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Your data matches 424 different statistics following compositions of up to 3 maps.
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Matching statistic: St001621
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,2,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,2,2,1,1] => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,2,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,1,1,1] => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1,1,2] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,2,1,1,1] => [4,1,3] => [[6,4,4],[3,3]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,2,2,1] => [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,2,1,1,2] => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,2,2,1,1] => [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[1,1,1,2,2,2] => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,3,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,1,1,1,2] => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,1,2,1,1,3] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,2,2,1,1,1] => [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[1,1,2,2,1,2] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,2,2,1] => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,2,2,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,3,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,3,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,1,1,1,2,1] => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,1,1,2,1,1] => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,2,1,1,2,2] => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,1,1,3,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,1,1,1,1] => [1,2,4] => [[5,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,1,1,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,2,2,2,1,1] => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,3,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,1,1,2,1,1] => [1,3,1,2] => [[4,3,3,1],[2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[2,1,1,1,2,2] => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1,2,1,1,1] => [1,2,1,3] => [[4,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1,2,2,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[2,1,1,3,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,2,2,1,1] => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1,1,1,2] => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,2,1,1,2,1] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,2,1,1,1] => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,2,1,1,2] => [4,1,2,1] => [[5,5,4,4],[4,3,3]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,2,2,1,1] => [4,2,2] => [[6,5,4],[4,3]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
Description
The number of atoms of a lattice.
An element of a lattice is an '''atom''' if it covers the least element.
Matching statistic: St001878
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,2,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,2,2,1,1] => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,2,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,1,1,1] => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1,1,2] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,2,1,1,1] => [4,1,3] => [[6,4,4],[3,3]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,2,2,1] => [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,2,1,1,2] => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,2,2,1,1] => [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[1,1,1,2,2,2] => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,3,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,1,1,1,2] => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,1,2,1,1,3] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,2,2,1,1,1] => [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[1,1,2,2,1,2] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,2,2,1] => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,2,2,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,3,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,3,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,1,1,1,2,1] => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,1,1,2,1,1] => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,2,1,1,2,2] => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,1,1,3,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,1,1,1,1] => [1,2,4] => [[5,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,1,1,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,2,2,2,1,1] => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,3,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,1,1,2,1,1] => [1,3,1,2] => [[4,3,3,1],[2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[2,1,1,1,2,2] => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1,2,1,1,1] => [1,2,1,3] => [[4,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1,2,2,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[2,1,1,3,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,2,2,1,1] => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1,1,1,2] => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,2,1,1,2,1] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,2,1,1,1] => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,2,1,1,2] => [4,1,2,1] => [[5,5,4,4],[4,3,3]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,2,2,1,1] => [4,2,2] => [[6,5,4],[4,3]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 50%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 50%
Values
[1,1,2,2,1] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,2,2,1,1] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[1,1,1,2,2,1] => [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,1,2,1,1,2] => [2,1,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,2,2,1,1] => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,2,1,1,2,1] => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,2,1,1,1] => [1,2,3] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[2,1,1,2,1,1] => [1,2,1,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[2,1,1,2,2] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,2,1,1,2] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 - 1
[1,1,1,1,2,2,1] => [4,2,1] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[1,1,1,2,1,1,2] => [3,1,2,1] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,1,2,2,1,1] => [3,2,2] => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,1,2,2,2] => [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,1,1,3,1,1,1] => [3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[1,1,2,1,1,1,2] => [2,1,3,1] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[1,1,2,1,1,3] => [2,1,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,2,2,1,1,1] => [2,2,3] => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[1,1,2,2,1,2] => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,1,2,2,2,1] => [2,3,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,2,2,3] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,1,3,1,1,2] => [2,1,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,1,3,3,1] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,2,1,1,1,2,1] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,2,1,1,2,1,1] => [1,1,2,1,2] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[1,2,1,1,2,2] => [1,1,2,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,2,1,1,3,1] => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,2,1,1,1,1] => [1,2,4] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[1,2,2,1,1,2] => [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[1,2,2,2,1,1] => [1,3,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[1,3,1,1,2,1] => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,3,3,1,1] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,1,1,1,2,1,1] => [1,3,1,2] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[2,1,1,1,2,2] => [1,3,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[2,1,1,2,1,1,1] => [1,2,1,3] => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
[2,1,1,2,2,1] => [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[2,1,1,3,1,1] => [1,2,1,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[2,1,2,2,1,1] => [1,1,2,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[2,2,1,1,1,2] => [2,3,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[2,2,1,1,2,1] => [2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[2,2,1,1,3] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,2,2,1,1,1] => [3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
[3,1,1,2,1,1] => [1,2,1,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
[3,1,1,2,2] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[3,2,2,1,1] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,1,1,1,2,1,1,2] => [4,1,2,1] => [1,1,1,3,2] => ([(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,1,1,1,2,2,1,1] => [4,2,2] => [1,1,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,1,1,1,2,2,2] => [4,3] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
[1,1,1,1,3,1,1,1] => [4,1,3] => [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 - 1
[1,1,1,2,1,1,1,2] => [3,1,3,1] => [1,1,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 - 1
[1,1,1,2,1,1,2,1] => [3,1,2,1,1] => [1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 - 1
