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Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St001659
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St001659: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 2
[3]
=> 3
[2,1]
=> 1
[1,1,1]
=> 3
[4]
=> 4
[3,1]
=> 2
[2,2]
=> 2
[2,1,1]
=> 2
[1,1,1,1]
=> 4
[5]
=> 5
[4,1]
=> 3
[3,2]
=> 4
[3,1,1]
=> 4
[2,2,1]
=> 4
[2,1,1,1]
=> 3
[1,1,1,1,1]
=> 5
[6]
=> 6
[5,1]
=> 4
[4,2]
=> 6
[4,1,1]
=> 6
[3,3]
=> 6
[3,2,1]
=> 1
[3,1,1,1]
=> 6
[2,2,2]
=> 6
[2,2,1,1]
=> 6
[2,1,1,1,1]
=> 4
[1,1,1,1,1,1]
=> 6
[7]
=> 7
[6,1]
=> 5
[5,2]
=> 8
[5,1,1]
=> 8
[4,3]
=> 9
[4,2,1]
=> 2
[4,1,1,1]
=> 9
[3,3,1]
=> 2
[3,2,2]
=> 2
[3,2,1,1]
=> 2
[3,1,1,1,1]
=> 8
[2,2,2,1]
=> 9
[2,2,1,1,1]
=> 8
[2,1,1,1,1,1]
=> 5
[1,1,1,1,1,1,1]
=> 7
[8]
=> 8
[7,1]
=> 6
[6,2]
=> 10
[6,1,1]
=> 10
[5,3]
=> 12
[5,2,1]
=> 3
Description
The number of ways to place as many non-attacking rooks as possible on a Ferrers board.
Matching statistic: St000531
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000531: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 34%
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000531: Integer partitions ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 34%
Values
[1]
=> [[1],[]]
=> ([],1)
=> [1]
=> 1
[2]
=> [[2],[]]
=> ([(0,1)],2)
=> [2]
=> 2
[1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> [2]
=> 2
[3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> [2,1]
=> 1
[1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> 2
[2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 2
[2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> [3,1]
=> 2
[1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[5]
=> [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> 3
[3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> 4
[3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> 4
[2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> 4
[2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> 3
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6
[5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5,1]
=> 4
[4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> [4,2]
=> 6
[4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> [4,2]
=> 6
[3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 6
[3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> [3,2,1]
=> 1
[3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> [4,2]
=> 6
[2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 6
[2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> [4,2]
=> 6
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> [5,1]
=> 4
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 6
[7]
=> [[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [7]
=> 7
[6,1]
=> [[6,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> [6,1]
=> 5
[5,2]
=> [[5,2],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> [5,2]
=> 8
[5,1,1]
=> [[5,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> [5,2]
=> 8
[4,3]
=> [[4,3],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> [4,3]
=> 9
[4,2,1]
=> [[4,2,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> [4,2,1]
=> 2
[4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> [4,3]
=> 9
[3,3,1]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> [4,2,1]
=> 2
[3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> [4,2,1]
=> 2
[3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> [4,2,1]
=> 2
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> [5,2]
=> 8
[2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> [4,3]
=> 9
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> [5,2]
=> 8
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> [6,1]
=> 5
[1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [7]
=> 7
[8]
=> [[8],[]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> [8]
=> 8
[7,1]
=> [[7,1],[]]
=> ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> [7,1]
=> 6
[6,2]
=> [[6,2],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> [6,2]
=> 10
[6,1,1]
=> [[6,1,1],[]]
=> ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
=> [6,2]
=> 10
[5,3]
=> [[5,3],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> [5,3]
=> 12
[5,2,1]
=> [[5,2,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
=> [5,2,1]
=> 3
[11]
=> [[11],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[10,1]
=> [[10,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[9,2]
=> [[9,2],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[9,1,1]
=> [[9,1,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[8,3]
=> [[8,3],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[8,2,1]
=> [[8,2,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[8,1,1,1]
=> [[8,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[7,4]
=> [[7,4],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[7,3,1]
=> [[7,3,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[7,2,2]
=> [[7,2,2],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[7,2,1,1]
=> [[7,2,1,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[7,1,1,1,1]
