2,-3,-4,1,5] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,3,5,2,1] => [4,3,5,2,1] => [-5,2,-3,-4,1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,5,3,1,2] => [4,5,3,1,2] => [5,-4,3,1,2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,1,2,4,3] => [5,1,2,4,3] => [1,4,2,-5,3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,1,3,4,2] => [5,1,3,4,2] => [1,3,4,-5,2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,1,4,2,3] => [5,1,4,2,3] => [1,-5,4,2,3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,2,3,1,4] => [5,2,3,1,4] => [2,-5,3,1,4] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,2,4,1,3] => [5,2,4,1,3] => [-5,2,4,1,3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
Description
The number of characters of the symmetric group whose value on the partition is zero.
The maximal value for any given size is recorded in [2].
Matching statistic: St001438
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 83%
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 83%
Values
[1] => [1,0]
=> [1,0]
=> [[1],[]]
=> 0
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> [[2],[]]
=> 0
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> [[1,1],[]]
=> 0
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> 0
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [[3],[]]
=> 0
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 0
[3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? ∊ {1,2}
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? ∊ {1,2}
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 0
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 0
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 0
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 2
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 2
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 2
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> 0
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> 0
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> 0
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 0
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 0
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 0
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 0
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 0
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 0
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 2
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 2
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> 0
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 3
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> 4
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> 4
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5}
Description
The number of missing boxes of a skew partition.
Matching statistic: St000941
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00194: Signed permutations —Foata-Han inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000941: Integer partitions ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 67%
Mp00194: Signed permutations —Foata-Han inverse⟶ Signed permutations
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St000941: Integer partitions ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1] => []
=> ? = 0
[1,2] => [1,2] => [1,2] => []
=> ? = 0
[2,1] => [2,1] => [-2,1] => [2]
=> 0
[1,2,3] => [1,2,3] => [1,2,3] => []
=> ? ∊ {0,0}
[1,3,2] => [1,3,2] => [-3,1,2] => [3]
=> 0