[1,2,1,1,1,3,1] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,2,1,1,2,1,2] => [1,1,2,1,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,2,1,1,2,3] => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,1,1,3,2] => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,1,1,4,1] => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,3,1,1,1,2,1] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,3,1,1,3,1] => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,4,1,1,2,1] => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[2,1,2,1,1,2,1] => [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[2,1,2,2,1,2] => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[2,3,1,1,2,1] => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[3,2,1,1,2,1] => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[2,1,3,3,1,2] => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[3,2,1,1,2,3] => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[3,1,2,2,1,3] => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,3,3,2,1] => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,3,2,2,3,1] => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,1,1,4,2] => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,1,1,2,4] => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[1,2,1,2,2,3,2] => [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,2,1,2,2,1,4] => [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,2,1,1,4,1,3] => [1,1,2,1,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,2,1,1,3,2,3] => [1,1,2,1,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,2,1,1,3,1,4] => [1,1,2,1,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,2,1,1,2,4,2] => [1,1,2,1,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,2,1,1,2,1,5] => [1,1,2,1,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,2,1,1,1,5,2] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,2,1,1,1,4,3] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,2,1,1,1,3,4] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,2,1,1,1,2,5] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[2,1,2,2,2,1,2] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[2,4,1,1,2,1] => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[4,2,1,1,2,1] => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 1 - 1
[2,3,2,2,1,2,1] => [1,1,2,1,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[4,1,2,2,1,2,1] => [1,1,2,1,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[3,1,4,1,1,2,1] => [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[3,2,3,1,1,2,1] => [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[4,1,3,1,1,2,1] => [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[2,4,2,1,1,2,1] => [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[5,1,2,1,1,2,1] => [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[2,5,1,1,1,2,1] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[3,4,1,1,1,2,1] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[4,3,1,1,1,2,1] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[5,2,1,1,1,2,1] => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,2,4,1,1,2,1] => [1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
[1,3,2,2,1,2,1] => [1,1,2,1,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 1 - 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000788
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000788: Perfect matchings ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000788: Perfect matchings ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
[1,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,14),(10,13),(11,12)]
=> ? = 1
[1,1,1,2,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 1
[1,1,2,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2
[1,1,2,2,1,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 2
[1,2,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[1,2,2,1,1,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> ? = 1
[2,1,1,2,1,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 2
[2,1,1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
[2,2,1,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
[1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14)]
=> ? = 1
[1,1,1,1,2,2,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11),(13,14)]
=> ? = 1
[1,1,1,2,1,1,2] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11),(13,14)]
=> ? = 2
[1,1,1,2,2,1,1] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,14),(12,13)]
=> ? = 2
[1,1,1,2,2,2] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> ? = 1
[1,1,1,3,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,14),(10,13),(11,12)]
=> ? = 1
[1,1,2,1,1,1,2] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10),(13,14)]
=> ? = 2
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1
[1,1,2,1,1,3] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2
[1,1,2,2,1,1,1] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,14),(10,13),(11,12)]
=> ? = 2
[1,1,2,2,1,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 1
[1,1,2,2,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> ? = 2
[1,1,2,2,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
[1,1,3,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2
[1,1,3,3,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
[1,2,1,1,1,2,1] => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12),(13,14)]
=> ? = 1
[1,2,1,1,2,1,1] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,14),(12,13)]
=> ? = 1
[1,2,1,1,2,2] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 1
[1,2,1,1,3,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[1,2,2,1,1,1,1] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,6),(4,5),(7,14),(8,13),(9,12),(10,11)]
=> ? = 1
[1,2,2,1,1,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1
[1,2,2,2,1,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 2
[1,3,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[1,3,3,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
[2,1,1,1,2,1,1] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,14),(12,13)]
=> ? = 2
[2,1,1,1,2,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 2
[2,1,1,2,1,1,1] => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,14),(10,13),(11,12)]
=> ? = 2
[2,1,1,2,2,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1
[2,1,1,3,1,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 2
[2,1,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 1
[2,2,1,1,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> ? = 2
[2,2,1,1,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 1
[2,2,1,1,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
[2,2,2,1,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> ? = 1
[3,1,1,2,1,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 2
[3,1,1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
[3,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
[1,1,1,1,2,1,1,2] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,14),(12,13),(15,16)]
=> ? = 2
[1,1,1,1,2,2,1,1] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11),(13,16),(14,15)]
=> ? = 2
[1,1,1,1,2,2,2] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,14),(10,13),(11,12)]
=> ? = 1
[1,1,1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14)]
=> ? = 1
[1,1,1,2,1,1,1,2] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,14),(10,13),(11,12),(15,16)]
=> ? = 2
[1,1,1,2,1,1,2,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11),(13,14),(15,16)]
=> ? = 1
[1,1,1,2,1,1,3] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11),(13,14)]
=> ? = 2
[1,1,1,2,2,1,2] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1
[1,1,1,2,2,2,1] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10),(13,14)]
=> ? = 2
[1,1,1,2,2,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 1
[1,1,1,3,1,1,2] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11),(13,14)]
=> ? = 2
[1,1,1,3,3,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 1
[1,1,1,4,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,14),(10,13),(11,12)]
=> ? = 1
[1,1,2,1,1,1,2,1] => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10),(13,14),(15,16)]
=> ? = 2
[1,1,2,1,1,1,3] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10),(13,14)]
=> ? = 2
[1,1,2,1,1,2,2] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,14),(12,13)]
=> ? = 2
[1,1,2,1,1,3,1] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1
[1,1,2,2,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
[1,1,3,3,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
[1,2,1,1,2,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[1,2,1,1,3,2] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[1,2,1,1,4,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[1,3,1,1,2,2] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 1
[1,3,1,1,3,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[1,3,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 1
[1,4,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[2,1,1,3,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
[2,1,2,2,1,2] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[2,2,1,1,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
[2,3,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[2,3,3,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
[3,1,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 1
[3,2,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[3,3,1,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
[4,1,1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
[4,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
[2,1,3,3,1,2] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[3,2,1,1,2,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[3,1,2,2,1,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[1,2,3,3,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[1,3,2,2,3,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[2,3,3,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
[2,2,3,3,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
[1,2,1,1,4,2] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[1,2,1,1,3,3] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 1
[1,2,1,1,2,4] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[2,4,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[4,2,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[2,3,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 1
[4,1,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 1
[1,4,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 1
Description
The number of nesting-similar perfect matchings of a perfect matching.
Consider the infinite tree $T$ defined in [1] as follows. $T$ has the perfect matchings on $\{1,\dots,2n\}$ on level $n$, with children obtained by inserting an arc with opener $1$. For example, the matching $[(1,2)]$ has the three children $[(1,2),(3,4)]$, $[(1,3),(2,4)]$ and $[(1,4),(2,3)]$.
Two perfect matchings $M$ and $N$ on $\{1,\dots,2n\}$ are nesting-similar, if the distribution of the number of nestings agrees on all levels of the subtrees of $T$ rooted at $M$ and $N$.
[thm 1.2, 1] shows that to find out whether $M$ and $N$ are nesting-similar, it is enough to check that $M$ and $N$ have the same number of nestings, and that the distribution of nestings agrees for their direct children.
[thm 3.5, 1], see also [2], gives the number of equivalence classes of nesting-similar matchings with $n$ arcs as $$2\cdot 4^{n-1} - \frac{3n-1}{2n+2}\binom{2n}{n}.$$ [prop 3.6, 1] has further interpretations of this number.