=> [[7,1,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[6,5]
=> [[6,5],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(3,9),(4,3),(4,10),(5,1),(5,8),(6,4),(6,7),(7,10),(9,8),(10,9)],11)
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[6,4,1]
=> [[6,4,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[6,3,2]
=> [[6,3,2],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[6,3,1,1]
=> [[6,3,1,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[6,2,2,1]
=> [[6,2,2,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[6,2,1,1,1]
=> [[6,2,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[6,1,1,1,1,1]
=> [[6,1,1,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[5,5,1]
=> [[5,5,1],[]]
=> ([(0,5),(0,6),(2,8),(3,2),(3,9),(4,3),(4,10),(5,4),(5,7),(6,1),(6,7),(7,10),(9,8),(10,9)],11)
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[5,4,2]
=> [[5,4,2],[]]
=> ([(0,5),(0,6),(2,9),(3,4),(3,10),(4,1),(4,8),(5,3),(5,7),(6,2),(6,7),(7,9),(7,10),(10,8)],11)
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[5,4,1,1]
=> [[5,4,1,1],[]]
=> ([(0,6),(0,7),(3,2),(4,5),(4,10),(5,1),(5,9),(6,4),(6,8),(7,3),(7,8),(8,10),(10,9)],11)
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[5,3,3]
=> [[5,3,3],[]]
=> ([(0,5),(0,6),(2,9),(3,1),(4,3),(4,8),(5,4),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10)],11)
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[5,3,2,1]
=> [[5,3,2,1],[]]
=> ([(0,6),(0,7),(3,2),(4,3),(4,9),(5,1),(5,10),(6,4),(6,8),(7,5),(7,8),(8,9),(8,10)],11)
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[5,3,1,1,1]
=> [[5,3,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(4,10),(5,1),(6,2),(7,3),(7,9),(8,4),(8,9),(9,10)],11)
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[5,2,2,2]
=> [[5,2,2,2],[]]
=> ([(0,6),(0,7),(2,9),(3,4),(4,1),(5,2),(5,10),(6,3),(6,8),(7,5),(7,8),(8,10),(10,9)],11)
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[5,2,2,1,1]
=> [[5,2,2,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(4,10),(5,1),(6,2),(7,3),(7,9),(8,4),(8,9),(9,10)],11)
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[5,2,1,1,1,1]
=> [[5,2,1,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[5,1,1,1,1,1,1]
=> [[5,1,1,1,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[4,4,2,1]
=> [[4,4,2,1],[]]
=> ([(0,5),(0,6),(2,8),(3,2),(3,10),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(10,8)],11)
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[4,4,1,1,1]
=> [[4,4,1,1,1],[]]
=> ([(0,6),(0,7),(2,9),(3,4),(4,1),(5,2),(5,10),(6,3),(6,8),(7,5),(7,8),(8,10),(10,9)],11)
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[4,3,3,1]
=> [[4,3,3,1],[]]
=> ([(0,5),(0,6),(3,2),(3,8),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,8),(7,9),(8,10),(9,10)],11)
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[4,3,2,2]
=> [[4,3,2,2],[]]
=> ([(0,5),(0,6),(2,8),(3,2),(3,10),(4,1),(4,9),(5,3),(5,7),(6,4),(6,7),(7,9),(7,10),(10,8)],11)
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[4,3,2,1,1]
=> [[4,3,2,1,1],[]]
=> ([(0,6),(0,7),(3,2),(4,3),(4,9),(5,1),(5,10),(6,4),(6,8),(7,5),(7,8),(8,9),(8,10)],11)
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[4,3,1,1,1,1]
=> [[4,3,1,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[4,2,2,2,1]
=> [[4,2,2,2,1],[]]
=> ([(0,6),(0,7),(3,2),(4,5),(4,10),(5,1),(5,9),(6,4),(6,8),(7,3),(7,8),(8,10),(10,9)],11)
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[4,2,2,1,1,1]
=> [[4,2,2,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[4,2,1,1,1,1,1]
=> [[4,2,1,1,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[4,1,1,1,1,1,1,1]
=> [[4,1,1,1,1,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[3,3,3,1,1]
=> [[3,3,3,1,1],[]]
=> ([(0,5),(0,6),(2,9),(3,1),(4,3),(4,8),(5,4),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10)],11)
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[3,3,2,2,1]
=> [[3,3,2,2,1],[]]
=> ([(0,5),(0,6),(2,9),(3,4),(3,10),(4,1),(4,8),(5,3),(5,7),(6,2),(6,7),(7,9),(7,10),(10,8)],11)
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[3,3,2,1,1,1]
=> [[3,3,2,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[3,3,1,1,1,1,1]
=> [[3,3,1,1,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[3,2,2,2,2]
=> [[3,2,2,2,2],[]]
=> ([(0,5),(0,6),(2,8),(3,2),(3,9),(4,3),(4,10),(5,4),(5,7),(6,1),(6,7),(7,10),(9,8),(10,9)],11)
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[3,2,2,2,1,1]
=> [[3,2,2,2,1,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[3,2,2,1,1,1,1]
=> [[3,2,2,1,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[3,2,1,1,1,1,1,1]
=> [[3,2,1,1,1,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[3,1,1,1,1,1,1,1,1]
=> [[3,1,1,1,1,1,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[2,2,2,2,2,1]
=> [[2,2,2,2,2,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
[2,2,2,2,1,1,1]
=> [[2,2,2,2,1,1,1],[]]
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,6,6,9,9,10,10,10,10,10,10,11,11,12,12,12,12,12,12,16,16,16,16,16,16,16,16,16,16,18,18,18,18,18,18,18,18,18,18,21,21,21,21,24,24,24,24,25,25,25}
Description
The leading coefficient of the rook polynomial of an integer partition.