[2,1,3] => [2,1,3] => [-2,1,3] => [2]
=> 0
[2,3,1] => [2,3,1] => [-3,-2,1] => [2,1]
=> 1
[3,1,2] => [3,1,2] => [3,1,2] => []
=> ? ∊ {0,0}
[3,2,1] => [3,2,1] => [2,-3,1] => [3]
=> 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => []
=> ? ∊ {0,0,0,0,0,0,0,2,2}
[1,2,4,3] => [1,2,4,3] => [-4,1,2,3] => [4]
=> 1
[1,3,2,4] => [1,3,2,4] => [-3,1,2,4] => [3]
=> 0
[1,3,4,2] => [1,3,4,2] => [-4,-3,1,2] => []
=> ? ∊ {0,0,0,0,0,0,0,2,2}
[1,4,2,3] => [1,4,2,3] => [4,1,2,3] => []
=> ? ∊ {0,0,0,0,0,0,0,2,2}
[1,4,3,2] => [1,4,3,2] => [3,-4,1,2] => [2]
=> 0
[2,1,3,4] => [2,1,3,4] => [-2,1,3,4] => [2]
=> 0
[2,1,4,3] => [2,1,4,3] => [-4,-2,1,3] => [3,1]
=> 2
[2,3,1,4] => [2,3,1,4] => [-3,-2,1,4] => [2,1]
=> 1
[2,3,4,1] => [2,3,4,1] => [-4,-3,-2,1] => [2]
=> 0
[2,4,1,3] => [2,4,1,3] => [2,-4,1,3] => [4]
=> 1
[2,4,3,1] => [2,4,3,1] => [3,-4,-2,1] => []
=> ? ∊ {0,0,0,0,0,0,0,2,2}
[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => []
=> ? ∊ {0,0,0,0,0,0,0,2,2}
[3,1,4,2] => [3,1,4,2] => [4,-3,1,2] => [4]
=> 1
[3,2,1,4] => [3,2,1,4] => [2,-3,1,4] => [3]
=> 0
[3,2,4,1] => [3,2,4,1] => [2,-4,-3,1] => [3,1]
=> 2
[3,4,1,2] => [3,4,1,2] => [3,4,1,2] => []
=> ? ∊ {0,0,0,0,0,0,0,2,2}
[3,4,2,1] => [3,4,2,1] => [2,4,-3,1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,2,2}
[4,1,2,3] => [4,1,2,3] => [1,-4,2,3] => [3]
=> 0
[4,1,3,2] => [4,1,3,2] => [-3,-4,1,2] => [2,2]
=> 1
[4,2,1,3] => [4,2,1,3] => [-2,-4,1,3] => []
=> ? ∊ {0,0,0,0,0,0,0,2,2}
[4,2,3,1] => [4,2,3,1] => [2,3,-4,1] => [4]
=> 1
[4,3,1,2] => [4,3,1,2] => [-4,3,1,2] => [4]
=> 1
[4,3,2,1] => [4,3,2,1] => [-3,2,-4,1] => []
=> ? ∊ {0,0,0,0,0,0,0,2,2}
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[1,2,3,5,4] => [1,2,3,5,4] => [-5,1,2,3,4] => [5]
=> 2
[1,2,4,3,5] => [1,2,4,3,5] => [-4,1,2,3,5] => [4]
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [-5,-4,1,2,3] => [3,2]
=> 1
[1,2,5,3,4] => [1,2,5,3,4] => [5,1,2,3,4] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[1,2,5,4,3] => [1,2,5,4,3] => [4,-5,1,2,3] => [5]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [-3,1,2,4,5] => [3]
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => [-5,-3,1,2,4] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,4,2,5] => [1,3,4,2,5] => [-4,-3,1,2,5] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,4,5,2] => [1,3,4,5,2] => [-5,-4,-3,1,2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[1,3,5,2,4] => [1,3,5,2,4] => [3,-5,1,2,4] => [3]
=> 0
[1,3,5,4,2] => [1,3,5,4,2] => [4,-5,-3,1,2] => [2,1]
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [4,1,2,3,5] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,2,5,3] => [1,4,2,5,3] => [5,-4,1,2,3] => [2]
=> 0
[1,4,3,2,5] => [1,4,3,2,5] => [3,-4,1,2,5] => [2]
=> 0
[1,4,3,5,2] => [1,4,3,5,2] => [3,-5,-4,1,2] => [3,2]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,1,2,3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[1,4,5,3,2] => [1,4,5,3,2] => [3,5,-4,1,2] => [3]
=> 0
[1,5,2,3,4] => [1,5,2,3,4] => [1,-5,2,3,4] => [4]
=> 1
[1,5,2,4,3] => [1,5,2,4,3] => [-4,-5,1,2,3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[1,5,3,2,4] => [1,5,3,2,4] => [-3,-5,1,2,4] => [3,2]
=> 1
[1,5,3,4,2] => [1,5,3,4,2] => [3,4,-5,1,2] => [5]
=> 2
[1,5,4,2,3] => [1,5,4,2,3] => [-5,4,1,2,3] => [3]
=> 0
[1,5,4,3,2] => [1,5,4,3,2] => [-4,3,-5,1,2] => [3,2]
=> 1
[2,1,3,4,5] => [2,1,3,4,5] => [-2,1,3,4,5] => [2]
=> 0
[2,1,3,5,4] => [2,1,3,5,4] => [-5,-2,1,3,4] => [4,1]
=> 3
[2,1,4,3,5] => [2,1,4,3,5] => [-4,-2,1,3,5] => [3,1]