Matching statistic: St001132
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001132: Perfect matchings ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001132: Perfect matchings ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
[1,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,14),(10,13),(11,12)]
=> ? = 1
[1,1,1,2,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 1
[1,1,2,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2
[1,1,2,2,1,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 2
[1,2,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[1,2,2,1,1,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> ? = 1
[2,1,1,2,1,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 2
[2,1,1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
[2,2,1,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
[1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14)]
=> ? = 1
[1,1,1,1,2,2,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11),(13,14)]
=> ? = 1
[1,1,1,2,1,1,2] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11),(13,14)]
=> ? = 2
[1,1,1,2,2,1,1] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,14),(12,13)]
=> ? = 2
[1,1,1,2,2,2] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> ? = 1
[1,1,1,3,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,14),(10,13),(11,12)]
=> ? = 1
[1,1,2,1,1,1,2] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10),(13,14)]
=> ? = 2
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1
[1,1,2,1,1,3] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2
[1,1,2,2,1,1,1] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,14),(10,13),(11,12)]
=> ? = 2
[1,1,2,2,1,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 1
[1,1,2,2,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> ? = 2
[1,1,2,2,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
[1,1,3,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2
[1,1,3,3,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
[1,2,1,1,1,2,1] => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12),(13,14)]
=> ? = 1
[1,2,1,1,2,1,1] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,14),(12,13)]
=> ? = 1
[1,2,1,1,2,2] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 1
[1,2,1,1,3,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[1,2,2,1,1,1,1] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,6),(4,5),(7,14),(8,13),(9,12),(10,11)]
=> ? = 1
[1,2,2,1,1,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1
[1,2,2,2,1,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 2
[1,3,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[1,3,3,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
[2,1,1,1,2,1,1] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,14),(12,13)]
=> ? = 2
[2,1,1,1,2,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 2
[2,1,1,2,1,1,1] => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,14),(10,13),(11,12)]
=> ? = 2
[2,1,1,2,2,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1
[2,1,1,3,1,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 2
[2,1,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 1
[2,2,1,1,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> ? = 2
[2,2,1,1,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 1
[2,2,1,1,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
[2,2,2,1,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> ? = 1
[3,1,1,2,1,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 2
[3,1,1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
[3,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
[1,1,1,1,2,1,1,2] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,14),(12,13),(15,16)]
=> ? = 2
[1,1,1,1,2,2,1,1] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11),(13,16),(14,15)]
=> ? = 2
[1,1,1,1,2,2,2] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,14),(10,13),(11,12)]
=> ? = 1
[1,1,1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14)]
=> ? = 1
[1,1,1,2,1,1,1,2] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,14),(10,13),(11,12),(15,16)]
=> ? = 2
[1,1,1,2,1,1,2,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11),(13,14),(15,16)]
=> ? = 1
[1,1,1,2,1,1,3] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11),(13,14)]
=> ? = 2
[1,1,1,2,2,1,2] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1
[1,1,1,2,2,2,1] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10),(13,14)]
=> ? = 2
[1,1,1,2,2,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 1
[1,1,1,3,1,1,2] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11),(13,14)]
=> ? = 2
[1,1,1,3,3,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 1
[1,1,1,4,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,14),(10,13),(11,12)]
=> ? = 1
[1,1,2,1,1,1,2,1] => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10),(13,14),(15,16)]
=> ? = 2
[1,1,2,1,1,1,3] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10),(13,14)]
=> ? = 2
[1,1,2,1,1,2,2] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,14),(12,13)]
=> ? = 2
[1,1,2,1,1,3,1] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1
[1,1,2,2,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
[1,1,3,3,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
[1,2,1,1,2,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[1,2,1,1,3,2] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[1,2,1,1,4,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[1,3,1,1,2,2] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 1
[1,3,1,1,3,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[1,3,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 1
[1,4,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[2,1,1,3,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
[2,1,2,2,1,2] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[2,2,1,1,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
[2,3,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[2,3,3,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
[3,1,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 1
[3,2,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[3,3,1,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
[4,1,1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
[4,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
[2,1,3,3,1,2] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[3,2,1,1,2,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[3,1,2,2,1,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[1,2,3,3,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[1,3,2,2,3,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[2,3,3,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 1
[2,2,3,3,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 1
[1,2,1,1,4,2] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[1,2,1,1,3,3] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 1
[1,2,1,1,2,4] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[2,4,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[4,2,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 1
[2,3,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 1
[4,1,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 1
[1,4,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 1
Description
The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching.
The bijection between perfect matchings of $\{1,\dots,2n\}$ and trees with $n+1$ leaves is described in Example 5.2.6 of [1].
Matching statistic: St000787
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000787: Perfect matchings ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000787: Perfect matchings ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 0 = 1 - 1
[1,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 0 = 1 - 1
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,14),(10,13),(11,12)]
=> ? = 1 - 1
[1,1,1,2,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 1 - 1
[1,1,2,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2 - 1
[1,1,2,2,1,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 2 - 1
[1,2,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
[1,2,2,1,1,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> ? = 1 - 1
[2,1,1,2,1,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 2 - 1
[2,1,1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 0 = 1 - 1
[2,2,1,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 0 = 1 - 1
[1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14)]
=> ? = 1 - 1
[1,1,1,1,2,2,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11),(13,14)]
=> ? = 1 - 1
[1,1,1,2,1,1,2] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11),(13,14)]
=> ? = 2 - 1
[1,1,1,2,2,1,1] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,14),(12,13)]
=> ? = 2 - 1
[1,1,1,2,2,2] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> ? = 1 - 1
[1,1,1,3,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,14),(10,13),(11,12)]
=> ? = 1 - 1
[1,1,2,1,1,1,2] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10),(13,14)]
=> ? = 2 - 1
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1 - 1
[1,1,2,1,1,3] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2 - 1