Let m be the minimum of the number of parts and the size of the first part of an integer partition λ. Then this statistic yields the number of ways to place m non-attacking rooks on the Ferrers board of λ.
Matching statistic: St001660
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001660: Skew partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 17%
St001660: Skew partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 17%
Values
[1]
=> [[1],[]]
=> 1
[2]
=> [[2],[]]
=> 2
[1,1]
=> [[1,1],[]]
=> 2
[3]
=> [[3],[]]
=> 3
[2,1]
=> [[2,1],[]]
=> 1
[1,1,1]
=> [[1,1,1],[]]
=> 3
[4]
=> [[4],[]]
=> 4
[3,1]
=> [[3,1],[]]
=> 2
[2,2]
=> [[2,2],[]]
=> 2
[2,1,1]
=> [[2,1,1],[]]
=> 2
[1,1,1,1]
=> [[1,1,1,1],[]]
=> 4
[5]
=> [[5],[]]
=> 5
[4,1]
=> [[4,1],[]]
=> 3
[3,2]
=> [[3,2],[]]
=> 4
[3,1,1]
=> [[3,1,1],[]]
=> 4
[2,2,1]
=> [[2,2,1],[]]
=> 4
[2,1,1,1]
=> [[2,1,1,1],[]]
=> 3
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 5
[6]
=> [[6],[]]
=> 6
[5,1]
=> [[5,1],[]]
=> 4
[4,2]
=> [[4,2],[]]
=> 6
[4,1,1]
=> [[4,1,1],[]]
=> 6
[3,3]
=> [[3,3],[]]
=> 6
[3,2,1]
=> [[3,2,1],[]]
=> 1
[3,1,1,1]
=> [[3,1,1,1],[]]
=> 6
[2,2,2]
=> [[2,2,2],[]]
=> 6
[2,2,1,1]
=> [[2,2,1,1],[]]
=> 6
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> 4
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> 6
[7]
=> [[7],[]]
=> 7
[6,1]
=> [[6,1],[]]
=> 5
[5,2]
=> [[5,2],[]]
=> 8
[5,1,1]
=> [[5,1,1],[]]
=> 8
[4,3]
=> [[4,3],[]]
=> 9
[4,2,1]
=> [[4,2,1],[]]
=> 2
[4,1,1,1]
=> [[4,1,1,1],[]]
=> 9
[3,3,1]
=> [[3,3,1],[]]
=> 2
[3,2,2]
=> [[3,2,2],[]]
=> 2
[3,2,1,1]
=> [[3,2,1,1],[]]
=> 2
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> 8
[2,2,2,1]
=> [[2,2,2,1],[]]
=> 9
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> 8
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> 5
[1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> 7
[8]
=> [[8],[]]
=> ? ∊ {3,3,4,4,4,4,4,4,6,6,8,8,10,10,10,10,12,12,12,12,12,12}
[7,1]
=> [[7,1],[]]
=> ? ∊ {3,3,4,4,4,4,4,4,6,6,8,8,10,10,10,10,12,12,12,12,12,12}
[6,2]
=> [[6,2],[]]
=> ? ∊ {3,3,4,4,4,4,4,4,6,6,8,8,10,10,10,10,12,12,12,12,12,12}
[6,1,1]
=> [[6,1,1],[]]
=> ? ∊ {3,3,4,4,4,4,4,4,6,6,8,8,10,10,10,10,12,12,12,12,12,12}
[5,3]
=> [[5,3],[]]
=> ? ∊ {3,3,4,4,4,4,4,4,6,6,8,8,10,10,10,10,12,12,12,12,12,12}
[5,2,1]
=> [[5,2,1],[]]
=> ? ∊ {3,3,4,4,4,4,4,4,6,6,8,8,10,10,10,10,12,12,12,12,12,12}
[5,1,1,1]
=> [[5,1,1,1],[]]
=> ? ∊ {3,3,4,4,4,4,4,4,6,6,8,8,10,10,10,10,12,12,12,12,12,12}
[4,4]
=> [[4,4],[]]
=> ? ∊ {3,3,4,4,4,4,4,4,6,6,8,8,10,10,10,10,12,12,12,12,12,12}
[4,3,1]
=> [[4,3,1],[]]
=> ? ∊ {3,3,4,4,4,4,4,4,6,6,8,8,10,10,10,10,12,12,12,12,12,12}
[4,2,2]
=> [[4,2,2],[]]
=> ? ∊ {3,3,4,4,4,4,4,4,6,6,8,8,10,10,10,10,12,12,12,12,12,12}
[4,2,1,1]
=> [[4,2,1,1],[]]
=> ? ∊ {3,3,4,4,4,4,4,4,6,6,8,8,10,10,10,10,12,12,12,12,12,12}
[4,1,1,1,1]
=> [[4,1,1,1,1],[]]
=> ? ∊ {3,3,4,4,4,4,4,4,6,6,8,8,10,10,10,10,12,12,12,12,12,12}
[3,3,2]
=> [[3,3,2],[]]
=> ? ∊ {3,3,4,4,4,4,4,4,6,6,8,8,10,10,10,10,12,12,12,12,12,12}
[3,3,1,1]
=> [[3,3,1,1],[]]
=> ? ∊ {3,3,4,4,4,4,4,4,6,6,8,8,10,10,10,10,12,12,12,12,12,12}
[3,2,2,1]
=> [[3,2,2,1],[]]
