searching the database
Your data matches 426 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001875
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,3,1,4] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[3,1,2,4] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 3
[3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[2,3,1,4,5] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[2,3,1,5,4] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[2,3,4,1,5] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[2,3,5,1,4] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[2,4,1,3,5] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[2,4,1,5,3] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[2,4,3,1,5] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[2,4,5,1,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[2,5,1,3,4] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[2,5,1,4,3] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[2,5,3,1,4] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[2,5,4,1,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[3,1,2,4,5] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[3,1,2,5,4] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[3,1,4,2,5] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[3,1,4,5,2] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[3,1,5,2,4] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[3,1,5,4,2] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[3,2,4,1,5] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[3,2,5,1,4] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[3,4,1,2,5] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[3,4,1,5,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[3,4,2,1,5] => [3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[3,4,2,5,1] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[3,4,5,1,2] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[3,5,1,2,4] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[3,5,1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[3,5,2,1,4] => [3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[3,5,2,4,1] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[3,5,4,1,2] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[4,1,2,3,5] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[4,1,2,5,3] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[4,1,3,2,5] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[4,1,3,5,2] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[4,1,5,2,3] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[4,1,5,3,2] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[4,2,1,3,5] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[4,2,1,5,3] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[4,2,3,1,5] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[4,2,3,5,1] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[4,2,5,1,3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[4,2,5,3,1] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[4,3,1,2,5] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[4,3,1,5,2] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(2,1)],3)
=> 3
[4,3,5,1,2] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St000455
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 33%
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 33%
Values
[2,3,1,4] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 3 - 3
[2,4,1,3] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 3 - 3
[3,1,2,4] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0 = 3 - 3
[3,1,4,2] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0 = 3 - 3
[3,4,1,2] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 0 = 3 - 3
[2,3,1,4,5] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 3 - 3
[2,3,1,5,4] => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[2,3,4,1,5] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 3 - 3
[2,3,5,1,4] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 3 - 3
[2,4,1,3,5] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 3 - 3
[2,4,1,5,3] => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[2,4,3,1,5] => [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[2,4,5,1,3] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 3 - 3
[2,5,1,3,4] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 3 - 3
[2,5,1,4,3] => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[2,5,3,1,4] => [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[2,5,4,1,3] => [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[3,1,2,4,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 3 - 3
[3,1,2,5,4] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[3,1,4,2,5] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[3,1,4,5,2] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[3,1,5,2,4] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[3,1,5,4,2] => [1,2,1,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[3,2,4,1,5] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[3,2,5,1,4] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[3,4,1,2,5] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 3 - 3
[3,4,1,5,2] => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[3,4,2,1,5] => [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[3,4,2,5,1] => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[3,4,5,1,2] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 0 = 3 - 3
[3,5,1,2,4] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 3 - 3
[3,5,1,4,2] => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[3,5,2,1,4] => [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[3,5,2,4,1] => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[3,5,4,1,2] => [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[4,1,2,3,5] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 3 - 3
[4,1,2,5,3] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[4,1,3,2,5] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[4,1,3,5,2] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[4,1,5,2,3] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[4,1,5,3,2] => [1,2,1,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[4,2,1,3,5] => [1,1,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[4,2,1,5,3] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[4,2,3,1,5] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[4,2,3,5,1] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[4,2,5,1,3] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[4,2,5,3,1] => [1,2,1,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[4,3,1,2,5] => [1,1,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[4,3,1,5,2] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[4,3,5,1,2] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[4,5,1,2,3] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 3 - 3
[4,5,1,3,2] => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[4,5,2,1,3] => [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[4,5,2,3,1] => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 3 - 3
[4,5,3,1,2] => [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[5,2,3,1,4] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[5,2,4,1,3] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[5,3,1,2,4] => [1,1,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[5,3,1,4,2] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[5,3,4,1,2] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 3
[2,3,1,4,5,6] => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[2,3,1,4,6,5] => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
[2,3,1,5,4,6] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
[2,3,1,5,6,4] => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
[2,3,1,6,4,5] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
[2,3,1,6,5,4] => [2,2,1,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
[2,3,4,1,5,6] => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[2,3,4,1,6,5] => [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
[2,3,4,5,1,6] => [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 0 = 3 - 3
[2,3,4,6,1,5] => [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 0 = 3 - 3
[2,3,5,1,4,6] => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[2,3,5,1,6,4] => [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
[2,3,5,4,1,6] => [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
[2,3,5,6,1,4] => [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 0 = 3 - 3
[2,3,6,1,4,5] => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[2,3,6,1,5,4] => [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
[2,3,6,4,1,5] => [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
[2,3,6,5,1,4] => [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
[2,4,1,3,5,6] => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 3 - 3
[2,4,1,3,6,5] => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
[2,4,1,5,3,6] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
[2,4,1,5,6,3] => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 3 - 3
[2,4,1,6,3,5] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
[2,4,3,1,5,6] => [2,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
[2,4,3,1,6,5] => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
[2,4,3,5,1,6] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
[2,4,3,6,1,5] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
[2,4,5,3,1,6] => [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
[2,4,6,3,1,5] => [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
[2,4,6,5,1,3] => [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
[2,5,1,4,3,6] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
[2,5,1,6,3,4] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
[2,5,3,1,4,6] => [2,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
[2,5,3,1,6,4] => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
[2,5,3,4,1,6] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
[2,5,3,6,1,4] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
[2,5,4,1,3,6] => [2,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
[2,5,4,1,6,3] => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
[2,5,4,3,1,6] => [2,1,1,2] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
[2,5,4,6,1,3] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 3
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: St001812
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001812: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001812: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 67%
Values
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 3 - 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 3 - 1
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 3 - 1
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 3 - 1
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 2 = 3 - 1
[2,3,1,4,5] => [2,5,1,4,3] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,3,1,5,4] => [2,5,1,4,3] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,3,4,1,5] => [2,5,4,1,3] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[2,3,5,1,4] => [2,5,4,1,3] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[2,4,1,3,5] => [2,5,1,4,3] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,4,1,5,3] => [2,5,1,4,3] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,4,3,1,5] => [2,5,4,1,3] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[2,4,5,1,3] => [2,5,4,1,3] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[2,5,1,3,4] => [2,5,1,4,3] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,5,1,4,3] => [2,5,1,4,3] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,5,3,1,4] => [2,5,4,1,3] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[2,5,4,1,3] => [2,5,4,1,3] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[3,1,2,4,5] => [3,1,5,4,2] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,1,2,5,4] => [3,1,5,4,2] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,1,4,2,5] => [3,1,5,4,2] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,1,4,5,2] => [3,1,5,4,2] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,1,5,2,4] => [3,1,5,4,2] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,1,5,4,2] => [3,1,5,4,2] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,2,4,1,5] => [3,2,5,1,4] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,2,5,1,4] => [3,2,5,1,4] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,4,1,2,5] => [3,5,1,4,2] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 3 - 1
[3,4,1,5,2] => [3,5,1,4,2] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 3 - 1
[3,4,2,1,5] => [3,5,2,1,4] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[3,4,2,5,1] => [3,5,2,4,1] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[3,4,5,1,2] => [3,5,4,1,2] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,5,1,2,4] => [3,5,1,4,2] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 3 - 1
[3,5,1,4,2] => [3,5,1,4,2] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 3 - 1
[3,5,2,1,4] => [3,5,2,1,4] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[3,5,2,4,1] => [3,5,2,4,1] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[3,5,4,1,2] => [3,5,4,1,2] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,1,2,3,5] => [4,1,5,3,2] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[4,1,2,5,3] => [4,1,5,3,2] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[4,1,3,2,5] => [4,1,5,3,2] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[4,1,3,5,2] => [4,1,5,3,2] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[4,1,5,2,3] => [4,1,5,3,2] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[4,1,5,3,2] => [4,1,5,3,2] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[4,2,1,3,5] => [4,2,1,5,3] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,2,1,5,3] => [4,2,1,5,3] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[4,2,3,1,5] => [4,2,5,1,3] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 3 - 1
[4,2,3,5,1] => [4,2,5,3,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[4,2,5,1,3] => [4,2,5,1,3] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 3 - 1
[4,2,5,3,1] => [4,2,5,3,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[4,3,1,2,5] => [4,3,1,5,2] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[4,3,1,5,2] => [4,3,1,5,2] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2 = 3 - 1
[4,3,5,1,2] => [4,3,5,1,2] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 2 = 3 - 1
[3,2,4,1,5,6] => [3,2,6,1,5,4] => [4,5,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[3,2,4,1,6,5] => [3,2,6,1,5,4] => [4,5,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[3,2,5,1,4,6] => [3,2,6,1,5,4] => [4,5,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[3,2,5,1,6,4] => [3,2,6,1,5,4] => [4,5,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[3,2,6,1,4,5] => [3,2,6,1,5,4] => [4,5,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[3,2,6,1,5,4] => [3,2,6,1,5,4] => [4,5,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[4,2,1,3,5,6] => [4,2,1,6,5,3] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[4,2,1,3,6,5] => [4,2,1,6,5,3] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[4,2,1,5,3,6] => [4,2,1,6,5,3] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[4,2,1,5,6,3] => [4,2,1,6,5,3] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[4,2,1,6,3,5] => [4,2,1,6,5,3] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[4,2,1,6,5,3] => [4,2,1,6,5,3] => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 - 1
[4,2,3,1,5,6] => [4,2,6,1,5,3] => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 3 - 1
[4,2,3,1,6,5] => [4,2,6,1,5,3] => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 3 - 1
[4,2,5,1,3,6] => [4,2,6,1,5,3] => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 3 - 1
[4,2,5,1,6,3] => [4,2,6,1,5,3] => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 3 - 1
[4,2,6,1,3,5] => [4,2,6,1,5,3] => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 3 - 1
[4,2,6,1,5,3] => [4,2,6,1,5,3] => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 3 - 1
[2,3,1,4,5,6,7] => [2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,1,4,5,7,6] => [2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,1,4,6,5,7] => [2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,1,4,6,7,5] => [2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,1,4,7,5,6] => [2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,1,4,7,6,5] => [2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,1,5,4,6,7] => [2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,1,5,4,7,6] => [2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,1,5,6,4,7] => [2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,1,5,6,7,4] => [2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,1,5,7,4,6] => [2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,1,5,7,6,4] => [2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,1,6,4,5,7] => [2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,1,6,4,7,5] => [2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,1,6,5,4,7] => [2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,1,6,5,7,4] => [2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,1,6,7,4,5] => [2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,1,6,7,5,4] => [2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,1,7,4,5,6] => [2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,1,7,4,6,5] => [2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,1,7,5,4,6] => [2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,1,7,5,6,4] => [2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,1,7,6,4,5] => [2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,1,7,6,5,4] => [2,7,1,6,5,4,3] => [3,4,5,6,1,7,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,4,1,5,6,7] => [2,7,6,1,5,4,3] => [3,4,5,1,6,7,2] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,4,1,5,7,6] => [2,7,6,1,5,4,3] => [3,4,5,1,6,7,2] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,4,1,6,5,7] => [2,7,6,1,5,4,3] => [3,4,5,1,6,7,2] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,4,1,6,7,5] => [2,7,6,1,5,4,3] => [3,4,5,1,6,7,2] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,4,1,7,5,6] => [2,7,6,1,5,4,3] => [3,4,5,1,6,7,2] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,4,1,7,6,5] => [2,7,6,1,5,4,3] => [3,4,5,1,6,7,2] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,4,5,1,6,7] => [2,7,6,5,1,4,3] => [3,4,1,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
[2,3,4,5,1,7,6] => [2,7,6,5,1,4,3] => [3,4,1,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3 - 1
Description
The biclique partition number of a graph.