[1,1,2,2,1,1,1] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,14),(10,13),(11,12)]
=> ? = 2 - 1
[1,1,2,2,1,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 1 - 1
[1,1,2,2,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> ? = 2 - 1
[1,1,2,2,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 0 = 1 - 1
[1,1,3,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2 - 1
[1,1,3,3,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 0 = 1 - 1
[1,2,1,1,1,2,1] => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12),(13,14)]
=> ? = 1 - 1
[1,2,1,1,2,1,1] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,14),(12,13)]
=> ? = 1 - 1
[1,2,1,1,2,2] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 0 = 1 - 1
[1,2,1,1,3,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
[1,2,2,1,1,1,1] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,6),(4,5),(7,14),(8,13),(9,12),(10,11)]
=> ? = 1 - 1
[1,2,2,1,1,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1 - 1
[1,2,2,2,1,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 2 - 1
[1,3,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
[1,3,3,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 0 = 1 - 1
[2,1,1,1,2,1,1] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,14),(12,13)]
=> ? = 2 - 1
[2,1,1,1,2,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 2 - 1
[2,1,1,2,1,1,1] => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,14),(10,13),(11,12)]
=> ? = 2 - 1
[2,1,1,2,2,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1 - 1
[2,1,1,3,1,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 2 - 1
[2,1,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 0 = 1 - 1
[2,2,1,1,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> ? = 2 - 1
[2,2,1,1,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 1 - 1
[2,2,1,1,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 0 = 1 - 1
[2,2,2,1,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> ? = 1 - 1
[3,1,1,2,1,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 2 - 1
[3,1,1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 0 = 1 - 1
[3,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 0 = 1 - 1
[1,1,1,1,2,1,1,2] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,14),(12,13),(15,16)]
=> ? = 2 - 1
[1,1,1,1,2,2,1,1] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11),(13,16),(14,15)]
=> ? = 2 - 1
[1,1,1,1,2,2,2] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,14),(10,13),(11,12)]
=> ? = 1 - 1
[1,1,1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14)]
=> ? = 1 - 1
[1,1,1,2,1,1,1,2] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,14),(10,13),(11,12),(15,16)]
=> ? = 2 - 1
[1,1,1,2,1,1,2,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11),(13,14),(15,16)]
=> ? = 1 - 1
[1,1,1,2,1,1,3] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11),(13,14)]
=> ? = 2 - 1
[1,1,1,2,2,1,2] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1 - 1
[1,1,1,2,2,2,1] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10),(13,14)]
=> ? = 2 - 1
[1,1,1,2,2,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 1 - 1
[1,1,1,3,1,1,2] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11),(13,14)]
=> ? = 2 - 1
[1,1,1,3,3,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 1 - 1
[1,1,1,4,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,14),(10,13),(11,12)]
=> ? = 1 - 1
[1,1,2,1,1,1,2,1] => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10),(13,14),(15,16)]
=> ? = 2 - 1
[1,1,2,1,1,1,3] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10),(13,14)]
=> ? = 2 - 1
[1,1,2,1,1,2,2] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,14),(12,13)]
=> ? = 2 - 1
[1,1,2,1,1,3,1] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1 - 1
[1,1,2,2,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 0 = 1 - 1
[1,1,3,3,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 0 = 1 - 1
[1,2,1,1,2,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
[1,2,1,1,3,2] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
[1,2,1,1,4,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
[1,3,1,1,2,2] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 0 = 1 - 1
[1,3,1,1,3,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
[1,3,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 0 = 1 - 1
[1,4,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
[2,1,1,3,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 0 = 1 - 1
[2,1,2,2,1,2] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
[2,2,1,1,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 0 = 1 - 1
[2,3,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
[2,3,3,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 0 = 1 - 1
[3,1,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 0 = 1 - 1
[3,2,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
[3,3,1,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 0 = 1 - 1
[4,1,1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 0 = 1 - 1
[4,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 0 = 1 - 1
[2,1,3,3,1,2] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
[3,2,1,1,2,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
[3,1,2,2,1,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
[1,2,3,3,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
[1,3,2,2,3,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
[2,3,3,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 0 = 1 - 1
[2,2,3,3,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 0 = 1 - 1
[1,2,1,1,4,2] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
[1,2,1,1,3,3] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 0 = 1 - 1
[1,2,1,1,2,4] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
[2,4,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
[4,2,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 0 = 1 - 1
[2,3,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 0 = 1 - 1
[4,1,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 0 = 1 - 1
[1,4,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 0 = 1 - 1
Description
The number of flips required to make a perfect matching noncrossing.
A crossing in a perfect matching is a pair of arcs $\{a,b\}$ and $\{c,d\}$ such that $a < c < b < d$. Replacing any such pair by either $\{a,c\}$ and $\{b,d\}$ or by $\{a,d\}$, $\{b,c\}$ produces a perfect matching with fewer crossings.
This statistic is the minimal number of such flips required to turn a given matching into a noncrossing matching.
Matching statistic: St001043
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001043: Perfect matchings ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001043: Perfect matchings ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 2 = 1 + 1
[1,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 2 = 1 + 1
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,14),(10,13),(11,12)]
=> ? = 1 + 1
[1,1,1,2,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 1 + 1
[1,1,2,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2 + 1
[1,1,2,2,1,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 2 + 1
[1,2,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
[1,2,2,1,1,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,12),(8,11),(9,10)]
=> ? = 1 + 1
[2,1,1,2,1,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 2 + 1
[2,1,1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 2 = 1 + 1
[2,2,1,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 2 = 1 + 1
[1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14)]
=> ? = 1 + 1
[1,1,1,1,2,2,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11),(13,14)]
=> ? = 1 + 1
[1,1,1,2,1,1,2] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11),(13,14)]
=> ? = 2 + 1
[1,1,1,2,2,1,1] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,14),(12,13)]
=> ? = 2 + 1
[1,1,1,2,2,2] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> ? = 1 + 1
[1,1,1,3,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,14),(10,13),(11,12)]
=> ? = 1 + 1
[1,1,2,1,1,1,2] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10),(13,14)]
=> ? = 2 + 1
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1 + 1
[1,1,2,1,1,3] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2 + 1
[1,1,2,2,1,1,1] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,14),(10,13),(11,12)]
=> ? = 2 + 1
[1,1,2,2,1,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 1 + 1
[1,1,2,2,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> ? = 2 + 1
[1,1,2,2,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 2 = 1 + 1
[1,1,3,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12)]
=> ? = 2 + 1
[1,1,3,3,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 2 = 1 + 1
[1,2,1,1,1,2,1] => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12),(13,14)]
=> ? = 1 + 1
[1,2,1,1,2,1,1] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,14),(12,13)]
=> ? = 1 + 1
[1,2,1,1,2,2] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 2 = 1 + 1
[1,2,1,1,3,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
[1,2,2,1,1,1,1] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,6),(4,5),(7,14),(8,13),(9,12),(10,11)]
=> ? = 1 + 1
