- FindStat

Your data matches 14 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000279
(load all 14 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000279: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [1,2] => 2
[1,2,3] => [1,2,3] => [2,3,1] => 0
[1,3,2] => [1,3,2] => [2,1,3] => 0
[2,1,3] => [2,1,3] => [3,2,1] => 0
[2,3,1] => [3,2,1] => [1,3,2] => 2
[3,1,2] => [3,2,1] => [1,3,2] => 2
[3,2,1] => [3,2,1] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 0
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => 0
[1,3,4,2] => [1,4,3,2] => [2,1,4,3] => 0
[1,4,2,3] => [1,4,3,2] => [2,1,4,3] => 0
[1,4,3,2] => [1,4,3,2] => [2,1,4,3] => 0
[2,1,3,4] => [2,1,3,4] => [3,2,4,1] => 0
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 0
[2,3,1,4] => [3,2,1,4] => [4,3,2,1] => 0
[2,3,4,1] => [4,2,3,1] => [1,3,4,2] => 2
[2,4,1,3] => [3,4,1,2] => [4,1,2,3] => 0
[2,4,3,1] => [4,3,2,1] => [1,4,3,2] => 2
[3,1,2,4] => [3,2,1,4] => [4,3,2,1] => 0
[3,1,4,2] => [4,2,3,1] => [1,3,4,2] => 2
[3,2,1,4] => [3,2,1,4] => [4,3,2,1] => 0
[3,2,4,1] => [4,2,3,1] => [1,3,4,2] => 2
[3,4,1,2] => [4,3,2,1] => [1,4,3,2] => 2
[3,4,2,1] => [4,3,2,1] => [1,4,3,2] => 2
[4,1,2,3] => [4,2,3,1] => [1,3,4,2] => 2
[4,1,3,2] => [4,2,3,1] => [1,3,4,2] => 2
[4,2,1,3] => [4,3,2,1] => [1,4,3,2] => 2
[4,2,3,1] => [4,3,2,1] => [1,4,3,2] => 2
[4,3,1,2] => [4,3,2,1] => [1,4,3,2] => 2
[4,3,2,1] => [4,3,2,1] => [1,4,3,2] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => 0
[1,2,4,5,3] => [1,2,5,4,3] => [2,3,1,5,4] => 0
[1,2,5,3,4] => [1,2,5,4,3] => [2,3,1,5,4] => 0
[1,2,5,4,3] => [1,2,5,4,3] => [2,3,1,5,4] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => 0
[1,3,4,2,5] => [1,4,3,2,5] => [2,5,4,3,1] => 0
[1,3,4,5,2] => [1,5,3,4,2] => [2,1,4,5,3] => 0
[1,3,5,2,4] => [1,4,5,2,3] => [2,5,1,3,4] => 0
[1,3,5,4,2] => [1,5,4,3,2] => [2,1,5,4,3] => 0
[1,4,2,3,5] => [1,4,3,2,5] => [2,5,4,3,1] => 0
[1,4,2,5,3] => [1,5,3,4,2] => [2,1,4,5,3] => 0
[1,4,3,2,5] => [1,4,3,2,5] => [2,5,4,3,1] => 0
[1,4,3,5,2] => [1,5,3,4,2] => [2,1,4,5,3] => 0
[1,4,5,2,3] => [1,5,4,3,2] => [2,1,5,4,3] => 0
[1,4,5,3,2] => [1,5,4,3,2] => [2,1,5,4,3] => 0

Description

The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations.

Matching statistic: St000260
(load all 5 compositions to match this statistic)

Mapping

Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 68%●distinct values known / distinct values provided: 50%