The biclique partition number of a graph is the minimum number of pairwise edge disjoint complete bipartite subgraphs so that each edge belongs to exactly one of them. A theorem of Graham and Pollak [1] asserts that the complete graph $K_n$ has biclique partition number $n - 1$.
Matching statistic: St001260
(load all 105 compositions to match this statistic)
(load all 105 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St001260: Alternating sign matrices ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St001260: Alternating sign matrices ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 1 = 3 - 2
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 1 = 3 - 2
[3,1,2,4] => [3,1,4,2] => [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 1 = 3 - 2
[3,1,4,2] => [3,1,4,2] => [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 1 = 3 - 2
[3,4,1,2] => [3,4,1,2] => [4,1,3,2] => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 1 = 3 - 2
[2,3,1,4,5] => [2,5,1,4,3] => [3,1,4,2,5] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1 = 3 - 2
[2,3,1,5,4] => [2,5,1,4,3] => [3,1,4,2,5] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1 = 3 - 2
[2,3,4,1,5] => [2,5,4,1,3] => [3,1,2,5,4] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1 = 3 - 2
[2,3,5,1,4] => [2,5,4,1,3] => [3,1,2,5,4] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1 = 3 - 2
[2,4,1,3,5] => [2,5,1,4,3] => [3,1,4,2,5] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1 = 3 - 2
[2,4,1,5,3] => [2,5,1,4,3] => [3,1,4,2,5] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1 = 3 - 2
[2,4,3,1,5] => [2,5,4,1,3] => [3,1,2,5,4] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1 = 3 - 2
[2,4,5,1,3] => [2,5,4,1,3] => [3,1,2,5,4] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1 = 3 - 2
[2,5,1,3,4] => [2,5,1,4,3] => [3,1,4,2,5] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1 = 3 - 2
[2,5,1,4,3] => [2,5,1,4,3] => [3,1,4,2,5] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1 = 3 - 2
[2,5,3,1,4] => [2,5,4,1,3] => [3,1,2,5,4] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1 = 3 - 2
[2,5,4,1,3] => [2,5,4,1,3] => [3,1,2,5,4] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1 = 3 - 2
[3,1,2,4,5] => [3,1,5,4,2] => [4,2,1,3,5] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 1 = 3 - 2
[3,1,2,5,4] => [3,1,5,4,2] => [4,2,1,3,5] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 1 = 3 - 2
[3,1,4,2,5] => [3,1,5,4,2] => [4,2,1,3,5] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 1 = 3 - 2
[3,1,4,5,2] => [3,1,5,4,2] => [4,2,1,3,5] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 1 = 3 - 2
[3,1,5,2,4] => [3,1,5,4,2] => [4,2,1,3,5] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 1 = 3 - 2
[3,1,5,4,2] => [3,1,5,4,2] => [4,2,1,3,5] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 1 = 3 - 2
[3,2,4,1,5] => [3,2,5,1,4] => [4,3,1,5,2] => [[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> 1 = 3 - 2
[3,2,5,1,4] => [3,2,5,1,4] => [4,3,1,5,2] => [[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> 1 = 3 - 2
[3,4,1,2,5] => [3,5,1,4,2] => [4,1,3,2,5] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 1 = 3 - 2
[3,4,1,5,2] => [3,5,1,4,2] => [4,1,3,2,5] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 1 = 3 - 2
[3,4,2,1,5] => [3,5,2,1,4] => [4,1,5,3,2] => [[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> 1 = 3 - 2
[3,4,2,5,1] => [3,5,2,4,1] => [4,1,5,2,3] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> 1 = 3 - 2
[3,4,5,1,2] => [3,5,4,1,2] => [4,1,2,5,3] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> 1 = 3 - 2
[3,5,1,2,4] => [3,5,1,4,2] => [4,1,3,2,5] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 1 = 3 - 2
[3,5,1,4,2] => [3,5,1,4,2] => [4,1,3,2,5] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 1 = 3 - 2
[3,5,2,1,4] => [3,5,2,1,4] => [4,1,5,3,2] => [[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]]
=> 1 = 3 - 2
[3,5,2,4,1] => [3,5,2,4,1] => [4,1,5,2,3] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> 1 = 3 - 2
[3,5,4,1,2] => [3,5,4,1,2] => [4,1,2,5,3] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> 1 = 3 - 2
[4,1,2,3,5] => [4,1,5,3,2] => [5,2,1,3,4] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 1 = 3 - 2
[4,1,2,5,3] => [4,1,5,3,2] => [5,2,1,3,4] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 1 = 3 - 2
[4,1,3,2,5] => [4,1,5,3,2] => [5,2,1,3,4] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 1 = 3 - 2
[4,1,3,5,2] => [4,1,5,3,2] => [5,2,1,3,4] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 1 = 3 - 2
[4,1,5,2,3] => [4,1,5,3,2] => [5,2,1,3,4] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 1 = 3 - 2
[4,1,5,3,2] => [4,1,5,3,2] => [5,2,1,3,4] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 1 = 3 - 2
[4,2,1,3,5] => [4,2,1,5,3] => [5,3,2,1,4] => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 1 = 3 - 2
[4,2,1,5,3] => [4,2,1,5,3] => [5,3,2,1,4] => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 1 = 3 - 2
[4,2,3,1,5] => [4,2,5,1,3] => [5,3,1,4,2] => [[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> 1 = 3 - 2
[4,2,3,5,1] => [4,2,5,3,1] => [5,3,1,2,4] => [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 1 = 3 - 2
[4,2,5,1,3] => [4,2,5,1,3] => [5,3,1,4,2] => [[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]]
=> 1 = 3 - 2
[4,2,5,3,1] => [4,2,5,3,1] => [5,3,1,2,4] => [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 1 = 3 - 2
[4,3,1,2,5] => [4,3,1,5,2] => [5,4,2,1,3] => [[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]
=> 1 = 3 - 2
[4,3,1,5,2] => [4,3,1,5,2] => [5,4,2,1,3] => [[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]
=> 1 = 3 - 2
[4,3,5,1,2] => [4,3,5,1,2] => [5,4,1,3,2] => [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
=> 1 = 3 - 2
[3,4,2,5,1,6] => [3,6,2,5,1,4] => [4,1,5,2,6,3] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 4 - 2
[3,4,2,6,1,5] => [3,6,2,5,1,4] => [4,1,5,2,6,3] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 4 - 2
[3,4,5,2,1,6] => [3,6,5,2,1,4] => [4,1,2,6,5,3] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 3 - 2
[3,4,5,2,6,1] => [3,6,5,2,4,1] => [4,1,2,6,3,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 3 - 2
[3,4,6,2,1,5] => [3,6,5,2,1,4] => [4,1,2,6,5,3] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 3 - 2
[3,4,6,2,5,1] => [3,6,5,2,4,1] => [4,1,2,6,3,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 3 - 2
[3,5,2,4,1,6] => [3,6,2,5,1,4] => [4,1,5,2,6,3] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 4 - 2
[3,5,2,6,1,4] => [3,6,2,5,1,4] => [4,1,5,2,6,3] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 4 - 2
[3,5,4,2,1,6] => [3,6,5,2,1,4] => [4,1,2,6,5,3] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 3 - 2
[3,5,4,2,6,1] => [3,6,5,2,4,1] => [4,1,2,6,3,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 3 - 2
[3,5,6,2,1,4] => [3,6,5,2,1,4] => [4,1,2,6,5,3] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 3 - 2
[3,5,6,2,4,1] => [3,6,5,2,4,1] => [4,1,2,6,3,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 3 - 2
[3,6,2,4,1,5] => [3,6,2,5,1,4] => [4,1,5,2,6,3] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 4 - 2
[3,6,2,5,1,4] => [3,6,2,5,1,4] => [4,1,5,2,6,3] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 4 - 2
[3,6,4,2,1,5] => [3,6,5,2,1,4] => [4,1,2,6,5,3] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 3 - 2
[3,6,4,2,5,1] => [3,6,5,2,4,1] => [4,1,2,6,3,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 3 - 2
[3,6,5,2,1,4] => [3,6,5,2,1,4] => [4,1,2,6,5,3] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 3 - 2
[3,6,5,2,4,1] => [3,6,5,2,4,1] => [4,1,2,6,3,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 3 - 2
[4,2,3,5,1,6] => [4,2,6,5,1,3] => [5,3,1,2,6,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 3 - 2
[4,2,3,6,1,5] => [4,2,6,5,1,3] => [5,3,1,2,6,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 3 - 2
[4,2,5,3,1,6] => [4,2,6,5,1,3] => [5,3,1,2,6,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 3 - 2
[4,2,5,6,1,3] => [4,2,6,5,1,3] => [5,3,1,2,6,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 3 - 2
[4,2,6,3,1,5] => [4,2,6,5,1,3] => [5,3,1,2,6,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 3 - 2
[4,2,6,5,1,3] => [4,2,6,5,1,3] => [5,3,1,2,6,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 3 - 2
[4,3,2,5,1,6] => [4,3,2,6,1,5] => [5,4,3,1,6,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 3 - 2
[4,3,2,6,1,5] => [4,3,2,6,1,5] => [5,4,3,1,6,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 3 - 2
[4,3,5,2,1,6] => [4,3,6,2,1,5] => [5,4,1,6,3,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 3 - 2
[4,3,5,2,6,1] => [4,3,6,2,5,1] => [5,4,1,6,2,3] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 3 - 2
[4,3,5,6,1,2] => [4,3,6,5,1,2] => [5,4,1,2,6,3] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 3 - 2
[4,3,6,2,1,5] => [4,3,6,2,1,5] => [5,4,1,6,3,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 3 - 2
[4,3,6,2,5,1] => [4,3,6,2,5,1] => [5,4,1,6,2,3] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 3 - 2
[4,3,6,5,1,2] => [4,3,6,5,1,2] => [5,4,1,2,6,3] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 3 - 2
[4,5,2,3,1,6] => [4,6,2,5,1,3] => [5,1,4,2,6,3] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 4 - 2
[4,5,2,6,1,3] => [4,6,2,5,1,3] => [5,1,4,2,6,3] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 4 - 2
[4,5,3,1,2,6] => [4,6,3,1,5,2] => [5,1,6,3,2,4] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 4 - 2
[4,5,3,1,6,2] => [4,6,3,1,5,2] => [5,1,6,3,2,4] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 4 - 2
[4,5,3,2,1,6] => [4,6,3,2,1,5] => [5,1,6,4,3,2] => [[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 3 - 2
[4,5,3,2,6,1] => [4,6,3,2,5,1] => [5,1,6,4,2,3] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 3 - 2
[4,5,3,6,1,2] => [4,6,3,5,1,2] => [5,1,6,2,4,3] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 4 - 2
[4,5,3,6,2,1] => [4,6,3,5,2,1] => [5,1,6,2,3,4] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 3 - 2
[4,5,6,2,1,3] => [4,6,5,2,1,3] => [5,1,2,6,4,3] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 3 - 2
[4,5,6,2,3,1] => [4,6,5,2,3,1] => [5,1,2,6,3,4] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 3 - 2
[4,5,6,3,1,2] => [4,6,5,3,1,2] => [5,1,2,3,6,4] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 3 - 2
[4,6,2,3,1,5] => [4,6,2,5,1,3] => [5,1,4,2,6,3] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 4 - 2
[4,6,2,5,1,3] => [4,6,2,5,1,3] => [5,1,4,2,6,3] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 4 - 2
[4,6,3,1,2,5] => [4,6,3,1,5,2] => [5,1,6,3,2,4] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 4 - 2
[4,6,3,1,5,2] => [4,6,3,1,5,2] => [5,1,6,3,2,4] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 4 - 2
[4,6,3,2,1,5] => [4,6,3,2,1,5] => [5,1,6,4,3,2] => [[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 3 - 2
[4,6,3,2,5,1] => [4,6,3,2,5,1] => [5,1,6,4,2,3] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 3 - 2
[4,6,3,5,1,2] => [4,6,3,5,1,2] => [5,1,6,2,4,3] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 4 - 2
Description
The permanent of an alternating sign matrix.