[1,2,2,1,1,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1 + 1
[1,2,2,2,1,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 2 + 1
[1,3,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
[1,3,3,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 2 = 1 + 1
[2,1,1,1,2,1,1] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,14),(12,13)]
=> ? = 2 + 1
[2,1,1,1,2,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 2 + 1
[2,1,1,2,1,1,1] => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,14),(10,13),(11,12)]
=> ? = 2 + 1
[2,1,1,2,2,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
=> ? = 1 + 1
[2,1,1,3,1,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 2 + 1
[2,1,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 2 = 1 + 1
[2,2,1,1,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8),(11,12)]
=> ? = 2 + 1
[2,2,1,1,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12)]
=> ? = 1 + 1
[2,2,1,1,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 2 = 1 + 1
[2,2,2,1,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> ? = 1 + 1
[3,1,1,2,1,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,12),(10,11)]
=> ? = 2 + 1
[3,1,1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 2 = 1 + 1
[3,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 2 = 1 + 1
[1,1,1,1,2,1,1,2] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,14),(12,13),(15,16)]
=> ? = 2 + 1
[1,1,1,1,2,2,1,1] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11),(13,16),(14,15)]
=> ? = 2 + 1
[1,1,1,1,2,2,2] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,14),(10,13),(11,12)]
=> ? = 1 + 1
[1,1,1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14)]
=> ? = 1 + 1
[1,1,1,2,1,1,1,2] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,14),(10,13),(11,12),(15,16)]
=> ? = 2 + 1
[1,1,1,2,1,1,2,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11),(13,14),(15,16)]
=> ? = 1 + 1
[1,1,1,2,1,1,3] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11),(13,14)]
=> ? = 2 + 1
[1,1,1,2,2,1,2] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1 + 1
[1,1,1,2,2,2,1] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10),(13,14)]
=> ? = 2 + 1
[1,1,1,2,2,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 1 + 1
[1,1,1,3,1,1,2] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11),(13,14)]
=> ? = 2 + 1
[1,1,1,3,3,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9),(11,12)]
=> ? = 1 + 1
[1,1,1,4,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,14),(10,13),(11,12)]
=> ? = 1 + 1
[1,1,2,1,1,1,2,1] => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10),(13,14),(15,16)]
=> ? = 2 + 1
[1,1,2,1,1,1,3] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10),(13,14)]
=> ? = 2 + 1
[1,1,2,1,1,2,2] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,14),(12,13)]
=> ? = 2 + 1
[1,1,2,1,1,3,1] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9),(11,12),(13,14)]
=> ? = 1 + 1
[1,1,2,2,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 2 = 1 + 1
[1,1,3,3,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 2 = 1 + 1
[1,2,1,1,2,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
[1,2,1,1,3,2] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
[1,2,1,1,4,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
[1,3,1,1,2,2] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 2 = 1 + 1
[1,3,1,1,3,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
[1,3,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 2 = 1 + 1
[1,4,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
[2,1,1,3,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 2 = 1 + 1
[2,1,2,2,1,2] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
[2,2,1,1,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 2 = 1 + 1
[2,3,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
[2,3,3,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 2 = 1 + 1
[3,1,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 2 = 1 + 1
[3,2,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
[3,3,1,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 2 = 1 + 1
[4,1,1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 2 = 1 + 1
[4,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 2 = 1 + 1
[2,1,3,3,1,2] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
[3,2,1,1,2,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
[3,1,2,2,1,3] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
[1,2,3,3,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
[1,3,2,2,3,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
[2,3,3,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 2 = 1 + 1
[2,2,3,3,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 2 = 1 + 1
[1,2,1,1,4,2] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
[1,2,1,1,3,3] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 2 = 1 + 1
[1,2,1,1,2,4] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
[2,4,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
[4,2,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> 2 = 1 + 1
[2,3,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 2 = 1 + 1
[4,1,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 2 = 1 + 1
[1,4,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> 2 = 1 + 1
Description
The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching.
The bijection between perfect matchings of $\{1,\dots,2n\}$ and trees with $n+1$ leaves is described in Example 5.2.6 of [1].
Matching statistic: St001964
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 50%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 50%
Values
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 1 - 1
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> ([(0,4),(1,3),(1,5),(3,6),(4,6),(5,2)],7)
=> ? = 1 - 1
[1,1,1,2,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
[1,1,2,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
[1,1,2,2,1,1] => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ? = 2 - 1
[1,2,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ? = 1 - 1
[1,2,2,1,1,1] => [1,2,3] => [[4,2,1],[1]]
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> ? = 1 - 1
[2,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> ? = 2 - 1
[2,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 1 - 1
[2,2,1,1,2] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[1,1,1,1,2,1,1,1] => [4,1,3] => [[6,4,4],[3,3]]
=> ([(0,6),(1,4),(1,5),(3,7),(4,7),(5,2),(6,3)],8)
=> ? = 1 - 1
[1,1,1,1,2,2,1] => [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7)
=> ? = 1 - 1
[1,1,1,2,1,1,2] => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ([(0,5),(1,3),(2,4),(2,5),(3,6),(4,6)],7)
=> ? = 2 - 1
[1,1,1,2,2,1,1] => [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
=> ? = 2 - 1
[1,1,1,2,2,2] => [3,3] => [[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ? = 1 - 1
[1,1,1,3,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> ([(0,4),(1,3),(1,5),(3,6),(4,6),(5,2)],7)
=> ? = 1 - 1
[1,1,2,1,1,1,2] => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2 - 1
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
=> ? = 1 - 1
[1,1,2,1,1,3] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
[1,1,2,2,1,1,1] => [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7)
=> ? = 2 - 1
[1,1,2,2,1,2] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
[1,1,2,2,2,1] => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
[1,1,2,2,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[1,1,3,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
[1,1,3,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[1,2,1,1,1,2,1] => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,4),(1,3),(1,5),(3,6),(4,6),(5,2)],7)
=> ? = 1 - 1
[1,2,1,1,2,1,1] => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,5),(0,6),(1,3),(1,4),(4,6),(5,2)],7)
=> ? = 1 - 1
[1,2,1,1,2,2] => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> ? = 1 - 1
[1,2,1,1,3,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ? = 1 - 1
[1,2,2,1,1,1,1] => [1,2,4] => [[5,2,1],[1]]
=> ([(0,5),(0,6),(1,3),(1,6),(4,2),(5,4)],7)
=> ? = 1 - 1
[1,2,2,1,1,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ? = 1 - 1
[1,2,2,2,1,1] => [1,3,2] => [[4,3,1],[2]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> ? = 2 - 1
[1,3,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ? = 1 - 1
[1,3,3,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 1 - 1
[2,1,1,1,2,1,1] => [1,3,1,2] => [[4,3,3,1],[2,2]]
=> ([(0,3),(0,5),(1,2),(1,4),(4,6),(5,6)],7)
=> ? = 2 - 1
[2,1,1,1,2,2] => [1,3,2] => [[4,3,1],[2]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> ? = 2 - 1
[2,1,1,2,1,1,1] => [1,2,1,3] => [[4,2,2,1],[1,1]]
=> ([(0,4),(0,6),(1,3),(1,5),(3,6),(5,2)],7)
=> ? = 2 - 1
[2,1,1,2,2,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
=> ? = 1 - 1
[2,1,1,3,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> ? = 2 - 1
[2,1,2,2,1,1] => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,3),(0,5),(1,4),(1,5),(4,2)],6)
=> ? = 1 - 1
[2,2,1,1,1,2] => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? = 2 - 1
[2,2,1,1,2,1] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
[2,2,1,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[2,2,2,1,1,1] => [3,3] => [[5,3],[2]]
=> ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
=> ? = 1 - 1
[3,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,4),(0,5),(1,2),(1,3),(3,5)],6)
=> ? = 2 - 1
[3,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 1 - 1
[3,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 1 - 1
[1,1,1,1,2,1,1,2] => [4,1,2,1] => [[5,5,4,4],[4,3,3]]
=> ([(0,6),(1,4),(2,3),(2,6),(3,7),(4,5),(5,7)],8)
=> ? = 2 - 1
[1,1,1,1,2,2,1,1] => [4,2,2] => [[6,5,4],[4,3]]
=> ([(0,6),(0,7),(1,4),(2,3),(2,6),(4,5),(5,7)],8)
=> ? = 2 - 1
[1,1,1,1,2,2,2] => [4,3] => [[6,4],[3]]
=> ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7)
=> ? = 1 - 1
[1,1,1,1,3,1,1,1] => [4,1,3] => [[6,4,4],[3,3]]
=> ([(0,6),(1,4),(1,5),(3,7),(4,7),(5,2),(6,3)],8)
=> ? = 1 - 1
[1,1,1,2,1,1,1,2] => [3,1,3,1] => [[5,5,3,3],[4,2,2]]
=> ([(0,6),(1,4),(2,3),(2,5),(3,7),(4,7),(5,6)],8)
=> ? = 2 - 1
[1,1,1,2,1,1,2,1] => [3,1,2,1,1] => [[4,4,4,3,3],[3,3,2,2]]
=> ([(0,3),(1,5),(2,4),(2,6),(3,7),(4,7),(5,6)],8)
=> ? = 1 - 1
[1,1,1,2,1,1,3] => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ([(0,5),(1,3),(2,4),(2,5),(3,6),(4,6)],7)
=> ? = 2 - 1
[1,1,1,2,2,1,2] => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> ([(0,5),(0,6),(1,4),(2,3),(3,5),(4,6)],7)
=> ? = 1 - 1
[1,1,1,2,2,2,1] => [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
=> ? = 2 - 1
[1,1,1,2,2,3] => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
[1,1,1,3,1,1,2] => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ([(0,5),(1,3),(2,4),(2,5),(3,6),(4,6)],7)
=> ? = 2 - 1
[1,1,1,3,3,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
[1,1,1,4,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> ([(0,4),(1,3),(1,5),(3,6),(4,6),(5,2)],7)
=> ? = 1 - 1
[1,1,2,1,1,1,2,1] => [2,1,3,1,1] => [[4,4,4,2,2],[3,3,1,1]]
=> ([(0,6),(1,4),(2,3),(2,5),(3,7),(4,7),(5,6)],8)
=> ? = 2 - 1
[1,1,2,1,1,1,3] => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ? = 2 - 1
[1,1,2,1,1,2,2] => [2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> ([(0,6),(1,3),(1,5),(2,4),(2,5),(4,6)],7)
=> ? = 2 - 1
[1,1,2,1,1,3,1] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7)
=> ? = 1 - 1
[1,1,2,2,1,3] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
[1,1,2,2,3,1] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
[1,1,2,2,4] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[1,1,3,3,2] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[2,1,1,3,3] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 1 - 1
[2,2,1,1,3,1] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
[2,2,1,1,4] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[2,2,2,1,1,2] => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
[2,3,3,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 1 - 1
[3,3,1,1,2] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[4,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 1 - 1
[4,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 1 - 1
[2,3,3,2,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0 = 1 - 1
[2,2,3,3,2] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0 = 1 - 1
[1,1,2,2,3,2] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
[1,1,2,2,1,4] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
[1,1,1,3,3,2] => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
[1,1,1,2,2,4] => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
[3,3,1,1,2,1] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
[1,1,2,2,4,1] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
=> 0 = 1 - 1
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: St001645
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 50%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 50%
Values
[1,1,2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[1,2,2,1,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[1,1,1,2,1,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 6
[1,1,1,2,2,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 6
[1,1,2,1,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 6
[1,1,2,2,1,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 6
[1,2,1,1,2,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[1,2,2,1,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 6
[2,1,1,2,1,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 6
[2,1,1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[2,2,1,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[1,1,1,1,2,1,1,1] => [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 + 6
[1,1,1,1,2,2,1] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 6
[1,1,1,2,1,1,2] => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 6
[1,1,1,2,2,1,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 6
[1,1,1,2,2,2] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 6
[1,1,1,3,1,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 6
[1,1,2,1,1,1,2] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 6
[1,1,2,1,1,2,1] => [2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 6
[1,1,2,1,1,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 6
[1,1,2,2,1,1,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 6
[1,1,2,2,1,2] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[1,1,2,2,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 6
[1,1,2,2,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[1,1,3,1,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 6
[1,1,3,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[1,2,1,1,1,2,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 6
[1,2,1,1,2,1,1] => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 6
[1,2,1,1,2,2] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 6
[1,2,1,1,3,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[1,2,2,1,1,1,1] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 6
[1,2,2,1,1,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 6
[1,2,2,2,1,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 6
[1,3,1,1,2,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[1,3,3,1,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[2,1,1,1,2,1,1] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 6
[2,1,1,1,2,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 6
[2,1,1,2,1,1,1] => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 6
[2,1,1,2,2,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 6
[2,1,1,3,1,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 6
[2,1,2,2,1,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 6
[2,2,1,1,1,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 6
[2,2,1,1,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[2,2,1,1,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[2,2,2,1,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 + 6
[3,1,1,2,1,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 + 6
[3,1,1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[3,2,2,1,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 6
[1,1,1,1,2,1,1,2] => [4,1,2,1] => ([(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2 + 6
[1,1,1,1,2,2,1,1] => [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2 + 6
[1,1,1,1,2,2,2] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1 + 6
[1,1,1,1,3,1,1,1] => [4,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 + 6
[1,1,1,2,1,1,1,2] => [3,1,3,1] => ([(0,7),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2 + 6
[1,1,1,2,1,1,2,1] => [3,1,2,1,1] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 1 + 6
[1,1,1,2,1,1,3] => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 6
[1,1,2,2,1,3] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[1,1,2,2,3,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[1,2,1,1,2,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[1,2,1,1,3,2] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[1,2,1,1,4,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[1,3,1,1,3,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[1,4,1,1,2,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[2,1,2,2,1,2] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[2,2,1,1,3,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[2,3,1,1,2,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[3,2,1,1,2,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[2,1,3,3,1,2] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[3,2,1,1,2,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[3,1,2,2,1,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[1,2,3,3,2,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[1,3,2,2,3,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[1,2,1,1,4,2] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[1,2,1,1,2,4] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[1,1,2,2,3,2] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[1,1,2,2,1,4] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[2,4,1,1,2,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[3,3,1,1,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[4,2,1,1,2,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
[1,1,2,2,4,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 1 + 6
Description
The pebbling number of a connected graph.
