- FindStat

Your data matches 105 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000750
(load all 2 compositions to match this statistic)

Mapping

Mp00066: Permutations —inverse⟶ Permutations
St000750: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the pattern 4213 in a permutation.

Matching statistic: St000406
(load all 2 compositions to match this statistic)

Mapping

St000406: Permutations ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%

Values

[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 0
[3,1,4,2] => 0
[3,2,1,4] => 0
[3,2,4,1] => 1
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 0
[1,3,5,4,2] => 0
[1,4,2,3,5] => 0
[1,4,2,5,3] => 0
[1,4,3,2,5] => 0
[1,4,3,5,2] => 1
[1,4,5,2,3] => 0
[1,4,5,3,2] => 0
[1,4,7,2,3,5,6] => ? = 0
[1,4,7,2,3,6,5] => ? = 0
[1,4,7,2,5,3,6] => ? = 0
[1,4,7,2,6,3,5] => ? = 0
[1,5,2,3,4,6,7] => ? = 0
[1,5,2,3,4,7,6] => ? = 0
[1,5,2,3,6,4,7] => ? = 0
[1,5,2,3,6,7,4] => ? = 0
[1,5,2,3,7,4,6] => ? = 0
[1,5,2,3,7,6,4] => ? = 0
[1,5,2,4,3,6,7] => ? = 0
[1,5,2,4,3,7,6] => ? = 0
[1,5,2,4,6,3,7] => ? = 1
[1,5,2,4,6,7,3] => ? = 2
[1,5,2,4,7,3,6] => ? = 1
[1,5,2,4,7,6,3] => ? = 2
[1,5,2,6,3,4,7] => ? = 0
[1,5,2,6,3,7,4] => ? = 0
[1,5,2,6,4,3,7] => ? = 0
[1,5,2,6,4,7,3] => ? = 2
[1,5,2,6,7,3,4] => ? = 0
[1,5,2,6,7,4,3] => ? = 0
[1,5,2,7,3,4,6] => ? = 0
[1,5,2,7,3,6,4] => ? = 0
[1,5,2,7,4,3,6] => ? = 0
[1,5,2,7,4,6,3] => ? = 1
[1,5,2,7,6,3,4] => ? = 0
[1,5,2,7,6,4,3] => ? = 0
[1,5,3,2,4,6,7] => ? = 0
[1,5,3,2,4,7,6] => ? = 0
[1,5,3,2,6,4,7] => ? = 0
[1,5,3,2,6,7,4] => ? = 0
[1,5,3,2,7,4,6] => ? = 0
[1,5,3,2,7,6,4] => ? = 0
[1,5,4,2,3,6,7] => ? = 0
[1,5,4,2,3,7,6] => ? = 0
[1,5,4,2,6,3,7] => ? = 1
[1,5,4,2,7,3,6] => ? = 1
[1,5,6,2,3,4,7] => ? = 0
[1,5,6,2,3,7,4] => ? = 0
[1,5,6,2,4,3,7] => ? = 0
[1,5,7,2,3,4,6] => ? = 0
[1,5,7,2,3,6,4] => ? = 0
[1,5,7,2,4,3,6] => ? = 0
[1,6,2,3,4,5,7] => ? = 0
[1,6,2,3,4,7,5] => ? = 0
[1,6,2,3,5,4,7] => ? = 0
[1,6,2,3,5,7,4] => ? = 1
[1,6,2,3,7,4,5] => ? = 0
[1,6,2,3,7,5,4] => ? = 0

Description

The number of occurrences of the pattern 3241 in a permutation.