Matching statistic: St000759
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Values
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1 = 3 - 2
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> 1 = 3 - 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 1 = 3 - 2
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> 1 = 3 - 2
[3,4,1,2] => ([(0,3),(1,2)],4)
=> [4,2]
=> 1 = 3 - 2
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1 = 3 - 2
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> 1 = 3 - 2
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1 = 3 - 2
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> 1 = 3 - 2
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> 1 = 3 - 2
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [5,3]
=> 1 = 3 - 2
[2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> 1 = 3 - 2
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> 1 = 3 - 2
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> 1 = 3 - 2
[2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [14]
=> ? = 3 - 2
[2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> 1 = 3 - 2
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> [10,4,4]
=> ? = 3 - 2
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 1 = 3 - 2
[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> 1 = 3 - 2
[3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> 1 = 3 - 2
[3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> 1 = 3 - 2
[3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [5,3]
=> 1 = 3 - 2
[3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [14]
=> ? = 3 - 2
[3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 1 = 3 - 2
[3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [14]
=> ? = 3 - 2
[3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 1 = 3 - 2
[3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> 1 = 3 - 2
[3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 1 = 3 - 2
[3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 3 - 2
[3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> [5,5]
=> 1 = 3 - 2
[3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> 1 = 3 - 2
[3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> [12,4]
=> ? = 3 - 2
[3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> [10,4,4]
=> ? = 3 - 2
[3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 3 - 2
[3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> [10,10]
=> ? = 3 - 2
[4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 1 = 3 - 2
[4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> 1 = 3 - 2
[4,1,3,2,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> 1 = 3 - 2
[4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> 1 = 3 - 2
[4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> 1 = 3 - 2
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> [10,4,4]
=> ? = 3 - 2
[4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 1 = 3 - 2
[4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [14]
=> ? = 3 - 2
[4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 1 = 3 - 2
[4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 3 - 2
[4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> [12,4]
=> ? = 3 - 2
[4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 3 - 2
[4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 1 = 3 - 2
[4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> [10,4,4]
=> ? = 3 - 2
[4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> [10,10]
=> ? = 3 - 2
[4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> [5,5]
=> 1 = 3 - 2
[4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> [10,10]
=> ? = 3 - 2
[4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> [10,10]
=> ? = 3 - 2
[4,5,2,3,1] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 3 - 2
[4,5,3,1,2] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 3 - 2
[5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 3 - 2
[5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 3 - 2
[5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 3 - 2
[5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 3 - 2
[5,3,4,1,2] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 3 - 2
[2,3,1,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 1 = 3 - 2
[2,3,1,4,6,5] => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
=> [6]
=> 1 = 3 - 2
[2,3,1,5,4,6] => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [6]
=> 1 = 3 - 2
[2,3,1,5,6,4] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> [3,3,3]
=> 1 = 3 - 2
[2,3,1,6,4,5] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> [3,3,3]
=> 1 = 3 - 2
[2,3,1,6,5,4] => ([(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [3,3,3,3,3,3]
=> ? = 3 - 2
[2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 1 = 3 - 2
[2,3,4,1,6,5] => ([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
=> [4,4]
=> 1 = 3 - 2
[2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [5]
=> 1 = 3 - 2
[2,3,4,6,1,5] => ([(0,5),(1,3),(3,4),(4,2),(4,5)],6)
=> [5,4]
=> 1 = 3 - 2
[2,3,5,1,4,6] => ([(0,5),(1,2),(2,3),(2,5),(3,4),(5,4)],6)
=> [7]
=> 1 = 3 - 2
[2,3,5,1,6,4] => ([(0,4),(0,5),(1,2),(2,3),(2,5),(3,4)],6)
=> [8,3]
=> 1 = 3 - 2
[2,3,5,4,1,6] => ([(0,5),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [5,5]
=> 1 = 3 - 2
[2,3,5,6,1,4] => ([(0,5),(1,4),(3,2),(4,3),(4,5)],6)
=> [9,3]
=> 1 = 3 - 2
[2,3,6,1,4,5] => ([(0,5),(1,4),(4,2),(4,5),(5,3)],6)
=> [4,3,3]
=> 1 = 3 - 2
[2,3,6,1,5,4] => ([(0,4),(0,5),(1,2),(2,3),(2,4),(2,5)],6)
=> [8,3,3,3,3]
=> ? = 3 - 2
[2,3,6,4,1,5] => ([(0,5),(1,4),(3,5),(4,2),(4,3)],6)
=> [14]
=> ? = 3 - 2
[2,3,6,5,1,4] => ([(0,5),(1,4),(4,2),(4,3),(4,5)],6)
=> [18,3,3]
=> ? = 3 - 2
[2,4,1,3,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [3,2]
=> 1 = 3 - 2
[2,4,1,6,3,5] => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> [8,3,2]
=> ? = 3 - 2
[2,4,1,6,5,3] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6)
=> [8,8,3,3]
=> ? = 3 - 2
[2,4,3,1,6,5] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [8,8]
=> ? = 3 - 2
[2,4,3,6,1,5] => ([(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [8,5,5]
=> ? = 3 - 2
[2,4,5,1,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(3,5)],6)
=> [6,4,3]
=> ? = 3 - 2
[2,4,5,3,1,6] => ([(0,5),(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [15]
=> ? = 3 - 2
[2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> [6,5,3]
=> ? = 3 - 2
[2,4,6,1,3,5] => ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> [8,4,2]
=> ? = 3 - 2
[2,4,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5)],6)
=> [8,6,4,3,3]
=> ? = 3 - 2
[2,4,6,3,1,5] => ([(0,5),(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> [18,5]
=> ? = 3 - 2
[2,4,6,5,1,3] => ([(0,5),(1,4),(1,5),(4,2),(4,3)],6)
=> [10,6,6,3,3]
=> ? = 3 - 2
[2,5,1,4,3,6] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [14]
=> ? = 3 - 2
[2,5,1,4,6,3] => ([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(5,3)],6)
=> [16,3]
=> ? = 3 - 2
[2,5,1,6,3,4] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6)
=> [13,2]
=> ? = 3 - 2
[2,5,1,6,4,3] => ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
=> [26,4]
=> ? = 3 - 2
[2,5,3,1,6,4] => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> [16,3]
=> ? = 3 - 2
[2,5,3,4,1,6] => ([(0,5),(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [15]
=> ? = 3 - 2
[2,5,3,6,1,4] => ([(0,4),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> [13,5,3]
=> ? = 3 - 2
[2,5,4,1,3,6] => ([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [10,4,4]
=> ? = 3 - 2
[2,5,4,1,6,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,5),(3,5)],6)
=> [6,6,4,4,3,3]
=> ? = 3 - 2
[2,5,4,3,1,6] => ([(0,5),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [15,15]
=> ? = 3 - 2
Description
The smallest missing part in an integer partition.
In [3], this is referred to as the mex, the minimal excluded part of the partition.
For compositions, this is studied in [sec.3.2., 1].