=> 2
[2,1,4,5,3] => [2,1,4,5,3] => [-5,-4,-2,1,3] => [5]
=> 2
[2,1,5,3,4] => [2,1,5,3,4] => [2,-5,1,3,4] => [5]
=> 2
[2,1,5,4,3] => [2,1,5,4,3] => [4,-5,-2,1,3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[2,3,1,4,5] => [2,3,1,4,5] => [-3,-2,1,4,5] => [2,1]
=> 1
[2,3,1,5,4] => [2,3,1,5,4] => [-5,-3,-2,1,4] => [3]
=> 0
[2,3,4,1,5] => [2,3,4,1,5] => [-4,-3,-2,1,5] => [2]
=> 0
[2,3,4,5,1] => [2,3,4,5,1] => [-5,-4,-3,-2,1] => [2,1]
=> 1
[2,3,5,1,4] => [2,3,5,1,4] => [3,-5,-2,1,4] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[2,3,5,4,1] => [2,3,5,4,1] => [4,-5,-3,-2,1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[2,4,1,3,5] => [2,4,1,3,5] => [2,-4,1,3,5] => [4]
=> 1
[2,4,1,5,3] => [2,4,1,5,3] => [2,-5,-4,1,3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[2,4,3,1,5] => [2,4,3,1,5] => [3,-4,-2,1,5] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[2,4,3,5,1] => [2,4,3,5,1] => [3,-5,-4,-2,1] => [5]
=> 2
[2,4,5,1,3] => [2,4,5,1,3] => [2,5,-4,1,3] => [5]
=> 2
[2,4,5,3,1] => [2,4,5,3,1] => [3,5,-4,-2,1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[2,5,1,3,4] => [2,5,1,3,4] => [-2,-5,1,3,4] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[2,5,1,4,3] => [2,5,1,4,3] => [2,4,-5,1,3] => [2]
=> 0
[2,5,3,1,4] => [2,5,3,1,4] => [2,3,-5,1,4] => [5]
=> 2
[3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[3,1,5,2,4] => [3,1,5,2,4] => [3,5,1,2,4] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[3,2,5,1,4] => [3,2,5,1,4] => [2,5,-3,1,4] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[3,2,5,4,1] => [3,2,5,4,1] => [2,4,-5,-3,1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[3,4,1,2,5] => [3,4,1,2,5] => [3,4,1,2,5] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[3,4,1,5,2] => [3,4,1,5,2] => [4,5,-3,1,2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[3,4,2,1,5] => [3,4,2,1,5] => [2,4,-3,1,5] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[3,4,2,5,1] => [3,4,2,5,1] => [2,5,-4,-3,1] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[3,4,5,1,2] => [3,4,5,1,2] => [3,4,5,1,2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[3,5,2,1,4] => [3,5,2,1,4] => [-3,2,-5,1,4] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[4,1,2,5,3] => [4,1,2,5,3] => [1,-5,-4,2,3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[4,1,5,3,2] => [4,1,5,3,2] => [-5,3,-4,1,2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[4,2,1,3,5] => [4,2,1,3,5] => [-2,-4,1,3,5] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[4,2,5,1,3] => [4,2,5,1,3] => [-5,2,-4,1,3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[4,2,5,3,1] => [4,2,5,3,1] => [2,3,5,-4,1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[4,3,1,5,2] => [4,3,1,5,2] => [-3,5,-4,1,2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[4,3,2,1,5] => [4,3,2,1,5] => [-3,2,-4,1,5] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[4,3,2,5,1] => [4,3,2,5,1] => [-3,2,-5,-4,1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[4,5,1,2,3] => [4,5,1,2,3] => [-5,1,-4,2,3] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[4,5,3,1,2] => [4,5,3,1,2] => [-5,-4,3,1,2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[5,1,2,3,4] => [5,1,2,3,4] => [1,5,2,3,4] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
[5,1,3,4,2] => [5,1,3,4,2] => [-3,4,-5,1,2] => []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5}
Description
The number of characters of the symmetric group whose value on the partition is even.