Values

[1,2] => [1,2] => [2] => ([],2)
 => ? = 0 - 1
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[1,2,3] => [1,2,3] => [3] => ([],3)
 => ? = 0 - 1
[1,3,2] => [3,1,2] => [1,2] => ([(1,2)],3)
 => ? = 0 - 1
[2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
 => ? = 0 - 1
[2,3,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[3,1,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[3,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
 => ? = 0 - 1
[1,2,4,3] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => ? = 0 - 1
[1,3,2,4] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => ? = 0 - 1
[1,3,4,2] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 - 1
[1,4,2,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 - 1
[1,4,3,2] => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[2,1,3,4] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => ? = 0 - 1
[2,1,4,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 - 1
[2,3,1,4] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 - 1
[2,3,4,1] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,4,1,3] => [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[2,4,3,1] => [4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,1,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 - 1
[3,1,4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 0 - 1
[3,2,4,1] => [3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,4,1,2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[3,4,2,1] => [3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[4,1,2,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 2 - 1
[4,1,3,2] => [4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[4,2,1,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[4,2,3,1] => [2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[4,3,1,2] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
 => ? = 0 - 1
[1,2,3,5,4] => [5,1,2,3,4] => [1,4] => ([(3,4)],5)
 => ? = 0 - 1
[1,2,4,3,5] => [4,1,2,3,5] => [1,4] => ([(3,4)],5)
 => ? = 0 - 1
[1,2,4,5,3] => [4,5,1,2,3] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0 - 1
[1,2,5,3,4] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0 - 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[1,3,2,4,5] => [3,1,2,4,5] => [1,4] => ([(3,4)],5)
 => ? = 0 - 1
[1,3,2,5,4] => [3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0 - 1
[1,3,4,2,5] => [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0 - 1
[1,3,4,5,2] => [3,4,5,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 - 1
[1,3,5,2,4] => [5,3,1,2,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[1,3,5,4,2] => [5,3,4,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[1,4,2,3,5] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0 - 1
[1,4,2,5,3] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 - 1
[1,4,3,2,5] => [4,3,1,2,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[1,4,3,5,2] => [4,3,5,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[1,4,5,2,3] => [4,1,5,2,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[1,4,5,3,2] => [4,5,3,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[1,5,2,3,4] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 - 1
[1,5,2,4,3] => [5,1,4,2,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[1,5,3,2,4] => [5,1,3,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[1,5,3,4,2] => [3,5,4,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[1,5,4,2,3] => [1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,4] => ([(3,4)],5)
 => ? = 0 - 1
[2,1,3,5,4] => [2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0 - 1
[2,1,4,3,5] => [2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0 - 1
[2,1,4,5,3] => [2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 - 1
[2,1,5,3,4] => [5,2,1,3,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[2,1,5,4,3] => [5,2,4,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[2,3,1,4,5] => [2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0 - 1
[2,3,1,5,4] => [2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 - 1
[2,3,4,1,5] => [2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 - 1
[2,3,4,5,1] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[2,3,5,1,4] => [5,2,3,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 - 1
[2,3,5,4,1] => [5,2,3,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[2,4,3,5,1] => [4,2,3,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[2,4,5,3,1] => [4,5,2,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[2,5,3,4,1] => [2,5,3,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[2,5,4,3,1] => [5,4,2,3,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[3,1,4,5,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[3,1,5,4,2] => [5,1,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[3,2,4,5,1] => [3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[3,2,5,4,1] => [3,5,2,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[3,4,1,5,2] => [3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[3,4,2,5,1] => [3,4,2,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[3,4,5,1,2] => [3,4,1,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[3,4,5,2,1] => [3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[3,5,1,4,2] => [3,5,1,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[3,5,2,4,1] => [5,3,2,4,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[3,5,4,1,2] => [5,3,1,4,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[3,5,4,2,1] => [5,3,4,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[4,1,2,5,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[4,1,3,5,2] => [4,1,3,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[4,1,5,2,3] => [4,1,2,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[4,1,5,3,2] => [4,5,1,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[4,2,1,5,3] => [2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[4,2,3,5,1] => [2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[4,2,5,1,3] => [2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[4,2,5,3,1] => [2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[4,3,1,5,2] => [1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[4,3,2,5,1] => [4,3,2,5,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[4,3,5,1,2] => [4,3,1,5,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[4,3,5,2,1] => [4,3,5,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[4,5,1,2,3] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[4,5,1,3,2] => [1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[4,5,2,1,3] => [4,2,1,5,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[4,5,2,3,1] => [4,2,5,3,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1

Description

The radius of a connected graph. This is the minimum eccentricity of any vertex.