Matching statistic: St000929
(load all 4 compositions to match this statistic)

Mapping

Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 32%●distinct values known / distinct values provided: 14%

Values

[1,2] => [1,2] => ([(0,1)],2)
 => [1]
 => ? = 0
[2,1] => [2,1] => ([],2)
 => [2]
 => 0
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => [1]
 => ? = 0
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => [2]
 => 0
[2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => [2]
 => 0
[2,3,1] => [3,2,1] => ([],3)
 => [3,3]
 => 0
[3,1,2] => [2,3,1] => ([(1,2)],3)
 => [3]
 => 0
[3,2,1] => [3,1,2] => ([(1,2)],3)
 => [3]
 => 0
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => ? = 0
[1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => [2]
 => 0
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => 0
[1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => 0
[1,4,2,3] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => [3]
 => 0
[1,4,3,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => [3]
 => 0
[2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => [2]
 => 0
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => 0
[2,3,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => 0
[2,3,4,1] => [4,2,3,1] => ([(2,3)],4)
 => [4,4,4]
 => 0
[2,4,1,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => [8]
 => 0
[2,4,3,1] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => [8]
 => 0
[3,1,2,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 0
[3,1,4,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => [4,2]
 => 0
[3,2,1,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 0
[3,2,4,1] => [4,3,2,1] => ([],4)
 => [4,4,4,4,4,4]
 => ? = 1
[3,4,1,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => [8]
 => 0
[3,4,2,1] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => [8]
 => 0
[4,1,2,3] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => [4]
 => 0
[4,1,3,2] => [3,4,2,1] => ([(2,3)],4)
 => [4,4,4]
 => 0
[4,2,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => 0
[4,2,3,1] => [4,3,1,2] => ([(2,3)],4)
 => [4,4,4]
 => 0
[4,3,1,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => 0
[4,3,2,1] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => [4]
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => ? = 0
[1,2,3,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => 0
[1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => 0
[1,2,4,5,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => 0
[1,2,5,3,4] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => 0
[1,2,5,4,3] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => 0
[1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => 0
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => 0
[1,3,4,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => 0
[1,3,4,5,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => 0
[1,3,5,2,4] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [8]
 => 0
[1,3,5,4,2] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [8]
 => 0
[1,4,2,3,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => 0
[1,4,2,5,3] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,2]
 => 0
[1,4,3,2,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => 0
[1,4,3,5,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => [4,4,4,4,4,4]
 => ? = 1
[1,4,5,2,3] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => [8]
 => 0
[1,4,5,3,2] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => [8]
 => 0
[1,5,2,3,4] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => 0
[1,5,2,4,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => 0
[1,5,3,2,4] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => 0
[1,5,3,4,2] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => 0
[1,5,4,2,3] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => 0
[1,5,4,3,2] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => 0
[2,3,4,5,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => [5,5,5,5]
 => ? = 0
[2,3,5,1,4] => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => [15]
 => ? = 0
[2,3,5,4,1] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
 => [15]
 => ? = 0
[2,4,1,5,3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [12,4]
 => ? = 0
[2,4,3,5,1] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
 => [10,10,10,10]
 => ? = 1
[2,4,5,1,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => ? = 0
[2,4,5,3,1] => [5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => ? = 0
[2,5,1,4,3] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => ? = 0
[2,5,3,1,4] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [14]
 => ? = 0
[2,5,3,4,1] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => ? = 0
[2,5,4,1,3] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [14]
 => ? = 0
[3,1,4,5,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => [12,4]
 => ? = 0
[3,2,4,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,4,4,4,4,4]
 => ? = 1
[3,2,4,5,1] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => [10,10,10,10]
 => ? = 2
[3,2,5,1,4] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => ? = 1
[3,2,5,4,1] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => ? = 2
[3,4,1,5,2] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => [5,5,5,5,5,5]
 => ? = 0
[3,4,2,5,1] => [5,4,3,2,1] => ([],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 2
[3,4,5,1,2] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => [15]
 => ? = 0
[3,4,5,2,1] => [5,1,3,4,2] => ([(1,3),(1,4),(4,2)],5)
 => [15]
 => ? = 0
[3,5,1,4,2] => [4,5,3,2,1] => ([(3,4)],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 0
[3,5,2,4,1] => [5,4,3,1,2] => ([(3,4)],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 1
[4,1,3,5,2] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => ? = 1
[4,1,5,2,3] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => ? = 0
[4,1,5,3,2] => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => ? = 0
[4,2,3,5,1] => [5,3,4,2,1] => ([(3,4)],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 2
[4,2,5,1,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => [5,5,5,5]
 => ? = 1
[4,2,5,3,1] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => [5,5,5,5]
 => ? = 1
[4,3,1,5,2] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => ? = 1
[4,3,2,5,1] => [5,4,2,3,1] => ([(3,4)],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 3
[4,3,5,1,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => ? = 2
[4,3,5,2,1] => [5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => ? = 2
[4,5,1,3,2] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => ? = 0
[4,5,2,1,3] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [14]
 => ? = 0
[4,5,2,3,1] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => ? = 0
[4,5,3,1,2] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [14]
 => ? = 0
[5,1,2,4,3] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
 => [15]
 => ? = 0
[5,1,3,2,4] => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => [15]
 => ? = 0
[5,1,3,4,2] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
 => [10,10,10,10]
 => ? = 0
[5,1,4,2,3] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => [10,10,10,10]
 => ? = 0
[5,1,4,3,2] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => [5,5,5,5,5,5]
 => ? = 0
[5,2,1,4,3] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => [10,4,4]
 => ? = 0
[5,2,3,1,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => [10,4,4]
 => ? = 0
[5,2,3,4,1] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => [5,5,5,5,5,5]
 => ? = 0