Matching statistic: St000475
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Values
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 3 - 3
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> 0 = 3 - 3
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 3 - 3
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> 0 = 3 - 3
[3,4,1,2] => ([(0,3),(1,2)],4)
=> [4,2]
=> 0 = 3 - 3
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 3 - 3
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> 0 = 3 - 3
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 3 - 3
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> 0 = 3 - 3
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> 0 = 3 - 3
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [5,3]
=> 0 = 3 - 3
[2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> 0 = 3 - 3
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> 0 = 3 - 3
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> 0 = 3 - 3
[2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [14]
=> ? = 3 - 3
[2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> 0 = 3 - 3
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> [10,4,4]
=> ? = 3 - 3
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 3 - 3
[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> 0 = 3 - 3
[3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> 0 = 3 - 3
[3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> 0 = 3 - 3
[3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [5,3]
=> 0 = 3 - 3
[3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [14]
=> ? = 3 - 3
[3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 0 = 3 - 3
[3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [14]
=> ? = 3 - 3
[3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0 = 3 - 3
[3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> 0 = 3 - 3
[3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 0 = 3 - 3
[3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 3 - 3
[3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> [5,5]
=> 0 = 3 - 3
[3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> 0 = 3 - 3
[3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> [12,4]
=> ? = 3 - 3
[3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> [10,4,4]
=> ? = 3 - 3
[3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 3 - 3
[3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> [10,10]
=> ? = 3 - 3
[4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 3 - 3
[4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> 0 = 3 - 3
[4,1,3,2,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> 0 = 3 - 3
[4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> 0 = 3 - 3
[4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> 0 = 3 - 3
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> [10,4,4]
=> ? = 3 - 3
[4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 0 = 3 - 3
[4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [14]
=> ? = 3 - 3
[4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 0 = 3 - 3
[4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 3 - 3
[4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> [12,4]
=> ? = 3 - 3
[4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 3 - 3
[4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 0 = 3 - 3
[4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> [10,4,4]
=> ? = 3 - 3
[4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> [10,10]
=> ? = 3 - 3
[4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> [5,5]
=> 0 = 3 - 3
[4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> [10,10]
=> ? = 3 - 3
[4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> [10,10]
=> ? = 3 - 3
[4,5,2,3,1] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 3 - 3
[4,5,3,1,2] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 3 - 3
[5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 3 - 3
[5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 3 - 3
[5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 3 - 3
[5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 3 - 3
[5,3,4,1,2] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 3 - 3
[2,3,1,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = 3 - 3
[2,3,1,4,6,5] => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
=> [6]
=> 0 = 3 - 3
[2,3,1,5,4,6] => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [6]
=> 0 = 3 - 3
[2,3,1,5,6,4] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> [3,3,3]
=> 0 = 3 - 3
[2,3,1,6,4,5] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> [3,3,3]
=> 0 = 3 - 3
[2,3,1,6,5,4] => ([(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [3,3,3,3,3,3]
=> ? = 3 - 3
[2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 3 - 3
[2,3,4,1,6,5] => ([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
=> [4,4]
=> 0 = 3 - 3
[2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [5]
=> 0 = 3 - 3
[2,3,4,6,1,5] => ([(0,5),(1,3),(3,4),(4,2),(4,5)],6)
=> [5,4]
=> 0 = 3 - 3
[2,3,5,1,4,6] => ([(0,5),(1,2),(2,3),(2,5),(3,4),(5,4)],6)
=> [7]
=> 0 = 3 - 3
[2,3,5,1,6,4] => ([(0,4),(0,5),(1,2),(2,3),(2,5),(3,4)],6)
=> [8,3]
=> 0 = 3 - 3
[2,3,5,4,1,6] => ([(0,5),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [5,5]
=> 0 = 3 - 3
[2,3,5,6,1,4] => ([(0,5),(1,4),(3,2),(4,3),(4,5)],6)
=> [9,3]
=> 0 = 3 - 3
[2,3,6,1,4,5] => ([(0,5),(1,4),(4,2),(4,5),(5,3)],6)
=> [4,3,3]
=> 0 = 3 - 3
[2,3,6,1,5,4] => ([(0,4),(0,5),(1,2),(2,3),(2,4),(2,5)],6)
=> [8,3,3,3,3]
=> ? = 3 - 3
[2,3,6,4,1,5] => ([(0,5),(1,4),(3,5),(4,2),(4,3)],6)
=> [14]
=> ? = 3 - 3
[2,3,6,5,1,4] => ([(0,5),(1,4),(4,2),(4,3),(4,5)],6)
=> [18,3,3]
=> ? = 3 - 3
[2,4,1,3,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [3,2]
=> 0 = 3 - 3
[2,4,1,6,3,5] => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> [8,3,2]
=> ? = 3 - 3
[2,4,1,6,5,3] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6)
=> [8,8,3,3]
=> ? = 3 - 3
[2,4,3,1,6,5] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [8,8]
=> ? = 3 - 3
[2,4,3,6,1,5] => ([(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [8,5,5]
=> ? = 3 - 3
[2,4,5,1,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(3,5)],6)
=> [6,4,3]
=> ? = 3 - 3
[2,4,5,3,1,6] => ([(0,5),(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [15]
=> ? = 3 - 3
[2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> [6,5,3]
=> ? = 3 - 3
[2,4,6,1,3,5] => ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> [8,4,2]
=> ? = 3 - 3
[2,4,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5)],6)
=> [8,6,4,3,3]
=> ? = 3 - 3
[2,4,6,3,1,5] => ([(0,5),(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> [18,5]
=> ? = 3 - 3
[2,4,6,5,1,3] => ([(0,5),(1,4),(1,5),(4,2),(4,3)],6)
=> [10,6,6,3,3]
=> ? = 3 - 3
[2,5,1,4,3,6] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [14]
=> ? = 3 - 3
[2,5,1,4,6,3] => ([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(5,3)],6)
=> [16,3]
=> ? = 3 - 3
[2,5,1,6,3,4] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6)
=> [13,2]
=> ? = 3 - 3
[2,5,1,6,4,3] => ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
=> [26,4]
=> ? = 3 - 3
[2,5,3,1,6,4] => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> [16,3]
=> ? = 3 - 3
[2,5,3,4,1,6] => ([(0,5),(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [15]
=> ? = 3 - 3
[2,5,3,6,1,4] => ([(0,4),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> [13,5,3]
=> ? = 3 - 3
[2,5,4,1,3,6] => ([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [10,4,4]
=> ? = 3 - 3
[2,5,4,1,6,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,5),(3,5)],6)
=> [6,6,4,4,3,3]
=> ? = 3 - 3
[2,5,4,3,1,6] => ([(0,5),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [15,15]
=> ? = 3 - 3
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000929
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Values
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 3 - 3
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> 0 = 3 - 3
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> 0 = 3 - 3
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> 0 = 3 - 3
[3,4,1,2] => ([(0,3),(1,2)],4)
=> [4,2]
=> 0 = 3 - 3
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 3 - 3
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> 0 = 3 - 3
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 3 - 3
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> 0 = 3 - 3
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> 0 = 3 - 3
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [5,3]
=> 0 = 3 - 3
[2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> 0 = 3 - 3
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> 0 = 3 - 3
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> 0 = 3 - 3
[2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [14]
=> ? = 3 - 3
[2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> 0 = 3 - 3
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> [10,4,4]
=> ? = 3 - 3
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> 0 = 3 - 3
[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> 0 = 3 - 3
[3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> 0 = 3 - 3
[3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> 0 = 3 - 3
[3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [5,3]
=> 0 = 3 - 3
[3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [14]
=> ? = 3 - 3
[3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 0 = 3 - 3
[3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [14]
=> ? = 3 - 3
[3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 0 = 3 - 3
[3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> 0 = 3 - 3
[3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 0 = 3 - 3
[3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 3 - 3
[3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> [5,5]
=> 0 = 3 - 3
[3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> 0 = 3 - 3
[3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> [12,4]
=> ? = 3 - 3
[3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> [10,4,4]
=> ? = 3 - 3
[3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 3 - 3
[3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> [10,10]
=> ? = 3 - 3
[4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 0 = 3 - 3
[4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> 0 = 3 - 3
[4,1,3,2,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> 0 = 3 - 3
[4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> 0 = 3 - 3
[4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> 0 = 3 - 3
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> [10,4,4]
=> ? = 3 - 3
[4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 0 = 3 - 3
[4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [14]
=> ? = 3 - 3
[4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 0 = 3 - 3
[4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 3 - 3
[4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> [12,4]
=> ? = 3 - 3
[4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 3 - 3
[4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 0 = 3 - 3
[4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> [10,4,4]
=> ? = 3 - 3
[4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> [10,10]
=> ? = 3 - 3
[4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> [5,5]
=> 0 = 3 - 3
[4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> [10,10]
=> ? = 3 - 3
[4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> [10,10]
=> ? = 3 - 3
[4,5,2,3,1] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 3 - 3
[4,5,3,1,2] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 3 - 3
[5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 3 - 3
[5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 3 - 3
[5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> ? = 3 - 3
[5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> ? = 3 - 3
[5,3,4,1,2] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> ? = 3 - 3
[2,3,1,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> 0 = 3 - 3
[2,3,1,4,6,5] => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
=> [6]
=> 0 = 3 - 3
[2,3,1,5,4,6] => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [6]
=> 0 = 3 - 3
[2,3,1,5,6,4] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> [3,3,3]
=> 0 = 3 - 3
[2,3,1,6,4,5] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> [3,3,3]
=> 0 = 3 - 3
[2,3,1,6,5,4] => ([(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [3,3,3,3,3,3]
=> ? = 3 - 3
[2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> 0 = 3 - 3
[2,3,4,1,6,5] => ([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
=> [4,4]
=> 0 = 3 - 3
[2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [5]
=> 0 = 3 - 3
[2,3,4,6,1,5] => ([(0,5),(1,3),(3,4),(4,2),(4,5)],6)
=> [5,4]
=> 0 = 3 - 3
[2,3,5,1,4,6] => ([(0,5),(1,2),(2,3),(2,5),(3,4),(5,4)],6)
=> [7]
=> 0 = 3 - 3
[2,3,5,1,6,4] => ([(0,4),(0,5),(1,2),(2,3),(2,5),(3,4)],6)
=> [8,3]
=> 0 = 3 - 3
[2,3,5,4,1,6] => ([(0,5),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [5,5]
=> 0 = 3 - 3
[2,3,5,6,1,4] => ([(0,5),(1,4),(3,2),(4,3),(4,5)],6)
=> [9,3]
=> 0 = 3 - 3
[2,3,6,1,4,5] => ([(0,5),(1,4),(4,2),(4,5),(5,3)],6)
=> [4,3,3]
=> 0 = 3 - 3
[2,3,6,1,5,4] => ([(0,4),(0,5),(1,2),(2,3),(2,4),(2,5)],6)
=> [8,3,3,3,3]
=> ? = 3 - 3
[2,3,6,4,1,5] => ([(0,5),(1,4),(3,5),(4,2),(4,3)],6)
=> [14]
=> ? = 3 - 3
[2,3,6,5,1,4] => ([(0,5),(1,4),(4,2),(4,3),(4,5)],6)
=> [18,3,3]
=> ? = 3 - 3
[2,4,1,3,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [3,2]
=> 0 = 3 - 3
[2,4,1,6,3,5] => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> [8,3,2]
=> ? = 3 - 3
[2,4,1,6,5,3] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6)
=> [8,8,3,3]
=> ? = 3 - 3
[2,4,3,1,6,5] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [8,8]
=> ? = 3 - 3
[2,4,3,6,1,5] => ([(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [8,5,5]
=> ? = 3 - 3
[2,4,5,1,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(3,5)],6)
=> [6,4,3]
=> ? = 3 - 3
[2,4,5,3,1,6] => ([(0,5),(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [15]
=> ? = 3 - 3
[2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> [6,5,3]
=> ? = 3 - 3
[2,4,6,1,3,5] => ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> [8,4,2]
=> ? = 3 - 3
[2,4,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5)],6)
=> [8,6,4,3,3]
=> ? = 3 - 3
[2,4,6,3,1,5] => ([(0,5),(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> [18,5]
=> ? = 3 - 3
[2,4,6,5,1,3] => ([(0,5),(1,4),(1,5),(4,2),(4,3)],6)
=> [10,6,6,3,3]
=> ? = 3 - 3
[2,5,1,4,3,6] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [14]
=> ? = 3 - 3
[2,5,1,4,6,3] => ([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(5,3)],6)
=> [16,3]
=> ? = 3 - 3
[2,5,1,6,3,4] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6)
=> [13,2]
=> ? = 3 - 3