Matching statistic: St000259

Mapping

Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 50%

Values

[1,2] => [1,2] => [2] => ([],2)
 => ? = 0
[2,1] => [2,1] => [2] => ([],2)
 => ? = 2
[1,2,3] => [1,2,3] => [3] => ([],3)
 => ? = 0
[1,3,2] => [3,1,2] => [3] => ([],3)
 => ? = 0
[2,1,3] => [2,1,3] => [3] => ([],3)
 => ? = 0
[2,3,1] => [2,3,1] => [3] => ([],3)
 => ? = 2
[3,1,2] => [1,3,2] => [1,2] => ([(1,2)],3)
 => ? = 2
[3,2,1] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
 => ? = 0
[1,2,4,3] => [4,1,2,3] => [4] => ([],4)
 => ? = 0
[1,3,2,4] => [3,1,2,4] => [4] => ([],4)
 => ? = 0
[1,3,4,2] => [3,4,1,2] => [4] => ([],4)
 => ? = 0
[1,4,2,3] => [1,4,2,3] => [1,3] => ([(2,3)],4)
 => ? = 0
[1,4,3,2] => [4,3,1,2] => [1,3] => ([(2,3)],4)
 => ? = 0
[2,1,3,4] => [2,1,3,4] => [4] => ([],4)
 => ? = 0
[2,1,4,3] => [2,4,1,3] => [4] => ([],4)
 => ? = 0
[2,3,1,4] => [2,3,1,4] => [4] => ([],4)
 => ? = 0
[2,3,4,1] => [2,3,4,1] => [4] => ([],4)
 => ? = 2
[2,4,1,3] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0
[2,4,3,1] => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[3,1,2,4] => [1,3,2,4] => [1,3] => ([(2,3)],4)
 => ? = 0
[3,1,4,2] => [1,3,4,2] => [1,3] => ([(2,3)],4)
 => ? = 2
[3,2,1,4] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0
[3,2,4,1] => [3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2
[3,4,1,2] => [3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2
[3,4,2,1] => [3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,1,2,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2
[4,1,3,2] => [4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,2,1,3] => [2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 2
[4,2,3,1] => [2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,3,1,2] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,3,2,1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
 => ? = 0
[1,2,3,5,4] => [5,1,2,3,4] => [5] => ([],5)
 => ? = 0
[1,2,4,3,5] => [4,1,2,3,5] => [5] => ([],5)
 => ? = 0
[1,2,4,5,3] => [4,5,1,2,3] => [5] => ([],5)
 => ? = 0
[1,2,5,3,4] => [1,5,2,3,4] => [1,4] => ([(3,4)],5)
 => ? = 0
[1,2,5,4,3] => [5,4,1,2,3] => [1,4] => ([(3,4)],5)
 => ? = 0
[1,3,2,4,5] => [3,1,2,4,5] => [5] => ([],5)
 => ? = 0
[1,3,2,5,4] => [3,5,1,2,4] => [5] => ([],5)
 => ? = 0
[1,3,4,2,5] => [3,4,1,2,5] => [5] => ([],5)
 => ? = 0
[1,3,4,5,2] => [3,4,5,1,2] => [5] => ([],5)
 => ? = 0
[1,3,5,2,4] => [5,3,1,2,4] => [1,4] => ([(3,4)],5)
 => ? = 0
[1,3,5,4,2] => [5,3,4,1,2] => [1,4] => ([(3,4)],5)
 => ? = 0
[1,4,2,3,5] => [1,4,2,3,5] => [1,4] => ([(3,4)],5)
 => ? = 0
[1,4,2,5,3] => [1,4,5,2,3] => [1,4] => ([(3,4)],5)
 => ? = 0
[1,4,3,2,5] => [4,3,1,2,5] => [1,4] => ([(3,4)],5)
 => ? = 0
[1,4,3,5,2] => [4,3,5,1,2] => [1,4] => ([(3,4)],5)
 => ? = 0
[1,4,5,2,3] => [4,1,5,2,3] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0
[1,4,5,3,2] => [4,5,3,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0
[1,5,2,3,4] => [1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0
[1,5,2,4,3] => [5,1,4,2,3] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0
[1,5,3,2,4] => [5,1,3,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0
[1,5,3,4,2] => [3,5,4,1,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0
[1,5,4,2,3] => [1,5,4,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 0
[1,5,4,3,2] => [5,4,3,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[2,1,3,4,5] => [2,1,3,4,5] => [5] => ([],5)
 => ? = 0
[2,3,5,4,1] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[2,4,5,3,1] => [4,5,2,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[2,5,3,4,1] => [2,5,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[2,5,4,3,1] => [5,4,2,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,1,5,4,2] => [5,1,3,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[3,2,5,4,1] => [3,5,2,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[3,4,5,2,1] => [3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[3,5,1,4,2] => [3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[3,5,2,4,1] => [5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,5,4,1,2] => [5,3,1,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[3,5,4,2,1] => [5,3,4,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[4,1,5,3,2] => [4,5,1,3,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[4,2,5,3,1] => [2,4,5,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[4,3,5,2,1] => [4,3,5,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[4,5,1,3,2] => [1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[4,5,2,3,1] => [4,2,5,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[4,5,3,1,2] => [4,1,5,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[4,5,3,2,1] => [4,5,3,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[5,1,2,4,3] => [5,1,2,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[5,1,3,4,2] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[5,1,4,2,3] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[5,1,4,3,2] => [5,4,1,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[5,2,1,4,3] => [2,5,1,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[5,2,3,4,1] => [2,3,5,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[5,2,4,1,3] => [5,2,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[5,2,4,3,1] => [5,2,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[5,3,1,4,2] => [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[5,3,2,4,1] => [3,2,5,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[5,3,4,1,2] => [3,1,5,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[5,3,4,2,1] => [3,5,4,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[5,4,1,2,3] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[5,4,1,3,2] => [5,1,4,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[5,4,2,1,3] => [2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[5,4,2,3,1] => [2,5,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[5,4,3,1,2] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[5,4,3,2,1] => [5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,3,4,6,5,1] => [6,2,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[2,3,5,6,4,1] => [5,6,2,3,4,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[2,3,6,4,5,1] => [2,6,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[2,3,6,5,4,1] => [6,5,2,3,4,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[2,4,3,6,5,1] => [4,6,2,3,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[2,4,5,6,3,1] => [4,5,6,2,3,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[2,4,6,3,5,1] => [6,4,2,3,5,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2