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: St001568
(load all 4 compositions to match this statistic)

Mapping

Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 26%●distinct values known / distinct values provided: 14%

Values

[1,2] => [1,2] => ([(0,1)],2)
 => [1]
 => ? = 0 + 1
[2,1] => [2,1] => ([],2)
 => [2]
 => 1 = 0 + 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => [1]
 => ? = 0 + 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => [2]
 => 1 = 0 + 1
[2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => [2]
 => 1 = 0 + 1
[2,3,1] => [3,2,1] => ([],3)
 => [3,3]
 => 1 = 0 + 1
[3,1,2] => [2,3,1] => ([(1,2)],3)
 => [3]
 => 1 = 0 + 1
[3,2,1] => [3,1,2] => ([(1,2)],3)
 => [3]
 => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => ? = 0 + 1
[1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => [2]
 => 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => 1 = 0 + 1
[1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => 1 = 0 + 1
[1,4,2,3] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => [3]
 => 1 = 0 + 1
[1,4,3,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => [3]
 => 1 = 0 + 1
[2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => [2]
 => 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => 1 = 0 + 1
[2,3,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => 1 = 0 + 1
[2,3,4,1] => [4,2,3,1] => ([(2,3)],4)
 => [4,4,4]
 => ? = 0 + 1
[2,4,1,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => [8]
 => 1 = 0 + 1
[2,4,3,1] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => [8]
 => 1 = 0 + 1
[3,1,2,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 1 = 0 + 1
[3,1,4,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => [4,2]
 => 1 = 0 + 1
[3,2,1,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 1 = 0 + 1
[3,2,4,1] => [4,3,2,1] => ([],4)
 => [4,4,4,4,4,4]
 => ? = 1 + 1
[3,4,1,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => [8]
 => 1 = 0 + 1
[3,4,2,1] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => [8]
 => 1 = 0 + 1
[4,1,2,3] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => [4]
 => 1 = 0 + 1
[4,1,3,2] => [3,4,2,1] => ([(2,3)],4)
 => [4,4,4]
 => ? = 0 + 1
[4,2,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => 1 = 0 + 1
[4,2,3,1] => [4,3,1,2] => ([(2,3)],4)
 => [4,4,4]
 => ? = 0 + 1
[4,3,1,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => 1 = 0 + 1
[4,3,2,1] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => [4]
 => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => ? = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => 1 = 0 + 1
[1,2,4,5,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => 1 = 0 + 1
[1,2,5,3,4] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => 1 = 0 + 1
[1,2,5,4,3] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => 1 = 0 + 1
[1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => 1 = 0 + 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => 1 = 0 + 1
[1,3,4,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => 1 = 0 + 1
[1,3,4,5,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => ? = 0 + 1
[1,3,5,2,4] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [8]
 => 1 = 0 + 1
[1,3,5,4,2] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [8]
 => 1 = 0 + 1
[1,4,2,3,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => 1 = 0 + 1
[1,4,2,5,3] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,2]
 => 1 = 0 + 1
[1,4,3,2,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => 1 = 0 + 1
[1,4,3,5,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => [4,4,4,4,4,4]
 => ? = 1 + 1
[1,4,5,2,3] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => [8]
 => 1 = 0 + 1
[1,4,5,3,2] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => [8]
 => 1 = 0 + 1
[1,5,2,3,4] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => 1 = 0 + 1
[1,5,2,4,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => ? = 0 + 1
[1,5,3,2,4] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => 1 = 0 + 1