[2,5,1,6,4,3] => ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
=> [26,4]
=> ? = 3 - 3
[2,5,3,1,6,4] => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> [16,3]
=> ? = 3 - 3
[2,5,3,4,1,6] => ([(0,5),(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [15]
=> ? = 3 - 3
[2,5,3,6,1,4] => ([(0,4),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> [13,5,3]
=> ? = 3 - 3
[2,5,4,1,3,6] => ([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [10,4,4]
=> ? = 3 - 3
[2,5,4,1,6,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,5),(3,5)],6)
=> [6,6,4,4,3,3]
=> ? = 3 - 3
[2,5,4,3,1,6] => ([(0,5),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [15,15]
=> ? = 3 - 3
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St000713
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000713: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000713: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Values
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 0 = 3 - 3
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [2,2,1]
=> 0 = 3 - 3
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 0 = 3 - 3
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [2,2,1]
=> 0 = 3 - 3
[3,4,1,2] => ([(0,3),(1,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 0 = 3 - 3
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 0 = 3 - 3
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 0 = 3 - 3
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0 = 3 - 3
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0 = 3 - 3
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 0 = 3 - 3
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [5,3]
=> [2,2,2,1,1]
=> 0 = 3 - 3
[2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0 = 3 - 3
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 0 = 3 - 3
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0 = 3 - 3
[2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> [3,3,3,2]
=> 0 = 3 - 3
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> [10,4,4]
=> [3,3,3,3,1,1,1,1,1,1]
=> ? = 3 - 3
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 0 = 3 - 3
[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 0 = 3 - 3
[3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 0 = 3 - 3
[3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0 = 3 - 3
[3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [5,3]
=> [2,2,2,1,1]
=> 0 = 3 - 3
[3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0 = 3 - 3
[3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> 0 = 3 - 3
[3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 0 = 3 - 3
[3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> [3,3,3,3]
=> 0 = 3 - 3
[3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> [5,5]
=> [2,2,2,2,2]
=> 0 = 3 - 3
[3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 0 = 3 - 3
[3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> [12,4]
=> [2,2,2,2,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> [10,4,4]
=> [3,3,3,3,1,1,1,1,1,1]
=> ? = 3 - 3
[3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> [3,3,3,3,3,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 3 - 3
[4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0 = 3 - 3
[4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0 = 3 - 3
[4,1,3,2,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0 = 3 - 3
[4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> [3,3,3,2]
=> 0 = 3 - 3
[4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 0 = 3 - 3
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> [10,4,4]
=> [3,3,3,3,1,1,1,1,1,1]
=> ? = 3 - 3
[4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0 = 3 - 3
[4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> [3,3,3,3]
=> 0 = 3 - 3
[4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> [12,4]
=> [2,2,2,2,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> [3,3,3,3,3,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> [3,3,3,3]
=> 0 = 3 - 3
[4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> [10,4,4]
=> [3,3,3,3,1,1,1,1,1,1]
=> ? = 3 - 3
[4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 3 - 3
[4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> [5,5]
=> [2,2,2,2,2]
=> 0 = 3 - 3
[4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 3 - 3
[4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 3 - 3
[4,5,2,3,1] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> [6,6,6,6,6]
=> ? = 3 - 3
[4,5,3,1,2] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> [6,6,6,6,6]
=> ? = 3 - 3
[5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> [3,3,3,3,3,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> [3,3,3,3,3,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[5,3,4,1,2] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> [6,6,6,6,6]
=> ? = 3 - 3
[2,3,1,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> [1,1,1]
=> 0 = 3 - 3
[2,3,1,4,6,5] => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 0 = 3 - 3
[2,3,1,5,4,6] => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 0 = 3 - 3
[2,3,1,5,6,4] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> [3,3,3]
=> [3,3,3]
=> 0 = 3 - 3
[2,3,1,6,4,5] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> [3,3,3]
=> [3,3,3]
=> 0 = 3 - 3
[2,3,1,6,5,4] => ([(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [3,3,3,3,3,3]
=> [6,6,6]
=> ? = 3 - 3
[2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> 0 = 3 - 3
[2,3,4,1,6,5] => ([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
=> [4,4]
=> [2,2,2,2]
=> 0 = 3 - 3
[2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [5]
=> [1,1,1,1,1]
=> 0 = 3 - 3
[2,3,4,6,1,5] => ([(0,5),(1,3),(3,4),(4,2),(4,5)],6)
=> [5,4]
=> [2,2,2,2,1]
=> 0 = 3 - 3
[2,3,5,1,4,6] => ([(0,5),(1,2),(2,3),(2,5),(3,4),(5,4)],6)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0 = 3 - 3
[2,3,5,1,6,4] => ([(0,4),(0,5),(1,2),(2,3),(2,5),(3,4)],6)
=> [8,3]
=> [2,2,2,1,1,1,1,1]
=> 0 = 3 - 3
[2,3,5,4,1,6] => ([(0,5),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [5,5]
=> [2,2,2,2,2]
=> 0 = 3 - 3
[2,3,5,6,1,4] => ([(0,5),(1,4),(3,2),(4,3),(4,5)],6)
=> [9,3]
=> [2,2,2,1,1,1,1,1,1]
=> 0 = 3 - 3
[2,3,6,1,4,5] => ([(0,5),(1,4),(4,2),(4,5),(5,3)],6)
=> [4,3,3]
=> [3,3,3,1]
=> 0 = 3 - 3
[2,3,6,1,5,4] => ([(0,4),(0,5),(1,2),(2,3),(2,4),(2,5)],6)
=> [8,3,3,3,3]
=> ?
=> ? = 3 - 3
[2,3,6,4,1,5] => ([(0,5),(1,4),(3,5),(4,2),(4,3)],6)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[2,3,6,5,1,4] => ([(0,5),(1,4),(4,2),(4,3),(4,5)],6)
=> [18,3,3]
=> ?
=> ? = 3 - 3
[2,4,1,3,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [3,2]
=> [2,2,1]
=> 0 = 3 - 3
[2,4,1,6,3,5] => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> [8,3,2]
=> [3,3,2,1,1,1,1,1]
=> ? = 3 - 3
[2,4,1,6,5,3] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6)
=> [8,8,3,3]
=> ?
=> ? = 3 - 3
[2,4,3,1,6,5] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [8,8]
=> [2,2,2,2,2,2,2,2]
=> ? = 3 - 3
[2,4,3,6,1,5] => ([(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [8,5,5]
=> ?
=> ? = 3 - 3
[2,4,5,1,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(3,5)],6)
=> [6,4,3]
=> [3,3,3,2,1,1]
=> ? = 3 - 3
[2,4,5,3,1,6] => ([(0,5),(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> [6,5,3]
=> [3,3,3,2,2,1]
=> ? = 3 - 3
[2,4,6,1,3,5] => ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> ? = 3 - 3
[2,4,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5)],6)
=> [8,6,4,3,3]
=> ?
=> ? = 3 - 3
[2,4,6,3,1,5] => ([(0,5),(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> [18,5]
=> ?
=> ? = 3 - 3
[2,4,6,5,1,3] => ([(0,5),(1,4),(1,5),(4,2),(4,3)],6)
=> [10,6,6,3,3]
=> ?
=> ? = 3 - 3
[2,5,1,4,3,6] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[2,5,1,4,6,3] => ([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(5,3)],6)
=> [16,3]
=> ?
=> ? = 3 - 3
[2,5,1,6,3,4] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6)
=> [13,2]
=> [2,2,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[2,5,1,6,4,3] => ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
=> [26,4]
=> ?
=> ? = 3 - 3
[2,5,3,1,6,4] => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> [16,3]
=> ?
=> ? = 3 - 3
[2,5,3,4,1,6] => ([(0,5),(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[2,5,3,6,1,4] => ([(0,4),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> [13,5,3]
=> ?
=> ? = 3 - 3
[2,5,4,1,3,6] => ([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [10,4,4]
=> [3,3,3,3,1,1,1,1,1,1]
=> ? = 3 - 3
[2,5,4,1,6,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,5),(3,5)],6)
=> [6,6,4,4,3,3]
=> ?
=> ? = 3 - 3
[2,5,4,3,1,6] => ([(0,5),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [15,15]
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ? = 3 - 3
Description
The dimension of the irreducible representation of Sp(4) labelled by an integer partition.
Consider the symplectic group $Sp(2n)$. Then the integer partition $(\mu_1,\dots,\mu_k)$ of length at most $n$ corresponds to the weight vector $(\mu_1-\mu_2,\dots,\mu_{k-2}-\mu_{k-1},\mu_n,0,\dots,0)$.
For example, the integer partition $(2)$ labels the symmetric square of the vector representation, whereas the integer partition $(1,1)$ labels the second fundamental representation.
Matching statistic: St000714
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000714: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000714: Integer partitions ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Values
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 0 = 3 - 3
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [2,2,1]
=> 0 = 3 - 3
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 0 = 3 - 3
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> [2,2,1]
=> 0 = 3 - 3
[3,4,1,2] => ([(0,3),(1,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 0 = 3 - 3
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 0 = 3 - 3
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 0 = 3 - 3
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0 = 3 - 3
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0 = 3 - 3
[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 0 = 3 - 3
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [5,3]
=> [2,2,2,1,1]
=> 0 = 3 - 3
[2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0 = 3 - 3
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 0 = 3 - 3
[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0 = 3 - 3
[2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> [3,3,3,2]
=> 0 = 3 - 3
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> [10,4,4]
=> [3,3,3,3,1,1,1,1,1,1]
=> ? = 3 - 3
[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 0 = 3 - 3
[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 0 = 3 - 3
[3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 0 = 3 - 3
[3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0 = 3 - 3
[3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [5,3]
=> [2,2,2,1,1]
=> 0 = 3 - 3
[3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0 = 3 - 3
[3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> 0 = 3 - 3
[3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 0 = 3 - 3
[3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> [3,3,3,3]
=> 0 = 3 - 3
[3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> [5,5]
=> [2,2,2,2,2]
=> 0 = 3 - 3
[3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 0 = 3 - 3
[3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> [12,4]
=> [2,2,2,2,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> [10,4,4]
=> [3,3,3,3,1,1,1,1,1,1]
=> ? = 3 - 3
[3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> [3,3,3,3,3,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 3 - 3
[4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0 = 3 - 3
[4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0 = 3 - 3
[4,1,3,2,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0 = 3 - 3
[4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> [3,3,3,2]
=> 0 = 3 - 3
[4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 0 = 3 - 3
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> [10,4,4]
=> [3,3,3,3,1,1,1,1,1,1]
=> ? = 3 - 3
[4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0 = 3 - 3
[4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> [3,3,3,3]
=> 0 = 3 - 3
[4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> [12,4]
=> [2,2,2,2,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> [3,3,3,3,3,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> [3,3,3,3]
=> 0 = 3 - 3
[4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> [10,4,4]
=> [3,3,3,3,1,1,1,1,1,1]
=> ? = 3 - 3
[4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 3 - 3
[4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> [5,5]
=> [2,2,2,2,2]
=> 0 = 3 - 3
[4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 3 - 3
[4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 3 - 3
[4,5,2,3,1] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> [6,6,6,6,6]
=> ? = 3 - 3
[4,5,3,1,2] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> [6,6,6,6,6]
=> ? = 3 - 3
[5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> [3,3,3,3,3,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> [3,3,3,3,3,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[5,3,4,1,2] => ([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> [6,6,6,6,6]
=> ? = 3 - 3
[2,3,1,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> [1,1,1]
=> 0 = 3 - 3
[2,3,1,4,6,5] => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 0 = 3 - 3
[2,3,1,5,4,6] => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 0 = 3 - 3
[2,3,1,5,6,4] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> [3,3,3]
=> [3,3,3]
=> 0 = 3 - 3
[2,3,1,6,4,5] => ([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
=> [3,3,3]
=> [3,3,3]
=> 0 = 3 - 3
[2,3,1,6,5,4] => ([(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [3,3,3,3,3,3]
=> [6,6,6]
=> ? = 3 - 3
[2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> 0 = 3 - 3
[2,3,4,1,6,5] => ([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
=> [4,4]
=> [2,2,2,2]
=> 0 = 3 - 3
[2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [5]
=> [1,1,1,1,1]
=> 0 = 3 - 3
[2,3,4,6,1,5] => ([(0,5),(1,3),(3,4),(4,2),(4,5)],6)
=> [5,4]
=> [2,2,2,2,1]
=> 0 = 3 - 3
[2,3,5,1,4,6] => ([(0,5),(1,2),(2,3),(2,5),(3,4),(5,4)],6)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0 = 3 - 3
[2,3,5,1,6,4] => ([(0,4),(0,5),(1,2),(2,3),(2,5),(3,4)],6)
=> [8,3]
=> [2,2,2,1,1,1,1,1]
=> 0 = 3 - 3
[2,3,5,4,1,6] => ([(0,5),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [5,5]
=> [2,2,2,2,2]
=> 0 = 3 - 3
[2,3,5,6,1,4] => ([(0,5),(1,4),(3,2),(4,3),(4,5)],6)
=> [9,3]
=> [2,2,2,1,1,1,1,1,1]
=> 0 = 3 - 3
[2,3,6,1,4,5] => ([(0,5),(1,4),(4,2),(4,5),(5,3)],6)
=> [4,3,3]
=> [3,3,3,1]
=> 0 = 3 - 3
[2,3,6,1,5,4] => ([(0,4),(0,5),(1,2),(2,3),(2,4),(2,5)],6)
=> [8,3,3,3,3]
=> ?