Description

The diameter of a connected graph. This is the greatest distance between any pair of vertices.

Matching statistic: St001498

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 50%

Values

[1,2] => [1,2] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 0
[2,1] => [2,1] => [1,1,0,0]
 => [1,1,1,0,0,0]
 => ? = 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 0
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 0
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 0
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 0
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 0
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 0
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 0
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 0
[2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0
[3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0
[3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 0
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 0
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 0
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 0
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 0
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 0
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 0
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 0
[1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 0
[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,5,2,3,4] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,5,2,4,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,5,3,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,5,4,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 0
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 0
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 0
[2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 0
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 0
[2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 0
[2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 0
[2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 0
[2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 0
[2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[2,3,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 0
[2,3,5,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[2,4,3,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[2,4,5,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[2,5,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[2,5,4,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[3,1,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[3,1,5,4,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[3,2,5,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[3,4,1,5,2] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[3,4,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[3,4,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[3,4,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[3,5,1,4,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[3,5,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[3,5,4,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[3,5,4,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[4,1,2,5,3] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[4,1,3,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[4,1,5,2,3] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[4,1,5,3,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[4,2,1,5,3] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[4,2,3,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[4,2,5,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[4,2,5,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[4,3,1,5,2] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[4,3,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[4,3,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[4,3,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[4,5,1,2,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[4,5,1,3,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[4,5,2,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[4,5,2,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2

Description

The normalised height of a Nakayama algebra with magnitude 1. We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.

Matching statistic: St001199
(load all 2 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 50%

Values

[1,2] => [1,2] => [1,2] => [1,0,1,0]
 => 1 = 0 + 1
[2,1] => [2,1] => [2,1] => [1,1,0,0]
 => ? = 2 + 1
[1,2,3] => [1,2,3] => [1,3,2] => [1,0,1,1,0,0]
 => 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 1 = 0 + 1
[2,1,3] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 1 = 0 + 1
[2,3,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => ? = 2 + 1
[3,1,2] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => ? = 2 + 1
[3,2,1] => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => ? = 2 + 1
[1,2,3,4] => [1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[2,1,3,4] => [2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[2,3,4,1] => [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[2,4,1,3] => [3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[3,1,4,2] => [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[3,2,4,1] => [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[4,1,2,3] => [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[4,1,3,2] => [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,5,2,3,4] => [1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,5,2,4,3] => [1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,5,3,2,4] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,5,4,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[2,1,5,3,4] => [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[2,3,4,1,5] => [4,2,3,1,5] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 0 + 1
[2,3,4,5,1] => [5,2,3,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[2,3,5,1,4] => [4,2,5,1,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
 => 1 = 0 + 1
[2,3,5,4,1] => [5,2,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[2,4,3,5,1] => [5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[2,4,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[2,5,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[2,5,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[3,1,4,5,2] => [5,2,3,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[3,1,5,4,2] => [5,2,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[3,2,4,5,1] => [5,2,3,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[3,2,5,4,1] => [5,2,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[3,4,1,5,2] => [5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[3,4,2,5,1] => [5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[3,4,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[3,4,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[3,5,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[3,5,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[3,5,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[3,5,4,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[4,1,2,5,3] => [5,2,3,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[4,1,3,5,2] => [5,2,3,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[4,1,5,2,3] => [5,2,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[4,1,5,3,2] => [5,2,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[4,2,1,5,3] => [5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[4,2,3,5,1] => [5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[4,2,5,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[4,2,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[4,3,1,5,2] => [5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[4,3,2,5,1] => [5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[4,3,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[4,3,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[4,5,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[4,5,1,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[4,5,2,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[4,5,2,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1

Description

The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ .

Matching statistic: St001198
(load all 6 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 50%

Values

[1,2] => [1,2] => [1,0,1,0]
 => [1,0,1,0]
 => 2 = 0 + 2
[2,1] => [2,1] => [1,1,0,0]
 => [1,1,0,0]
 => ? = 2 + 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 2 = 0 + 2
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 0 + 2
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2 = 0 + 2
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => ? = 2 + 2
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => ? = 2 + 2
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => ? = 2 + 2
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 0 + 2
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 0 + 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
[3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
[3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 0 + 2
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 0 + 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
[1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,5,2,3,4] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,5,2,4,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,5,3,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,5,4,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 0 + 2
[2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 0 + 2
[2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 0 + 2
[2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 0 + 2
[2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[2,3,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[2,3,5,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[2,4,3,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[2,4,5,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[2,5,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[2,5,4,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[3,1,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[3,1,5,4,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[3,2,5,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[3,4,1,5,2] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[3,4,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[3,4,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[3,4,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[3,5,1,4,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[3,5,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[3,5,4,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[3,5,4,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,1,2,5,3] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,1,3,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,1,5,2,3] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,1,5,3,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,2,1,5,3] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,2,3,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,2,5,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,2,5,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,3,1,5,2] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,3,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,3,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,3,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,5,1,2,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,5,1,3,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,5,2,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,5,2,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2

Description

The 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

$eA$ .

Matching statistic: St001206
(load all 6 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 50%

Values

[1,2] => [1,2] => [1,0,1,0]
 => [1,0,1,0]
 => 2 = 0 + 2
[2,1] => [2,1] => [1,1,0,0]
 => [1,1,0,0]
 => ? = 2 + 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => 2 = 0 + 2
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 0 + 2
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2 = 0 + 2
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => ? = 2 + 2
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => ? = 2 + 2
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => ? = 2 + 2
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 0 + 2
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 0 + 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
[3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
[3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 0 + 2
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 0 + 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 0 + 2
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
[1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 0 + 2
[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,5,2,3,4] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,5,2,4,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,5,3,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,5,4,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 0 + 2
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 0 + 2
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2 = 0 + 2
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 0 + 2
[2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 0 + 2
[2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 0 + 2
[2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 0 + 2
[2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[2,3,5,1,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 0 + 2
[2,3,5,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[2,4,3,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[2,4,5,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[2,5,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[2,5,4,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[3,1,4,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[3,1,5,4,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[3,2,5,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[3,4,1,5,2] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[3,4,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[3,4,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[3,4,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[3,5,1,4,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[3,5,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[3,5,4,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[3,5,4,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,1,2,5,3] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,1,3,5,2] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,1,5,2,3] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,1,5,3,2] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,2,1,5,3] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,2,3,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,2,5,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,2,5,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,3,1,5,2] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,3,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,3,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,3,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,5,1,2,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,5,1,3,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,5,2,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2
[4,5,2,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 2

Description

The 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

$eA$ .

Matching statistic: St000205

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 50%

Values

[1,2] => ([],2)
 => [1,1]
 => [1]
 => 0
[2,1] => ([(0,1)],2)
 => [2]
 => []
 => ? = 2
[1,2,3] => ([],3)
 => [1,1,1]
 => [1,1]
 => 0
[1,3,2] => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0
[2,1,3] => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0
[2,3,1] => ([(0,2),(1,2)],3)
 => [3]
 => []
 => ? = 2
[3,1,2] => ([(0,2),(1,2)],3)
 => [3]
 => []
 => ? = 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? = 2
[1,2,3,4] => ([],4)
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,3] => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 0
[1,3,2,4] => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 0
[1,3,4,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
[1,4,2,3] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
[2,1,3,4] => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 0
[2,3,1,4] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 0
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2
[3,1,2,4] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? = 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2
[1,2,3,4,5] => ([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
[1,3,2,4,5] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => 0
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[2,1,3,4,5] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 0
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ? = 0
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 0
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ? = 0
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ? = 2
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 0
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ? = 2
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2

Description

Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. Given

$\lambda$ count how many ''integer partitions''

$w$ (weight) there are, such that

$P_{\lambda,w}$ is non-integral, i.e.,

$w$ such that the Gelfand-Tsetlin polytope

$P_{\lambda,w}$ has at least one non-integral vertex.