[1,5,3,4,2] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => ? = 0 + 1
[1,5,4,2,3] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => 1 = 0 + 1
[1,5,4,3,2] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => 1 = 0 + 1
[2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => 1 = 0 + 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => 1 = 0 + 1
[2,1,4,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => 1 = 0 + 1
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [6,6]
 => ? = 0 + 1
[2,1,5,3,4] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [6]
 => 1 = 0 + 1
[2,1,5,4,3] => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [6]
 => 1 = 0 + 1
[2,3,1,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => [3,3]
 => 1 = 0 + 1
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6,6]
 => ? = 0 + 1
[2,3,4,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => ? = 0 + 1
[2,3,4,5,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => [5,5,5,5]
 => ? = 0 + 1
[2,3,5,1,4] => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => [15]
 => ? = 0 + 1
[2,3,5,4,1] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
 => [15]
 => ? = 0 + 1
[2,4,1,5,3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [12,4]
 => ? = 0 + 1
[2,4,3,5,1] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
 => [10,10,10,10]
 => ? = 1 + 1
[2,4,5,1,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => ? = 0 + 1
[2,4,5,3,1] => [5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => ? = 0 + 1
[2,5,1,4,3] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => ? = 0 + 1
[2,5,3,1,4] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [14]
 => ? = 0 + 1
[2,5,3,4,1] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
 => [15,5,5]
 => ? = 0 + 1
[2,5,4,1,3] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [14]
 => ? = 0 + 1
[3,1,4,5,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => [12,4]
 => ? = 0 + 1
[3,2,4,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,4,4,4,4,4]
 => ? = 1 + 1
[3,2,4,5,1] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => [10,10,10,10]
 => ? = 2 + 1
[3,2,5,1,4] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => ? = 1 + 1
[3,2,5,4,1] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => [15,15]
 => ? = 2 + 1
[3,4,1,5,2] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => [5,5,5,5,5,5]
 => ? = 0 + 1
[3,4,2,5,1] => [5,4,3,2,1] => ([],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 2 + 1
[3,4,5,1,2] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => [15]
 => ? = 0 + 1
[3,4,5,2,1] => [5,1,3,4,2] => ([(1,3),(1,4),(4,2)],5)
 => [15]
 => ? = 0 + 1
[3,5,1,4,2] => [4,5,3,2,1] => ([(3,4)],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 0 + 1
[3,5,2,1,4] => [4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
 => [4,4,3]
 => ? = 0 + 1
[3,5,2,4,1] => [5,4,3,1,2] => ([(3,4)],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 1 + 1
[3,5,4,1,2] => [2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5)
 => [4,4,3]
 => ? = 0 + 1
[4,1,3,2,5] => [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => ? = 0 + 1
[4,1,3,5,2] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => ? = 1 + 1
[4,1,5,2,3] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => ? = 0 + 1
[4,1,5,3,2] => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
 => [10,10]
 => ? = 0 + 1
[4,2,3,1,5] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => ? = 0 + 1
[4,2,3,5,1] => [5,3,4,2,1] => ([(3,4)],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 2 + 1
[4,2,5,1,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => [5,5,5,5]
 => ? = 1 + 1
[4,2,5,3,1] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => [5,5,5,5]
 => ? = 1 + 1
[4,3,1,5,2] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => ? = 1 + 1
[4,3,2,5,1] => [5,4,2,3,1] => ([(3,4)],5)
 => [5,5,5,5,5,5,5,5,5,5,5,5]
 => ? = 3 + 1
[4,3,5,1,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => ? = 2 + 1

Description

The smallest positive integer that does not appear twice in the partition.