Matching statistic: St000031
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Values
[1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 1
[1,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 1
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,6,1,4,7,8,5] => ? = 1
[1,1,1,2,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 1
[1,1,2,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 2
[1,1,2,2,1,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,6,3,7,5] => ? = 2
[1,2,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 1
[1,2,2,1,1,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,1,5,2,6,7,4] => ? = 1
[2,1,1,2,1,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 2
[2,1,1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 1
[2,2,1,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 1
[1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,4,7,1,5,8,9,6] => ? = 1
[1,1,1,1,2,2,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [2,3,4,6,1,8,5,7] => ? = 1
[1,1,1,2,1,1,2] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,6,1,4,8,5,7] => ? = 2
[1,1,1,2,2,1,1] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,3,5,1,7,4,8,6] => ? = 2
[1,1,1,2,2,2] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 1
[1,1,1,3,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,6,1,4,7,8,5] => ? = 1
[1,1,2,1,1,1,2] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,5,1,3,6,8,4,7] => ? = 2
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,5,1,3,8,4,6,7] => ? = 1
[1,1,2,1,1,3] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 2
[1,1,2,2,1,1,1] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,6,3,7,8,5] => ? = 2
[1,1,2,2,1,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 1
[1,1,2,2,2,1] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 2
[1,1,2,2,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 1
[1,1,3,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,5,1,3,7,4,6] => ? = 2
[1,1,3,3,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 1
[1,2,1,1,1,2,1] => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [4,1,2,5,8,3,6,7] => ? = 1
[1,2,1,1,2,1,1] => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,1,2,7,3,5,8,6] => ? = 1
[1,2,1,1,2,2] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ? = 1
[1,2,1,1,3,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 1
[1,2,2,1,1,1,1] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,1,5,2,6,7,8,4] => ? = 1
[1,2,2,1,1,2] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ? = 1
[1,2,2,2,1,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 2
[1,3,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ? = 1
[1,3,3,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 1
[2,1,1,1,2,1,1] => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,4,7,2,5,8,6] => ? = 2
[2,1,1,1,2,2] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [3,1,4,6,2,7,5] => ? = 2
[2,1,1,2,1,1,1] => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,1,6,2,4,7,8,5] => ? = 2
[2,1,1,2,2,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,5,2,7,4,6] => ? = 1
[2,1,1,3,1,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 2
[2,1,2,2,1,1] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ? = 1
[2,2,1,1,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,4,1,5,7,3,6] => ? = 2
[2,2,1,1,2,1] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,4,1,7,3,5,6] => ? = 1
[2,2,1,1,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 1
[2,2,2,1,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => ? = 1
[3,1,1,2,1,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [3,1,6,2,4,7,5] => ? = 2
[3,1,1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 1
[3,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 1
[1,1,1,1,2,1,1,2] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,4,7,1,5,9,6,8] => ? = 2
[1,1,1,1,2,2,1,1] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [2,3,4,6,1,8,5,9,7] => ? = 2
[1,1,1,1,2,2,2] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [2,3,4,6,1,7,8,5] => ? = 1
[1,1,1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,4,7,1,5,8,9,6] => ? = 1
[1,1,1,2,1,1,1,2] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,6,1,4,7,9,5,8] => ? = 2
[1,1,1,2,1,1,2,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,6,1,4,9,5,7,8] => ? = 1
[1,1,1,2,1,1,3] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,6,1,4,8,5,7] => ? = 2
[1,1,1,2,2,1,2] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [2,3,5,1,8,4,6,7] => ? = 1
[1,1,1,2,2,2,1] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [2,3,5,1,6,8,4,7] => ? = 2
[1,1,1,2,2,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 1
[1,1,1,3,1,1,2] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,6,1,4,8,5,7] => ? = 2
[1,1,1,3,3,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,3,5,1,7,4,6] => ? = 1
[1,1,2,2,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 1
[1,1,3,3,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 1
[2,1,1,2,2,1,1] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,8,6] => 1
[2,1,1,3,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 1
[2,2,1,1,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 1
[2,3,3,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 1
[3,3,1,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 1
[4,1,1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 1
[4,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 1
[2,3,3,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 1
[2,2,3,3,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 1
[1,2,2,1,1,3,3] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,8,6] => 1
[5,1,1,2,2,1,1] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,8,6] => 1
[4,1,1,2,2,1,1] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,7,4,8,6] => 1
Description
The number of cycles in the cycle decomposition of a permutation.