Matching statistic: St000772
(load all 45 compositions to match this statistic)
(load all 45 compositions to match this statistic)
Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 67%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 67%
Values
[1] => {{1}}
=> [1] => ([],1)
=> 1 = 0 + 1
[1,2] => {{1},{2}}
=> [1,2] => ([],2)
=> ? = 0 + 1
[2,1] => {{1,2}}
=> [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => {{1},{2},{3}}
=> [1,2,3] => ([],3)
=> ? ∊ {0,0,0} + 1
[1,3,2] => {{1},{2,3}}
=> [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,0} + 1
[2,1,3] => {{1,2},{3}}
=> [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,0} + 1
[2,3,1] => {{1,2,3}}
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,1,2] => {{1,2,3}}
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,2,1] => {{1,3},{2}}
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,3,4] => {{1},{2},{3},{4}}
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2} + 1
[1,2,4,3] => {{1},{2},{3,4}}
=> [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2} + 1
[1,3,2,4] => {{1},{2,3},{4}}
=> [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2} + 1
[1,3,4,2] => {{1},{2,3,4}}
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2} + 1
[1,4,2,3] => {{1},{2,3,4}}
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2} + 1
[1,4,3,2] => {{1},{2,4},{3}}
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2} + 1
[2,1,3,4] => {{1,2},{3},{4}}
=> [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2} + 1
[2,1,4,3] => {{1,2},{3,4}}
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2} + 1
[2,3,1,4] => {{1,2,3},{4}}
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2} + 1
[2,3,4,1] => {{1,2,3,4}}
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,4,1,3] => {{1,2,3,4}}
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,4,3,1] => {{1,2,4},{3}}
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,1,2,4] => {{1,2,3},{4}}
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2} + 1
[3,1,4,2] => {{1,2,3,4}}
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1,4] => {{1,3},{2},{4}}
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2} + 1
[3,2,4,1] => {{1,3,4},{2}}
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,4,1,2] => {{1,3},{2,4}}
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[3,4,2,1] => {{1,2,3,4}}
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,2,3] => {{1,2,3,4}}
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,3,2] => {{1,2,4},{3}}
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,2,1,3] => {{1,3,4},{2}}
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,2,3,1] => {{1,4},{2},{3}}
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,3,1,2] => {{1,2,3,4}}
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => {{1,4},{2,3}}
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,2,4,5,3] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,2,5,3,4] => {{1},{2},{3,4,5}}
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,3,2,5,4] => {{1},{2,3},{4,5}}
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,3,4,2,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,3,4,5,2] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,3,5,2,4] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,3,5,4,2] => {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,4,2,3,5] => {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,4,2,5,3] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,4,3,5,2] => {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,4,5,2,3] => {{1},{2,4},{3,5}}
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,4,5,3,2] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,5,2,3,4] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,5,2,4,3] => {{1},{2,3,5},{4}}
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,5,3,2,4] => {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,5,4,2,3] => {{1},{2,3,4,5}}
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[1,5,4,3,2] => {{1},{2,5},{3,4}}
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[2,1,3,5,4] => {{1,2},{3},{4,5}}
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[2,1,4,3,5] => {{1,2},{3,4},{5}}
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[2,1,4,5,3] => {{1,2},{3,4,5}}
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[2,1,5,3,4] => {{1,2},{3,4,5}}
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[2,1,5,4,3] => {{1,2},{3,5},{4}}
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[2,3,1,4,5] => {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[2,3,1,5,4] => {{1,2,3},{4,5}}
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[2,3,4,1,5] => {{1,2,3,4},{5}}