Matching statistic: St000475
(load all 2 compositions to match this statistic)

Mapping

Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 25%●distinct values known / distinct values provided: 14%

Values

[1,2] => [1,2] => ([(0,1)],2)
 => [3]
 => 0
[2,1] => [2,1] => ([],2)
 => [2,2]
 => 0
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => [4]
 => 0
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => [3,2]
 => 0
[2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => [3,2]
 => 0
[2,3,1] => [3,2,1] => ([],3)
 => [2,2,2,2]
 => 0
[3,1,2] => [2,3,1] => ([(1,2)],3)
 => [6]
 => 0
[3,2,1] => [3,1,2] => ([(1,2)],3)
 => [6]
 => 0
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => [5]
 => 0
[1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => 0
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => 0
[1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => 0
[1,4,2,3] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => [7]
 => 0
[1,4,3,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => [7]
 => 0
[2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => 0
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => 0
[2,3,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => 0
[2,3,4,1] => [4,2,3,1] => ([(2,3)],4)
 => [6,6]
 => 0
[2,4,1,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => [6,2,2]
 => 0
[2,4,3,1] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => [6,2,2]
 => 0
[3,1,2,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => [7]
 => 0
[3,1,4,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => [3,3,3]
 => 0
[3,2,1,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => [7]
 => 0
[3,2,4,1] => [4,3,2,1] => ([],4)
 => [2,2,2,2,2,2,2,2]
 => ? = 1
[3,4,1,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => [6,2,2]
 => 0
[3,4,2,1] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => [6,2,2]
 => 0
[4,1,2,3] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => [4,4]
 => 0
[4,1,3,2] => [3,4,2,1] => ([(2,3)],4)
 => [6,6]
 => 0
[4,2,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => 0
[4,2,3,1] => [4,3,1,2] => ([(2,3)],4)
 => [6,6]
 => 0
[4,3,1,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => 0
[4,3,2,1] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => [4,4]
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => 0
[1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => 0
[1,2,4,5,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => 0
[1,2,5,3,4] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => [8]
 => 0
[1,2,5,4,3] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => [8]
 => 0
[1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => 0
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => 0
[1,3,4,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => 0
[1,3,4,5,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => ? = 0
[1,3,5,2,4] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => 0
[1,3,5,4,2] => [1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => 0
[1,4,2,3,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => 0
[1,4,2,5,3] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => 0
[1,4,3,2,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => 0
[1,4,3,5,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => ? = 1
[1,4,5,2,3] => [1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => 0
[1,4,5,3,2] => [1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => 0
[1,5,2,3,4] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => 0
[1,5,2,4,3] => [1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => ? = 0
[1,5,3,2,4] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => 0