=> ? = 3 - 3
[2,3,6,4,1,5] => ([(0,5),(1,4),(3,5),(4,2),(4,3)],6)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[2,3,6,5,1,4] => ([(0,5),(1,4),(4,2),(4,3),(4,5)],6)
=> [18,3,3]
=> ?
=> ? = 3 - 3
[2,4,1,3,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [3,2]
=> [2,2,1]
=> 0 = 3 - 3
[2,4,1,6,3,5] => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> [8,3,2]
=> [3,3,2,1,1,1,1,1]
=> ? = 3 - 3
[2,4,1,6,5,3] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6)
=> [8,8,3,3]
=> ?
=> ? = 3 - 3
[2,4,3,1,6,5] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [8,8]
=> [2,2,2,2,2,2,2,2]
=> ? = 3 - 3
[2,4,3,6,1,5] => ([(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [8,5,5]
=> ?
=> ? = 3 - 3
[2,4,5,1,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(3,5)],6)
=> [6,4,3]
=> [3,3,3,2,1,1]
=> ? = 3 - 3
[2,4,5,3,1,6] => ([(0,5),(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6)
=> [6,5,3]
=> [3,3,3,2,2,1]
=> ? = 3 - 3
[2,4,6,1,3,5] => ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> ? = 3 - 3
[2,4,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5)],6)
=> [8,6,4,3,3]
=> ?
=> ? = 3 - 3
[2,4,6,3,1,5] => ([(0,5),(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> [18,5]
=> ?
=> ? = 3 - 3
[2,4,6,5,1,3] => ([(0,5),(1,4),(1,5),(4,2),(4,3)],6)
=> [10,6,6,3,3]
=> ?
=> ? = 3 - 3
[2,5,1,4,3,6] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[2,5,1,4,6,3] => ([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(5,3)],6)
=> [16,3]
=> ?
=> ? = 3 - 3
[2,5,1,6,3,4] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6)
=> [13,2]
=> [2,2,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[2,5,1,6,4,3] => ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
=> [26,4]
=> ?
=> ? = 3 - 3
[2,5,3,1,6,4] => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> [16,3]
=> ?
=> ? = 3 - 3
[2,5,3,4,1,6] => ([(0,5),(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 3 - 3
[2,5,3,6,1,4] => ([(0,4),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
=> [13,5,3]
=> ?
=> ? = 3 - 3
[2,5,4,1,3,6] => ([(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [10,4,4]
=> [3,3,3,3,1,1,1,1,1,1]
=> ? = 3 - 3
[2,5,4,1,6,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,5),(3,5)],6)
=> [6,6,4,4,3,3]
=> ?
=> ? = 3 - 3
[2,5,4,3,1,6] => ([(0,5),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [15,15]
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> ? = 3 - 3
Description
The number of semistandard Young tableau of given shape, with entries at most 2.
This is also the dimension of the corresponding irreducible representation of $GL_2$.
Matching statistic: St001651
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 3 - 3
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 3 - 3
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 3 - 3
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 3 - 3
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 3 - 3
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 3 - 3
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 3 - 3
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 3 - 3
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0 = 3 - 3
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 3 - 3
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0 = 3 - 3
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 3 - 3
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 3 - 3
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0 = 3 - 3
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 3
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 3
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 3 - 3
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 3 - 3
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 3 - 3
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 3 - 3
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0 = 3 - 3
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0 = 3 - 3
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 3
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 3 - 3
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 3
[3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 3 - 3
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 3 - 3
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 3 - 3
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 3 - 3
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 3 - 3
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 3 - 3
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,16),(1,17),(1,18),(1,25),(2,13),(2,14),(2,15),(2,25),(3,10),(3,11),(3,12),(3,25),(4,8),(4,9),(4,12),(4,15),(4,18),(5,7),(5,9),(5,11),(5,14),(5,17),(6,7),(6,8),(6,10),(6,13),(6,16),(7,21),(7,24),(7,29),(8,19),(8,22),(8,29),(9,20),(9,23),(9,29),(10,26),(10,29),(11,27),(11,29),(12,28),(12,29),(13,19),(13,21),(13,26),(14,20),(14,21),(14,27),(15,19),(15,20),(15,28),(16,22),(16,24),(16,26),(17,23),(17,24),(17,27),(18,22),(18,23),(18,28),(19,30),(20,30),(21,30),(22,30),(23,30),(24,30),(25,26),(25,27),(25,28),(26,30),(27,30),(28,30),(29,30)],31)
=> ? = 3 - 3
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 3 - 3
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 3 - 3
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,28),(1,29),(2,12),(2,16),(2,20),(2,22),(2,29),(3,11),(3,15),(3,20),(3,21),(3,28),(4,13),(4,17),(4,19),(4,21),(4,29),(5,14),(5,18),(5,19),(5,22),(5,28),(6,8),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,27),(8,34),(8,35),(9,27),(9,30),(9,31),(10,27),(10,32),(10,33),(11,23),(11,30),(11,34),(12,24),(12,31),(12,34),(13,23),(13,31),(13,35),(14,24),(14,30),(14,35),(15,25),(15,32),(15,34),(16,26),(16,33),(16,34),(17,25),(17,33),(17,35),(18,26),(18,32),(18,35),(19,35),(19,36),(20,34),(20,36),(21,23),(21,25),(21,36),(22,24),(22,26),(22,36),(23,37),(24,37),(25,37),(26,37),(27,37),(28,30),(28,32),(28,36),(29,31),(29,33),(29,36),(30,37),(31,37),(32,37),(33,37),(34,37),(35,37),(36,37)],38)
=> ? = 3 - 3
[4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 3 - 3
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0 = 3 - 3
[4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 3 - 3
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 3
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 3 - 3
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 3 - 3
[4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 3 - 3
[4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 3
[4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 3 - 3
[4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 3 - 3
[4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,16),(1,17),(1,18),(1,25),(2,13),(2,14),(2,15),(2,25),(3,10),(3,11),(3,12),(3,25),(4,8),(4,9),(4,12),(4,15),(4,18),(5,7),(5,9),(5,11),(5,14),(5,17),(6,7),(6,8),(6,10),(6,13),(6,16),(7,21),(7,24),(7,29),(8,19),(8,22),(8,29),(9,20),(9,23),(9,29),(10,26),(10,29),(11,27),(11,29),(12,28),(12,29),(13,19),(13,21),(13,26),(14,20),(14,21),(14,27),(15,19),(15,20),(15,28),(16,22),(16,24),(16,26),(17,23),(17,24),(17,27),(18,22),(18,23),(18,28),(19,30),(20,30),(21,30),(22,30),(23,30),(24,30),(25,26),(25,27),(25,28),(26,30),(27,30),(28,30),(29,30)],31)
=> ? = 3 - 3
[4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 3 - 3
[4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 3 - 3
[4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 3 - 3
[4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,28),(1,29),(2,12),(2,16),(2,20),(2,22),(2,29),(3,11),(3,15),(3,20),(3,21),(3,28),(4,13),(4,17),(4,19),(4,21),(4,29),(5,14),(5,18),(5,19),(5,22),(5,28),(6,8),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,27),(8,34),(8,35),(9,27),(9,30),(9,31),(10,27),(10,32),(10,33),(11,23),(11,30),(11,34),(12,24),(12,31),(12,34),(13,23),(13,31),(13,35),(14,24),(14,30),(14,35),(15,25),(15,32),(15,34),(16,26),(16,33),(16,34),(17,25),(17,33),(17,35),(18,26),(18,32),(18,35),(19,35),(19,36),(20,34),(20,36),(21,23),(21,25),(21,36),(22,24),(22,26),(22,36),(23,37),(24,37),(25,37),(26,37),(27,37),(28,30),(28,32),(28,36),(29,31),(29,33),(29,36),(30,37),(31,37),(32,37),(33,37),(34,37),(35,37),(36,37)],38)
=> ? = 3 - 3
[4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,15),(1,20),(1,21),(2,9),(2,10),(2,14),(2,18),(2,19),(3,8),(3,13),(3,17),(3,19),(3,21),(4,8),(4,12),(4,16),(4,18),(4,20),(5,7),(5,14),(5,15),(5,16),(5,17),(6,7),(6,10),(6,11),(6,12),(6,13),(7,31),(7,32),(8,30),(8,32),(9,30),(9,31),(10,22),(10,23),(10,31),(11,24),(11,25),(11,31),(12,22),(12,24),(12,32),(13,23),(13,25),(13,32),(14,26),(14,27),(14,31),(15,28),(15,29),(15,31),(16,26),(16,28),(16,32),(17,27),(17,29),(17,32),(18,22),(18,26),(18,30),(19,23),(19,27),(19,30),(20,24),(20,28),(20,30),(21,25),(21,29),(21,30),(22,33),(23,33),(24,33),(25,33),(26,33),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 3 - 3
[4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,28),(1,29),(2,12),(2,16),(2,20),(2,22),(2,29),(3,11),(3,15),(3,20),(3,21),(3,28),(4,13),(4,17),(4,19),(4,21),(4,29),(5,14),(5,18),(5,19),(5,22),(5,28),(6,8),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,27),(8,34),(8,35),(9,27),(9,30),(9,31),(10,27),(10,32),(10,33),(11,23),(11,30),(11,34),(12,24),(12,31),(12,34),(13,23),(13,31),(13,35),(14,24),(14,30),(14,35),(15,25),(15,32),(15,34),(16,26),(16,33),(16,34),(17,25),(17,33),(17,35),(18,26),(18,32),(18,35),(19,35),(19,36),(20,34),(20,36),(21,23),(21,25),(21,36),(22,24),(22,26),(22,36),(23,37),(24,37),(25,37),(26,37),(27,37),(28,30),(28,32),(28,36),(29,31),(29,33),(29,36),(30,37),(31,37),(32,37),(33,37),(34,37),(35,37),(36,37)],38)
=> ? = 3 - 3
[4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,28),(1,29),(2,12),(2,16),(2,20),(2,22),(2,29),(3,11),(3,15),(3,20),(3,21),(3,28),(4,13),(4,17),(4,19),(4,21),(4,29),(5,14),(5,18),(5,19),(5,22),(5,28),(6,8),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,27),(8,34),(8,35),(9,27),(9,30),(9,31),(10,27),(10,32),(10,33),(11,23),(11,30),(11,34),(12,24),(12,31),(12,34),(13,23),(13,31),(13,35),(14,24),(14,30),(14,35),(15,25),(15,32),(15,34),(16,26),(16,33),(16,34),(17,25),(17,33),(17,35),(18,26),(18,32),(18,35),(19,35),(19,36),(20,34),(20,36),(21,23),(21,25),(21,36),(22,24),(22,26),(22,36),(23,37),(24,37),(25,37),(26,37),(27,37),(28,30),(28,32),(28,36),(29,31),(29,33),(29,36),(30,37),(31,37),(32,37),(33,37),(34,37),(35,37),(36,37)],38)
=> ? = 3 - 3
[4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,16),(1,20),(1,24),(1,30),(1,32),(2,15),(2,19),(2,23),(2,30),(2,31),(3,17),(3,21),(3,23),(3,29),(3,32),(4,18),(4,22),(4,24),(4,29),(4,31),(5,10),(5,13),(5,14),(5,17),(5,18),(5,30),(6,10),(6,11),(6,12),(6,15),(6,16),(6,29),(7,9),(7,11),(7,13),(7,19),(7,22),(7,32),(8,9),(8,12),(8,14),(8,20),(8,21),(8,31),(9,35),(9,36),(9,41),(10,33),(10,34),(10,41),(11,25),(11,38),(11,41),(12,26),(12,37),(12,41),(13,28),(13,39),(13,41),(14,27),(14,40),(14,41),(15,25),(15,33),(15,37),(16,26),(16,33),(16,38),(17,27),(17,34),(17,39),(18,28),(18,34),(18,40),(19,25),(19,35),(19,39),(20,26),(20,36),(20,40),(21,27),(21,36),(21,37),(22,28),(22,35),(22,38),(23,37),(23,39),(24,38),(24,40),(25,42),(26,42),(27,42),(28,42),(29,34),(29,37),(29,38),(30,33),(30,39),(30,40),(31,35),(31,37),(31,40),(32,36),(32,38),(32,39),(33,42),(34,42),(35,42),(36,42),(37,42),(38,42),(39,42),(40,42),(41,42)],43)
=> ? = 3 - 3