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[2,3,4,5,1] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,5,1,4] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,5,4,1] => {{1,2,3,5},{4}}
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,4,1,3,5] => {{1,2,3,4},{5}}
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[2,4,1,5,3] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,4,3,1,5] => {{1,2,4},{3},{5}}
=> [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5} + 1
[2,4,3,5,1] => {{1,2,4,5},{3}}
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,4,5,1,3] => {{1,2,4},{3,5}}
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,4,5,3,1] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,5,1,3,4] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,5,1,4,3] => {{1,2,3,5},{4}}
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,5,3,1,4] => {{1,2,4,5},{3}}
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,5,3,4,1] => {{1,2,5},{3},{4}}
=> [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,5,4,1,3] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,5,4,3,1] => {{1,2,5},{3,4}}
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,1,4,5,2] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,1,5,2,4] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,1,5,4,2] => {{1,2,3,5},{4}}
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2,4,5,1] => {{1,3,4,5},{2}}
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2,5,1,4] => {{1,3,4,5},{2}}
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,2,5,4,1] => {{1,3,5},{2},{4}}
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,4,1,5,2] => {{1,3},{2,4,5}}
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,4,2,5,1] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,4,5,1,2] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,4,5,2,1] => {{1,3,5},{2,4}}
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,5,1,2,4] => {{1,3},{2,4,5}}
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,5,1,4,2] => {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,5,2,1,4] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,5,2,4,1] => {{1,2,3,5},{4}}
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,5,4,1,2] => {{1,3,4},{2,5}}
=> [3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1 = 0 + 1
[3,5,4,2,1] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,1,2,5,3] => {{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,1,3,5,2] => {{1,2,4,5},{3}}
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[4,1,5,2,3] => {{1,2,4},{3,5}}
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St000512
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000512: Integer partitions ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 67%
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000512: Integer partitions ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [-1] => []
=> ? = 0
[1,2] => [1,2] => [2,-1] => []
=> ? ∊ {0,0}
[2,1] => [2,1] => [1,-2] => [1]
=> ? ∊ {0,0}
[1,2,3] => [1,2,3] => [2,3,-1] => []
=> ? ∊ {0,0,0,1}
[1,3,2] => [1,3,2] => [3,2,-1] => [1]
=> ? ∊ {0,0,0,1}
[2,1,3] => [2,1,3] => [1,3,-2] => [1]
=> ? ∊ {0,0,0,1}
[2,3,1] => [2,3,1] => [1,2,-3] => [1,1]
=> 0
[3,1,2] => [3,1,2] => [3,1,-2] => []
=> ? ∊ {0,0,0,1}
[3,2,1] => [3,2,1] => [2,1,-3] => [2]
=> 0
[1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => [1,1]
=> 0
[1,4,2,3] => [1,4,2,3] => [3,4,2,-1] => []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[1,4,3,2] => [1,4,3,2] => [4,3,2,-1] => [2]
=> 0
[2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => [1,1]
=> 0
[2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => [1,1]
=> 0
[2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => [1,1,1]
=> 1
[2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[2,4,3,1] => [2,4,3,1] => [1,3,2,-4] => [2,1]
=> 1
[3,1,2,4] => [3,1,2,4] => [3,1,4,-2] => []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[3,1,4,2] => [3,1,4,2] => [4,1,3,-2] => [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => [2]
=> 0
[3,2,4,1] => [3,2,4,1] => [2,1,3,-4] => [2,1]
=> 1
[3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[3,4,2,1] => [3,4,2,1] => [3,1,2,-4] => [3]
=> 1
[4,1,2,3] => [4,1,2,3] => [3,4,1,-2] => [2]
=> 0
[4,1,3,2] => [4,1,3,2] => [4,3,1,-2] => []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => [3]
=> 1
[4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => [1]
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[4,3,2,1] => [4,3,2,1] => [3,2,1,-4] => [2,1]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,-1] => [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,-1] => [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,-1] => [1,1]
=> 0
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,-1] => []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[1,2,5,4,3] => [1,2,5,4,3] => [2,5,4,3,-1] => [2]