Matching statistic: St000206

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 50%

Values

[1,2] => ([],2)
 => [1,1]
 => [1]
 => 0
[2,1] => ([(0,1)],2)
 => [2]
 => []
 => ? = 2
[1,2,3] => ([],3)
 => [1,1,1]
 => [1,1]
 => 0
[1,3,2] => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0
[2,1,3] => ([(1,2)],3)
 => [2,1]
 => [1]
 => 0
[2,3,1] => ([(0,2),(1,2)],3)
 => [3]
 => []
 => ? = 2
[3,1,2] => ([(0,2),(1,2)],3)
 => [3]
 => []
 => ? = 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? = 2
[1,2,3,4] => ([],4)
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,3] => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 0
[1,3,2,4] => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 0
[1,3,4,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
[1,4,2,3] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
[2,1,3,4] => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 0
[2,3,1,4] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 0
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2
[3,1,2,4] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 0
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? = 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2
[1,2,3,4,5] => ([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
[1,3,2,4,5] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => 0
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[2,1,3,4,5] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 0
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 0
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 0
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ? = 0
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 0
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 0
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ? = 0
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ? = 2
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 0
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ? = 2
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2

Description

Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. Given

$\lambda$ count how many ''integer compositions''

$w$ (weight) there are, such that

$P_{\lambda,w}$ is non-integral, i.e.,

$w$ such that the Gelfand-Tsetlin polytope

$P_{\lambda,w}$ has at least one non-integral vertex. See also [[St000205]]. Each value in this statistic is greater than or equal to corresponding value in [[St000205]].

Matching statistic: St000781

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 50%

Values

[1,2] => ([],2)
 => [1,1]
 => [1]
 => 1 = 0 + 1
[2,1] => ([(0,1)],2)
 => [2]
 => []
 => ? = 2 + 1
[1,2,3] => ([],3)
 => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1 = 0 + 1
[2,1,3] => ([(1,2)],3)
 => [2,1]
 => [1]
 => 1 = 0 + 1
[2,3,1] => ([(0,2),(1,2)],3)
 => [3]
 => []
 => ? = 2 + 1
[3,1,2] => ([(0,2),(1,2)],3)
 => [3]
 => []
 => ? = 2 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? = 2 + 1
[1,2,3,4] => ([],4)
 => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,3,2,4] => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1 = 0 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1 = 0 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1 = 0 + 1
[2,1,3,4] => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 1 = 0 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 1 = 0 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1 = 0 + 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2 + 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 0 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1 = 0 + 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? = 2 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => 1 = 0 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2 + 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? = 2 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2 + 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 2 + 1
[1,2,3,4,5] => ([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,2,4,3,5] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,3,2,4,5] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 1 = 0 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1 = 0 + 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 1 = 0 + 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1 = 0 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 1 = 0 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1 = 0 + 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => 1 = 0 + 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1 = 0 + 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1 = 0 + 1
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1 = 0 + 1
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1 = 0 + 1
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1 = 0 + 1
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1 = 0 + 1
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1 = 0 + 1
[2,1,3,4,5] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 1 = 0 + 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 1 = 0 + 1
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1 = 0 + 1
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1 = 0 + 1
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1 = 0 + 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1 = 0 + 1
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1 = 0 + 1
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1 = 0 + 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 + 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ? = 0 + 1
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 + 1
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => 1 = 0 + 1
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 0 + 1
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => 1 = 0 + 1
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 + 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 + 1
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 + 1
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ? = 0 + 1
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 + 1
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 + 1
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 + 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 + 1
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 + 1
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ? = 2 + 1
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 0 + 1
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 + 1
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 + 1
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 + 1
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 + 1
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 + 1
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 + 1
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 + 1
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 + 1
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 + 1
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 + 1
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 + 1
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 + 1
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => []
 => ? = 2 + 1
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 + 1
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ? = 2 + 1
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 + 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 + 1
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 2 + 1

Description

The number of proper colouring schemes of a Ferrers diagram. A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic is the number of distinct such integer partitions that occur.

The following 4 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000264The girth of a graph, which is not a tree. 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

$eA$ . St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.