[1,5,3,4,2] => [1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => ? = 0
[1,5,4,2,3] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => 0
[2,3,4,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => ? = 0
[2,3,4,5,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => [4,4,4,4]
 => ? = 0
[2,3,5,1,4] => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => [14]
 => ? = 0
[2,3,5,4,1] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
 => [14]
 => ? = 0
[2,4,1,5,3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => ? = 0
[2,4,3,5,1] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => ? = 1
[2,4,5,1,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [6,2,2,2,2]
 => ? = 0
[2,4,5,3,1] => [5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [6,2,2,2,2]
 => ? = 0
[2,5,1,4,3] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => [10,6]
 => ? = 0
[2,5,3,4,1] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
 => [10,6]
 => ? = 0
[3,1,4,5,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => [8,3,2]
 => ? = 0
[3,2,4,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => ? = 1
[3,2,4,5,1] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => ? = 2
[3,2,5,1,4] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => ? = 1
[3,2,5,4,1] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => ? = 2
[3,4,1,5,2] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => [6,6,6]
 => ? = 0
[3,4,2,5,1] => [5,4,3,2,1] => ([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 2
[3,4,5,1,2] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => [14]
 => ? = 0
[3,4,5,2,1] => [5,1,3,4,2] => ([(1,3),(1,4),(4,2)],5)
 => [14]
 => ? = 0
[3,5,1,4,2] => [4,5,3,2,1] => ([(3,4)],5)
 => [6,6,6,6]
 => ? = 0
[3,5,2,4,1] => [5,4,3,1,2] => ([(3,4)],5)
 => [6,6,6,6]
 => ? = 1
[4,1,3,2,5] => [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => ? = 0
[4,1,3,5,2] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
 => [6,3,3,3]
 => ? = 1
[4,1,5,2,3] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => ? = 0
[4,1,5,3,2] => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => ? = 0
[4,2,3,1,5] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => ? = 0
[4,2,3,5,1] => [5,3,4,2,1] => ([(3,4)],5)
 => [6,6,6,6]
 => ? = 2
[4,2,5,1,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => [4,4,4,4]
 => ? = 1
[4,2,5,3,1] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => [4,4,4,4]
 => ? = 1
[4,3,1,5,2] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5)
 => [6,3,3,3]
 => ? = 1
[4,3,2,5,1] => [5,4,2,3,1] => ([(3,4)],5)
 => [6,6,6,6]
 => ? = 3
[4,3,5,1,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => ? = 2
[4,3,5,2,1] => [5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => ? = 2
[4,5,1,3,2] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => [10,6]
 => ? = 0
[4,5,2,3,1] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
 => [10,6]
 => ? = 0
[5,1,2,4,3] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
 => [14]
 => ? = 0
[5,1,3,2,4] => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => [14]
 => ? = 0
[5,1,3,4,2] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => ? = 0
[5,1,4,2,3] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => ? = 0
[5,1,4,3,2] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => [6,6,6]
 => ? = 0
[5,2,1,4,3] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => [6,5,3]
 => ? = 0
[5,2,3,1,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => ? = 0
[5,2,3,4,1] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => [6,6,6]
 => ? = 0
[5,2,4,3,1] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => ? = 0
[5,3,2,4,1] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => ? = 1