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,-1] => [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,-1] => [1,1]
=> 0
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,-1] => [1,1]
=> 0
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,-1] => [1,1,1]
=> 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,-1] => [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[1,3,5,4,2] => [1,3,5,4,2] => [5,2,4,3,-1] => [2,1]
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,-1] => []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,-1] => [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,2,5,-1] => [2]
=> 0
[1,4,3,5,2] => [1,4,3,5,2] => [5,3,2,4,-1] => [2,1]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,-1] => []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[1,4,5,3,2] => [1,4,5,3,2] => [5,4,2,3,-1] => [3]
=> 1
[1,5,2,3,4] => [1,5,2,3,4] => [3,4,5,2,-1] => [2]
=> 0
[1,5,2,4,3] => [1,5,2,4,3] => [3,5,4,2,-1] => []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[1,5,3,2,4] => [1,5,3,2,4] => [4,3,5,2,-1] => []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[1,5,3,4,2] => [1,5,3,4,2] => [5,3,4,2,-1] => [3]
=> 1
[1,5,4,2,3] => [1,5,4,2,3] => [4,5,3,2,-1] => [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,-1] => [2,1]
=> 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,-2] => [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,-2] => [1,1]
=> 0
[2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,-2] => [1,1]
=> 0
[2,1,4,5,3] => [2,1,4,5,3] => [1,5,3,4,-2] => [1,1,1]
=> 1
[2,1,5,3,4] => [2,1,5,3,4] => [1,4,5,3,-2] => [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,-2] => [2,1]
=> 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,2,4,5,-3] => [1,1]
=> 0
[2,3,1,5,4] => [2,3,1,5,4] => [1,2,5,4,-3] => [1,1,1]
=> 1
[2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,5,-4] => [1,1,1]
=> 1
[2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,-5] => [1,1,1,1]
=> 4
[2,3,5,1,4] => [2,3,5,1,4] => [1,2,5,3,-4] => [1,1]
=> 0
[2,3,5,4,1] => [2,3,5,4,1] => [1,2,4,3,-5] => [2,1,1]
=> 2
[2,4,1,3,5] => [2,4,1,3,5] => [1,4,2,5,-3] => [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[2,4,1,5,3] => [2,4,1,5,3] => [1,5,2,4,-3] => [1,1]
=> 0
[2,4,3,1,5] => [2,4,3,1,5] => [1,3,2,5,-4] => [2,1]
=> 1
[2,4,3,5,1] => [2,4,3,5,1] => [1,3,2,4,-5] => [2,1,1]
=> 2
[2,4,5,1,3] => [2,4,5,1,3] => [1,5,2,3,-4] => [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[2,4,5,3,1] => [2,4,5,3,1] => [1,4,2,3,-5] => [3,1]
=> 1
[2,5,1,3,4] => [2,5,1,3,4] => [1,4,5,2,-3] => [2,1]
=> 1
[2,5,1,4,3] => [2,5,1,4,3] => [1,5,4,2,-3] => [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[2,5,3,1,4] => [2,5,3,1,4] => [1,3,5,2,-4] => [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[2,5,3,4,1] => [2,5,3,4,1] => [1,3,4,2,-5] => [3,1]
=> 1
[2,5,4,1,3] => [2,5,4,1,3] => [1,5,3,2,-4] => [1,1]
=> 0
[2,5,4,3,1] => [2,5,4,3,1] => [1,4,3,2,-5] => [2,1,1]
=> 2
[3,1,2,4,5] => [3,1,2,4,5] => [3,1,4,5,-2] => []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[3,1,2,5,4] => [3,1,2,5,4] => [3,1,5,4,-2] => [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[3,1,4,2,5] => [3,1,4,2,5] => [4,1,3,5,-2] => [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[3,1,4,5,2] => [3,1,4,5,2] => [5,1,3,4,-2] => [1,1]
=> 0
[3,1,5,2,4] => [3,1,5,2,4] => [4,1,5,3,-2] => []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[3,1,5,4,2] => [3,1,5,4,2] => [5,1,4,3,-2] => [2]
=> 0
[3,2,1,4,5] => [3,2,1,4,5] => [2,1,4,5,-3] => [2]
=> 0
[3,2,1,5,4] => [3,2,1,5,4] => [2,1,5,4,-3] => [2,1]
=> 1
[3,2,4,1,5] => [3,2,4,1,5] => [2,1,3,5,-4] => [2,1]
=> 1
[3,2,4,5,1] => [3,2,4,5,1] => [2,1,3,4,-5] => [2,1,1]
=> 2
[3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[3,4,1,5,2] => [3,4,1,5,2] => [5,1,2,4,-3] => [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[3,4,5,1,2] => [3,4,5,1,2] => [5,1,2,3,-4] => []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[3,5,1,4,2] => [3,5,1,4,2] => [5,1,4,2,-3] => []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[3,5,2,1,4] => [3,5,2,1,4] => [3,1,5,2,-4] => []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[3,5,4,1,2] => [3,5,4,1,2] => [5,1,3,2,-4] => [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[4,1,3,2,5] => [4,1,3,2,5] => [4,3,1,5,-2] => []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[4,1,3,5,2] => [4,1,3,5,2] => [5,3,1,4,-2] => [1]
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
[4,1,5,3,2] => [4,1,5,3,2] => [5,4,1,3,-2] => []
=> ? ∊ {0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5}
Description
The number of invariant subsets of size 3 when acting with a permutation of given cycle type.