The following 414 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000237The number of small exceedances. St000352The Elizalde-Pak rank of a permutation. St000696The number of cycles in the breakpoint graph of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000007The number of saliances of the permutation. St000022The number of fixed points of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000141The maximum drop size of a permutation. St000153The number of adjacent cycles of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000223The number of nestings in the permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000402Half the size of the symmetry class of a permutation. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000441The number of successions of a permutation. St000546The number of global descents of a permutation. St000662The staircase size of the code of a permutation. St000665The number of rafts of a permutation. St000862The number of parts of the shifted shape of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000451The length of the longest pattern of the form k 1 2. St000842The breadth of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001490The number of connected components of a skew partition. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001703The villainy of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001488The number of corners of a skew partition. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St001243The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path. St001242The toal dimension of certain Sn modules determined by LLT polynomials associated with a Dyck path. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St000056The decomposition (or block) number of a permutation. St000117The number of centered tunnels of a Dyck path. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000287The number of connected components of a graph. St000458The number of permutations obtained by switching adjacencies or successions. St000486The number of cycles of length at least 3 of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000694The number of affine bounded permutations that project to a given permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000864The number of circled entries of the shifted recording tableau of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001081The number of minimal length factorizations of a permutation into star transpositions. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001274The number of indecomposable injective modules with projective dimension equal to two. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001435The number of missing boxes in the first row. St001461The number of topologically connected components of the chord diagram of a permutation. St001487The number of inner corners of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001518The number of graphs with the same ordinary spectrum as the given graph. St001569The maximal modular displacement of a permutation. St001578The minimal number of edges to add or remove to make a graph a line graph. St001590The crossing number of a perfect matching. St001652The length of a longest interval of consecutive numbers. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001856The number of edges in the reduced word graph of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001871The number of triconnected components of a graph. St001890The maximum magnitude of the Möbius function of a poset. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001960The number of descents of a permutation minus one if its first entry is not one. St000023The number of inner peaks of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000064The number of one-box pattern of a permutation. St000134The size of the orbit of an alternating sign matrix under gyration. St000217The number of occurrences of the pattern 312 in a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000221The number of strong fixed points of a permutation. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000309The number of vertices with even degree. St000315The number of isolated vertices of a graph. St000317The cycle descent number of a permutation. St000322The skewness of a graph. St000335The difference of lower and upper interactions. St000353The number of inner valleys of a permutation. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000542The number of left-to-right-minima of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000779The tier of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000886The number of permutations with the same antidiagonal sums. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000956The maximal displacement of a permutation. St000958The number of Bruhat factorizations of a permutation. St000962The 3-shifted major index of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001060The distinguishing index of a graph. St001061The number of indices that are both descents and recoils of a permutation. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001082The number of boxed occurrences of 123 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001152The number of pairs with even minimum in a perfect matching. St001160The number of proper blocks (or intervals) of a permutations. St001162The minimum jump of a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001285The number of primes in the column sums of the two line notation of a permutation. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001330The hat guessing number of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001381The fertility of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001403The number of vertical separators in a permutation. St001405The number of bonds in a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001438The number of missing boxes of a skew partition. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001469The holeyness of a permutation. St001481The minimal height of a peak of a Dyck path. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001513The number of nested exceedences of a permutation. St001520The number of strict 3-descents. St001530The depth of a Dyck path. St001536The number of cyclic misalignments of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001577The minimal number of edges to add or remove to make a graph a cograph. St001589The nesting number of a perfect matching. St001637The number of (upper) dissectors of a poset. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001668The number of points of the poset minus the width of the poset. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001715The number of non-records in a permutation. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St001734The lettericity of a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001835The number of occurrences of a 231 pattern in the restricted growth word of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000015The number of peaks of a Dyck path. St000037The sign of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000155The number of exceedances (also excedences) of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000407The number of occurrences of the pattern 2143 in a permutation. St000439The position of the first down step of a Dyck path. St000702The number of weak deficiencies of a permutation. St000741The Colin de Verdière graph invariant. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000991The number of right-to-left minima of a permutation. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001220The width of a permutation. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001298The number of repeated entries in the Lehmer code of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001517The length of a longest pair of twins in a permutation. St001537The number of cyclic crossings of a permutation. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001667The maximal size of a pair of weak twins for a permutation. St001684The reduced word complexity of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001726The number of visible inversions of a permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001874Lusztig's a-function for the symmetric group. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000124The cardinality of the preimage of the Simion-Schmidt map. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000222The number of alignments in the permutation. St000299The number of nonisomorphic vertex-induced subtrees. St000311The number of vertices of odd degree in a graph. St000327The number of cover relations in a poset. St000401The size of the symmetry class of a permutation. St000429The number of occurrences of the pattern 123 or of the pattern 321 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000443The number of long tunnels of a Dyck path. St000545The number of parabolic double cosets with minimal element being the given permutation. St000822The Hadwiger number of the graph. St000863The length of the first row of the shifted shape of a permutation. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001180Number of indecomposable injective modules with projective dimension at most 1. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001402The number of separators in a permutation. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001535The number of cyclic alignments of a permutation. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000029The depth of a permutation. St000144The pyramid weight of the Dyck path. St000197The number of entries equal to positive one in the alternating sign matrix. St000209Maximum difference of elements in cycles. St000210Minimum over maximum difference of elements in cycles. St000216The absolute length of a permutation. St000235The number of indices that are not cyclical small weak excedances. St000238The number of indices that are not small weak excedances. St000240The number of indices that are not small excedances. St000242The number of indices that are not cyclical small weak excedances. St000494The number of inversions of distance at most 3 of a permutation. St000809The reduced reflection length of the permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000831The number of indices that are either descents or recoils. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001388The number of non-attacking neighbors of a permutation. St001480The number of simple summands of the module J^2/J^3. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001958The degree of the polynomial interpolating the values of a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000673The number of non-fixed points of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001345The Hamming dimension of a graph. St001375The pancake length of a permutation. St000224The sorting index of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001468The smallest fixpoint of a permutation. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000055The inversion sum of a permutation. St000301The number of facets of the stable set polytope of a graph. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000830The total displacement of a permutation. St000945The number of matchings in the dihedral orbit of a perfect matching. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000110The number of permutations less than or equal to a permutation in left weak order. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001639The number of alternating subsets such that applying the permutation does not yield an alternating subset. St000001The number of reduced words for a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St001959The product of the heights of the peaks of a Dyck path. St001138The number of indecomposable modules with projective dimension or injective dimension at most one in the corresponding Nakayama algebra. St000616The inversion index of a permutation. St000635The number of strictly order preserving maps of a poset into itself. St000881The number of short braid edges in the graph of braid moves of a permutation. St000341The non-inversion sum of a permutation. St000520The number of patterns in a permutation. St000109The number of elements less than or equal to the given element in Bruhat order. St000079The number of alternating sign matrices for a given Dyck path. St001731The factorization defect of a permutation. St000530The number of permutations with the same descent word as the given permutation. St000342The cosine of a permutation. St001531Number of partial orders contained in the poset determined by the Dyck path. St000690The size of the conjugacy class of a permutation. St000959The number of strong Bruhat factorizations of a permutation. St001560The product of the cardinalities of the lower order ideal and upper order ideal generated by a permutation in weak order. St001528The number of permutations such that the product with the permutation has the same number of fixed points. St001293The sum of all $1/(i+\pi(i))$ for a permutation $\pi$ times the lcm of all possible values among permutations of the same length.
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