=> ? ∊ {3,3,4,4,4,4,4,4,6,6,8,8,10,10,10,10,12,12,12,12,12,12}
[3,2,1,1,1]
=> [[3,2,1,1,1],[]]
=> ? ∊ {3,3,4,4,4,4,4,4,6,6,8,8,10,10,10,10,12,12,12,12,12,12}
[3,1,1,1,1,1]
=> [[3,1,1,1,1,1],[]]
=> ? ∊ {3,3,4,4,4,4,4,4,6,6,8,8,10,10,10,10,12,12,12,12,12,12}
[2,2,2,2]
=> [[2,2,2,2],[]]
=> ? ∊ {3,3,4,4,4,4,4,4,6,6,8,8,10,10,10,10,12,12,12,12,12,12}
[2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? ∊ {3,3,4,4,4,4,4,4,6,6,8,8,10,10,10,10,12,12,12,12,12,12}
[2,2,1,1,1,1]
=> [[2,2,1,1,1,1],[]]
=> ? ∊ {3,3,4,4,4,4,4,4,6,6,8,8,10,10,10,10,12,12,12,12,12,12}
[2,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1],[]]
=> ? ∊ {3,3,4,4,4,4,4,4,6,6,8,8,10,10,10,10,12,12,12,12,12,12}
[1,1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1,1],[]]
=> ? ∊ {3,3,4,4,4,4,4,4,6,6,8,8,10,10,10,10,12,12,12,12,12,12}
[9]
=> [[9],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[8,1]
=> [[8,1],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[7,2]
=> [[7,2],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[7,1,1]
=> [[7,1,1],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[6,3]
=> [[6,3],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[6,2,1]
=> [[6,2,1],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[6,1,1,1]
=> [[6,1,1,1],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[5,4]
=> [[5,4],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[5,3,1]
=> [[5,3,1],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[5,2,2]
=> [[5,2,2],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[5,2,1,1]
=> [[5,2,1,1],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[5,1,1,1,1]
=> [[5,1,1,1,1],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[4,4,1]
=> [[4,4,1],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[4,3,2]
=> [[4,3,2],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[4,3,1,1]
=> [[4,3,1,1],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[4,2,2,1]
=> [[4,2,2,1],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[4,2,1,1,1]
=> [[4,2,1,1,1],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[4,1,1,1,1,1]
=> [[4,1,1,1,1,1],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[3,3,3]
=> [[3,3,3],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[3,3,2,1]
=> [[3,3,2,1],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[3,3,1,1,1]
=> [[3,3,1,1,1],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[3,2,2,2]
=> [[3,2,2,2],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[3,2,2,1,1]
=> [[3,2,2,1,1],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[3,2,1,1,1,1]
=> [[3,2,1,1,1,1],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[3,1,1,1,1,1,1]
=> [[3,1,1,1,1,1,1],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[2,2,2,2,1]
=> [[2,2,2,2,1],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[2,2,2,1,1,1]
=> [[2,2,2,1,1,1],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
[2,2,1,1,1,1,1]
=> [[2,2,1,1,1,1,1],[]]
=> ? ∊ {4,4,6,6,6,6,6,6,6,6,6,7,7,8,8,8,8,9,9,12,12,12,12,15,15,15,15,16,16,16}
Description
The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board.
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