[4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,16),(1,20),(1,24),(1,30),(1,32),(2,15),(2,19),(2,23),(2,30),(2,31),(3,17),(3,21),(3,23),(3,29),(3,32),(4,18),(4,22),(4,24),(4,29),(4,31),(5,10),(5,13),(5,14),(5,17),(5,18),(5,30),(6,10),(6,11),(6,12),(6,15),(6,16),(6,29),(7,9),(7,11),(7,13),(7,19),(7,22),(7,32),(8,9),(8,12),(8,14),(8,20),(8,21),(8,31),(9,35),(9,36),(9,41),(10,33),(10,34),(10,41),(11,25),(11,38),(11,41),(12,26),(12,37),(12,41),(13,28),(13,39),(13,41),(14,27),(14,40),(14,41),(15,25),(15,33),(15,37),(16,26),(16,33),(16,38),(17,27),(17,34),(17,39),(18,28),(18,34),(18,40),(19,25),(19,35),(19,39),(20,26),(20,36),(20,40),(21,27),(21,36),(21,37),(22,28),(22,35),(22,38),(23,37),(23,39),(24,38),(24,40),(25,42),(26,42),(27,42),(28,42),(29,34),(29,37),(29,38),(30,33),(30,39),(30,40),(31,35),(31,37),(31,40),(32,36),(32,38),(32,39),(33,42),(34,42),(35,42),(36,42),(37,42),(38,42),(39,42),(40,42),(41,42)],43)
=> ? = 3 - 3
[5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 3 - 3
[5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 3 - 3
[5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 3 - 3
[5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 3 - 3
[5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,16),(1,20),(1,24),(1,30),(1,32),(2,15),(2,19),(2,23),(2,30),(2,31),(3,17),(3,21),(3,23),(3,29),(3,32),(4,18),(4,22),(4,24),(4,29),(4,31),(5,10),(5,13),(5,14),(5,17),(5,18),(5,30),(6,10),(6,11),(6,12),(6,15),(6,16),(6,29),(7,9),(7,11),(7,13),(7,19),(7,22),(7,32),(8,9),(8,12),(8,14),(8,20),(8,21),(8,31),(9,35),(9,36),(9,41),(10,33),(10,34),(10,41),(11,25),(11,38),(11,41),(12,26),(12,37),(12,41),(13,28),(13,39),(13,41),(14,27),(14,40),(14,41),(15,25),(15,33),(15,37),(16,26),(16,33),(16,38),(17,27),(17,34),(17,39),(18,28),(18,34),(18,40),(19,25),(19,35),(19,39),(20,26),(20,36),(20,40),(21,27),(21,36),(21,37),(22,28),(22,35),(22,38),(23,37),(23,39),(24,38),(24,40),(25,42),(26,42),(27,42),(28,42),(29,34),(29,37),(29,38),(30,33),(30,39),(30,40),(31,35),(31,37),(31,40),(32,36),(32,38),(32,39),(33,42),(34,42),(35,42),(36,42),(37,42),(38,42),(39,42),(40,42),(41,42)],43)
=> ? = 3 - 3
[2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 3 - 3
[2,3,1,4,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 3 - 3
[2,3,1,5,4,6] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 3 - 3
[2,3,1,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0 = 3 - 3
[2,3,1,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0 = 3 - 3
[2,3,1,6,5,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 3
[2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 3 - 3
[2,3,4,1,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0 = 3 - 3
[2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0 = 3 - 3
[2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> 0 = 3 - 3
[2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0 = 3 - 3
[2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> 0 = 3 - 3
[2,3,5,4,1,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 3 - 3
[2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(1,11),(1,12),(1,16),(1,20),(2,8),(2,9),(2,12),(2,15),(2,19),(3,7),(3,9),(3,11),(3,14),(3,18),(4,7),(4,8),(4,10),(4,13),(4,17),(5,17),(5,18),(5,19),(5,20),(5,21),(6,13),(6,14),(6,15),(6,16),(6,21),(7,24),(7,30),(7,44),(8,22),(8,28),(8,44),(9,23),(9,29),(9,44),(10,25),(10,31),(10,44),(11,26),(11,32),(11,44),(12,27),(12,33),(12,44),(13,22),(13,24),(13,25),(13,34),(14,23),(14,24),(14,26),(14,35),(15,22),(15,23),(15,27),(15,36),(16,25),(16,26),(16,27),(16,37),(17,28),(17,30),(17,31),(17,34),(18,29),(18,30),(18,32),(18,35),(19,28),(19,29),(19,33),(19,36),(20,31),(20,32),(20,33),(20,37),(21,34),(21,35),(21,36),(21,37),(22,38),(22,45),(23,39),(23,45),(24,40),(24,45),(25,41),(25,45),(26,42),(26,45),(27,43),(27,45),(28,38),(28,46),(29,39),(29,46),(30,40),(30,46),(31,41),(31,46),(32,42),(32,46),(33,43),(33,46),(34,38),(34,40),(34,41),(35,39),(35,40),(35,42),(36,38),(36,39),(36,43),(37,41),(37,42),(37,43),(38,47),(39,47),(40,47),(41,47),(42,47),(43,47),(44,45),(44,46),(45,47),(46,47)],48)
=> ? = 3 - 3
[2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> 0 = 3 - 3
[2,3,6,1,5,4] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,16),(1,17),(1,18),(1,29),(2,13),(2,14),(2,15),(2,29),(3,10),(3,11),(3,12),(3,29),(4,8),(4,9),(4,12),(4,15),(4,18),(5,7),(5,9),(5,11),(5,14),(5,17),(6,7),(6,8),(6,10),(6,13),(6,16),(7,19),(7,22),(7,25),(7,28),(8,19),(8,20),(8,23),(8,26),(9,19),(9,21),(9,24),(9,27),(10,20),(10,22),(10,30),(11,21),(11,22),(11,31),(12,20),(12,21),(12,32),(13,23),(13,25),(13,30),(14,24),(14,25),(14,31),(15,23),(15,24),(15,32),(16,26),(16,28),(16,30),(17,27),(17,28),(17,31),(18,26),(18,27),(18,32),(19,33),(19,34),(19,35),(20,33),(20,36),(21,33),(21,37),(22,33),(22,38),(23,34),(23,36),(24,34),(24,37),(25,34),(25,38),(26,35),(26,36),(27,35),(27,37),(28,35),(28,38),(29,30),(29,31),(29,32),(30,36),(30,38),(31,37),(31,38),(32,36),(32,37),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
=> ? = 3 - 3
[2,3,6,4,1,5] => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,16),(1,17),(1,18),(1,29),(2,13),(2,14),(2,15),(2,29),(3,10),(3,11),(3,12),(3,29),(4,8),(4,9),(4,12),(4,15),(4,18),(5,7),(5,9),(5,11),(5,14),(5,17),(6,7),(6,8),(6,10),(6,13),(6,16),(7,19),(7,22),(7,25),(7,28),(8,19),(8,20),(8,23),(8,26),(9,19),(9,21),(9,24),(9,27),(10,20),(10,22),(10,30),(11,21),(11,22),(11,31),(12,20),(12,21),(12,32),(13,23),(13,25),(13,30),(14,24),(14,25),(14,31),(15,23),(15,24),(15,32),(16,26),(16,28),(16,30),(17,27),(17,28),(17,31),(18,26),(18,27),(18,32),(19,33),(19,34),(19,35),(20,33),(20,36),(21,33),(21,37),(22,33),(22,38),(23,34),(23,36),(24,34),(24,37),(25,34),(25,38),(26,35),(26,36),(27,35),(27,37),(28,35),(28,38),(29,30),(29,31),(29,32),(30,36),(30,38),(31,37),(31,38),(32,36),(32,37),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
=> ? = 3 - 3
[2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(1,9),(1,36),(1,37),(2,12),(2,16),(2,21),(2,22),(2,37),(3,11),(3,15),(3,19),(3,20),(3,37),(4,14),(4,18),(4,20),(4,22),(4,36),(5,13),(5,17),(5,19),(5,21),(5,36),(6,9),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,10),(7,11),(7,12),(7,13),(7,14),(8,35),(8,38),(8,39),(9,35),(9,40),(9,41),(10,31),(10,32),(10,33),(10,34),(10,35),(11,23),(11,24),(11,31),(11,38),(12,25),(12,26),(12,32),(12,38),(13,23),(13,25),(13,33),(13,39),(14,24),(14,26),(14,34),(14,39),(15,27),(15,28),(15,31),(15,40),(16,29),(16,30),(16,32),(16,40),(17,27),(17,29),(17,33),(17,41),(18,28),(18,30),(18,34),(18,41),(19,23),(19,27),(19,48),(20,24),(20,28),(20,48),(21,25),(21,29),(21,48),(22,26),(22,30),(22,48),(23,42),(23,49),(24,43),(24,49),(25,44),(25,49),(26,45),(26,49),(27,42),(27,50),(28,43),(28,50),(29,44),(29,50),(30,45),(30,50),(31,42),(31,43),(31,46),(32,44),(32,45),(32,46),(33,42),(33,44),(33,47),(34,43),(34,45),(34,47),(35,46),(35,47),(36,39),(36,41),(36,48),(37,38),(37,40),(37,48),(38,46),(38,49),(39,47),(39,49),(40,46),(40,50),(41,47),(41,50),(42,51),(43,51),(44,51),(45,51),(46,51),(47,51),(48,49),(48,50),(49,51),(50,51)],52)
=> ? = 3 - 3
[2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 3 - 3
[2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0 = 3 - 3
[2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0 = 3 - 3
[2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> 0 = 3 - 3
[2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> 0 = 3 - 3
[2,4,1,6,5,3] => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,16),(1,17),(1,18),(1,29),(2,13),(2,14),(2,15),(2,29),(3,10),(3,11),(3,12),(3,29),(4,8),(4,9),(4,12),(4,15),(4,18),(5,7),(5,9),(5,11),(5,14),(5,17),(6,7),(6,8),(6,10),(6,13),(6,16),(7,19),(7,22),(7,25),(7,28),(8,19),(8,20),(8,23),(8,26),(9,19),(9,21),(9,24),(9,27),(10,20),(10,22),(10,30),(11,21),(11,22),(11,31),(12,20),(12,21),(12,32),(13,23),(13,25),(13,30),(14,24),(14,25),(14,31),(15,23),(15,24),(15,32),(16,26),(16,28),(16,30),(17,27),(17,28),(17,31),(18,26),(18,27),(18,32),(19,33),(19,34),(19,35),(20,33),(20,36),(21,33),(21,37),(22,33),(22,38),(23,34),(23,36),(24,34),(24,37),(25,34),(25,38),(26,35),(26,36),(27,35),(27,37),(28,35),(28,38),(29,30),(29,31),(29,32),(30,36),(30,38),(31,37),(31,38),(32,36),(32,37),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
=> ? = 3 - 3
[2,4,3,1,5,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 3 - 3
[2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0 = 3 - 3
[2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> 0 = 3 - 3
[2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> 0 = 3 - 3
[3,1,2,4,5,6] => ([(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 3 - 3
[3,1,2,4,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 3 - 3
[3,1,2,5,4,6] => ([(1,2),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 3 - 3
[3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0 = 3 - 3
[3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0 = 3 - 3
[3,1,4,2,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> 0 = 3 - 3
[3,1,4,2,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0 = 3 - 3
[3,1,4,5,2,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0 = 3 - 3
[3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> 0 = 3 - 3
[3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> 0 = 3 - 3
[3,1,5,2,4,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> 0 = 3 - 3
[3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
=> 0 = 3 - 3
Description
The Frankl number of a lattice.
For a lattice $L$ on at least two elements, this is
$$
\max_x(|L|-2|[x, 1]|),
$$
where we maximize over all join irreducible elements and $[x, 1]$ denotes the interval from $x$ to the top element. Frankl's conjecture asserts that this number is non-negative, and zero if and only if $L$ is a Boolean lattice.
The following 416 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001568The smallest positive integer that does not appear twice in the partition. 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 $eA$. 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 $eA$. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001330The hat guessing number of a graph. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St000788The number of nesting-similar perfect matchings of a perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000787The number of flips required to make a perfect matching noncrossing. St000717The number of ordinal summands of a poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000068The number of minimal elements in a poset. St001116The game chromatic number of a graph. St000487The length of the shortest cycle of a permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St000210Minimum over maximum difference of elements in cycles. St000407The number of occurrences of the pattern 2143 in a permutation. St000516The number of stretching pairs of a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001513The number of nested exceedences of a permutation. St001536The number of cyclic misalignments of a permutation. 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. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001845The number of join irreducibles minus the rank of a lattice. St000069The number of maximal elements of a poset. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001964The interval resolution global dimension of a poset. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St001618The cardinality of the Frattini sublattice of a lattice. St001434The number of negative sum pairs of a signed permutation. St000741The Colin de Verdière graph invariant. St000097The order of the largest clique of the graph. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001381The fertility of a permutation. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St000322The skewness of a graph. St001569The maximal modular displacement of a permutation. 