Matching statistic: St001176
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 67%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 67%
Values
[1] => [1,0]
=> []
=> ?
=> ? = 0
[1,2] => [1,0,1,0]
=> [1]
=> []
=> ? ∊ {0,0}
[2,1] => [1,1,0,0]
=> []
=> ?
=> ? ∊ {0,0}
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 0
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> []
=> ? ∊ {0,0,0,1}
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,1}
[3,1,2] => [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,1}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,1}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 0
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> 3
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> 3
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> 3
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> 3
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 3
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 3
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> 2
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> 2
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> 2
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> 2
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> 1
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> 1
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> 1
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> 1
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> 1
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,5}
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St001384
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001384: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 56%●distinct values known / distinct values provided: 50%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001384: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 56%●distinct values known / distinct values provided: 50%
Values
[1] => [1,0]
=> []
=> ?
=> ? = 0
[1,2] => [1,0,1,0]
=> [1]
=> []
=> ? ∊ {0,0}
[2,1] => [1,1,0,0]
=> []
=> ?
=> ? ∊ {0,0}
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 0
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> []
=> ? ∊ {0,0,0,1}
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,1}
[3,1,2] => [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,1}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,1}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> 0
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> 2
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> 1
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> 1
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> 1
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> 0
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> 0
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> 0
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> 0
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
Description
The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains.
The following 116 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001435The number of missing boxes in the first row. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000478Another weight of a partition according to Alladi. St000567The sum of the products of all pairs of parts. St000674The number of hills of a Dyck path. St000693The modular (standard) major index of a standard tableau. St000946The sum of the skew hook positions in a Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000454The largest eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000681The Grundy value of Chomp on Ferrers diagrams. St000937The number of positive values of the symmetric group character corresponding to the partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001525The number of symmetric hooks on the diagonal of a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001651The Frankl number of a lattice. St001498The normalised height of a Nakayama algebra with magnitude 1. St000939The number of characters of the symmetric group whose value on the partition is positive. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001845The number of join irreducibles minus the rank of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000260The radius of a connected graph. St000909The number of maximal chains of maximal size in a poset. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001846The number of elements which do not have a complement in the lattice. St000527The width of the poset. St001060The distinguishing index of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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. St000632The jump number of the poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite 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". St001902The number of potential covers of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000908The length of the shortest maximal antichain in a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001472The permanent of the Coxeter matrix of the poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001645The pebbling number of a connected graph. St001779The order of promotion on the set of linear extensions of a poset. St001330The hat guessing number of a graph. St000284The Plancherel distribution on integer partitions. 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. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001875The number of simple modules with projective dimension at most 1. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001487The number of inner corners of a skew partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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. St001095The number of non-isomorphic posets with precisely one further covering relation. St001545The second Elser number of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000914The sum of the values of the Möbius function of a poset.
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