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 $K[x]/(x^n)$. St001890The maximum magnitude of the Möbius function of a poset. St001429The number of negative entries in a signed permutation. 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 $eA$. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000183The side length of the Durfee square of an integer partition. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001668The number of points of the poset minus the width of the poset. St000056The decomposition (or block) number of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000694The number of affine bounded permutations that project to a given permutation. 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 $c_{n−1}$. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001518The number of graphs with the same ordinary spectrum as the given graph. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000096The number of spanning trees of a graph. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000221The number of strong fixed points of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000943The number of spots the most unlucky car had to go further in a parking function. 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. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001430The number of positive entries in a signed permutation. St001435The number of missing boxes in the first row. 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. St001520The number of strict 3-descents. St001549The number of restricted non-inversions between exceedances. St001552The number of inversions between excedances and fixed points of a permutation. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St001948The number of augmented double ascents of a permutation. St000822The Hadwiger number of the graph. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001488The number of corners of a skew partition. St000062The length of the longest increasing subsequence of the permutation. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000308The height of the tree associated to a permutation. St000485The length of the longest cycle of a permutation. St000542The number of left-to-right-minima of a permutation. St000733The row containing the largest entry of a standard tableau. St000862The number of parts of the shifted shape of a permutation. St000897The number of different multiplicities of parts of an integer partition. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St000990The first ascent of a permutation. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001060The distinguishing index of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001469The holeyness of a permutation. St001481The minimal height of a peak of a Dyck path. St001517The length of a longest pair of twins in a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001555The order of a signed permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001667The maximal size of a pair of weak twins for a permutation. St001734The lettericity of a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001933The largest multiplicity of a part in an integer partition. St000007The number of saliances of the permutation. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000061The number of nodes on the left branch of a binary tree. St000084The number of subtrees. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000326The position of the first one in a binary word after appending a 1 at the end. St000450The number of edges minus the number of vertices plus 2 of a graph. 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. St000545The number of parabolic double cosets with minimal element being the given permutation. St000570The Edelman-Greene number of a permutation. St000627The exponent of a binary word. St000654The first descent of a permutation. St000667The greatest common divisor of the parts of the partition. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000843The decomposition number of a perfect matching. St000864The number of circled entries of the shifted recording tableau of a permutation. St000873The aix statistic of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. 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. St000958The number of Bruhat factorizations of a permutation. St000991The number of right-to-left minima of a permutation. St000993The multiplicity of the largest part of an integer partition. 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. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001048The number of leaves in the subtree containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001081The number of minimal length factorizations of a permutation into star transpositions. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. 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. St001273The projective dimension of the first term in an injective coresolution of the regular module. 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. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of 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. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001577The minimal number of edges to add or remove to make a graph a cograph. St001589The nesting number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001665The number of pure excedances of a permutation. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001765The number of connected components of the friends and strangers graph. St001828The Euler characteristic of a graph. St001871The number of triconnected components of a graph. 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. St000002The number of occurrences of the pattern 123 in a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000042The number of crossings of a perfect matching. St000051The size of the left subtree of a binary tree. St000095The number of triangles of a graph. St000117The number of centered tunnels of a Dyck path. St000124The cardinality of the preimage of the Simion-Schmidt map. St000133The "bounce" of a permutation. St000142The number of even parts of a partition. St000148The number of odd parts of a partition. St000217The number of occurrences of the pattern 312 in a permutation. St000234The number of global ascents of a permutation. St000241The number of cyclical small excedances. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000317The cycle descent number of a permutation. St000338The number of pixed points of a permutation. St000357The number of occurrences of the pattern 12-3. 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. St000405The number of occurrences of the pattern 1324 in a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000478Another weight of a partition according to Alladi. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000531The leading coefficient of the rook polynomial of an integer partition. St000546The number of global descents of a permutation. St000549The number of odd partial sums of an integer partition. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000664The number of right ropes of a permutation. St000674The number of hills of a Dyck path. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000754The Grundy value for the game of removing nestings in a perfect matching. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000879The number of long braid edges in the graph of braid moves of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St000895The number of ones on the main diagonal of an alternating sign matrix. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000962The 3-shifted major index of a permutation. St000989The number of final rises of a permutation. St000995The largest even part 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. St001008Number of indecomposable injective modules with projective dimension 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. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 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. St001092The number of distinct even parts of a partition. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001139The number of occurrences of hills of size 2 in a Dyck path. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. 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. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-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. 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. 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. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001252Half the sum of the even parts of a partition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001371The length of the longest Yamanouchi prefix of a binary word. St001411The number of patterns 321 or 3412 in a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001537The number of cyclic crossings of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001570The minimal number of edges to add to make a graph Hamiltonian. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001856The number of edges in the reduced word graph of a permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001866The nesting alignments of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001175The size of a partition minus the hook length of the base cell. St001248Sum of the even parts of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000889The number of alternating sign matrices with the same antidiagonal sums. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St000635The number of strictly order preserving maps of a poset into itself. St000454The largest eigenvalue of a graph if it is integral. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St000907The number of maximal antichains of minimal length in a poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001301The first Betti number of the order complex associated with the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001095The number of non-isomorphic posets with precisely one further covering relation. St001396Number of triples of incomparable elements in a finite poset. St000160The multiplicity of the smallest part of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001625The Möbius invariant of a lattice. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001249Sum of the odd parts of a partition. St001383The BG-rank of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000477The weight of a partition according to Alladi. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000846The maximal number of elements covering an element of a poset. St001597The Frobenius rank of a skew partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000633The size of the automorphism group of a poset. St001399The distinguishing number of a poset. St000850The number of 1/2-balanced pairs in a poset. St001472The permanent of the Coxeter matrix of the poset. St000312The number of leaves in a graph. St000390The number of runs of ones in a binary word. St000291The number of descents of a binary word. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000629The defect of a binary word. St001851The number of Hecke atoms of a signed permutation. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001754The number of tolerances of a finite lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001862The number of crossings of a signed permutation.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!