Description

The number of parts equal to 1 in a partition.

Matching statistic: St000455

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00259: Graphs —vertex addition⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 22%●distinct values known / distinct values provided: 14%

Values

[1,2] => [2] => ([],2)
 => ([],3)
 => ? = 0
[2,1] => [1,1] => ([(0,1)],2)
 => ([(1,2)],3)
 => 0
[1,2,3] => [3] => ([],3)
 => ([],4)
 => ? = 0
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => ([(1,3),(2,3)],4)
 => 0
[2,1,3] => [1,2] => ([(1,2)],3)
 => ([(2,3)],4)
 => 0
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => ([(1,3),(2,3)],4)
 => 0
[3,1,2] => [1,2] => ([(1,2)],3)
 => ([(2,3)],4)
 => 0
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(1,2),(1,3),(2,3)],4)
 => 0
[1,2,3,4] => [4] => ([],4)
 => ([],5)
 => ? = 0
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ([(2,4),(3,4)],5)
 => 0
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ([(2,4),(3,4)],5)
 => 0
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[2,1,3,4] => [1,3] => ([(2,3)],4)
 => ([(3,4)],5)
 => 0
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ([(2,4),(3,4)],5)
 => 0
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ([(2,4),(3,4)],5)
 => 0
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[3,1,2,4] => [1,3] => ([(2,3)],4)
 => ([(3,4)],5)
 => 0
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(2,3),(2,4),(3,4)],5)
 => 0
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => ([(2,4),(3,4)],5)
 => 0
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[4,1,2,3] => [1,3] => ([(2,3)],4)
 => ([(3,4)],5)
 => 0
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(2,3),(2,4),(3,4)],5)
 => 0
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(2,3),(2,4),(3,4)],5)
 => 0
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,2,3,4,5] => [5] => ([],5)
 => ([],6)
 => ? = 0
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => 0
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => 0
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ([(3,5),(4,5)],6)
 => 0
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => 0
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => 0
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => ([(3,5),(4,5)],6)
 => 0
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => 0
[1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ([(3,5),(4,5)],6)
 => 0
[1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[2,1,3,4,5] => [1,4] => ([(3,4)],5)
 => ([(4,5)],6)
 => 0
[2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[2,1,4,3,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[2,1,5,3,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ([(3,5),(4,5)],6)
 => 0
[2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => 0
[2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0
[2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => 0
[2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => ([(3,5),(4,5)],6)
 => 0
[2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[2,4,3,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[2,5,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[2,5,3,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[3,1,4,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[3,1,5,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[3,1,5,4,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[3,2,1,5,4] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[3,2,4,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[3,2,5,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[3,2,5,4,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[3,4,1,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[3,4,2,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[3,5,1,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[3,5,2,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[4,1,2,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[4,1,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[4,1,3,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[4,1,5,2,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[4,1,5,3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[4,2,1,5,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[4,2,3,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0
[4,2,3,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[4,2,5,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 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)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[4,3,1,5,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 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)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 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: St001934

Mapping

Mp00065: Permutations —permutation poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 16%●distinct values known / distinct values provided: 14%

Values

[1,2] => ([(0,1)],2)
 => [1]
 => [1]
 => 1 = 0 + 1
[2,1] => ([],2)
 => [2]
 => [1,1]
 => 1 = 0 + 1
[1,2,3] => ([(0,2),(2,1)],3)
 => [1]
 => [1]
 => 1 = 0 + 1
[1,3,2] => ([(0,1),(0,2)],3)
 => [2]
 => [1,1]
 => 1 = 0 + 1
[2,1,3] => ([(0,2),(1,2)],3)
 => [2]
 => [1,1]
 => 1 = 0 + 1
[2,3,1] => ([(1,2)],3)
 => [3]
 => [1,1,1]
 => 1 = 0 + 1
[3,1,2] => ([(1,2)],3)
 => [3]
 => [1,1,1]
 => 1 = 0 + 1
[3,2,1] => ([],3)
 => [3,3]
 => [2,2,2]
 => 1 = 0 + 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => [1]
 => 1 = 0 + 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => [2]
 => [1,1]
 => 1 = 0 + 1
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => [2]
 => [1,1]
 => 1 = 0 + 1
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,1,1]
 => 1 = 0 + 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => [3]
 => [1,1,1]
 => 1 = 0 + 1
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => [3,3]
 => [2,2,2]
 => 1 = 0 + 1
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => [2]
 => [1,1]
 => 1 = 0 + 1
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2]
 => [2,2]
 => 1 = 0 + 1
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1]
 => 1 = 0 + 1
[2,3,4,1] => ([(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 1 = 0 + 1
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [2,2,1]
 => 1 = 0 + 1
[2,4,3,1] => ([(1,2),(1,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => ? = 0 + 1
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1]
 => 1 = 0 + 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => [3,2]
 => [2,2,1]
 => 1 = 0 + 1
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => [3,3]
 => [2,2,2]
 => 1 = 0 + 1
[3,2,4,1] => ([(1,3),(2,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => ? = 1 + 1
[3,4,1,2] => ([(0,3),(1,2)],4)
 => [4,2]
 => [2,2,1,1]
 => 1 = 0 + 1
[3,4,2,1] => ([(2,3)],4)
 => [4,4,4]
 => [3,3,3,3]
 => ? = 0 + 1
[4,1,2,3] => ([(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 1 = 0 + 1
[4,1,3,2] => ([(1,2),(1,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => ? = 0 + 1
[4,2,1,3] => ([(1,3),(2,3)],4)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => ? = 0 + 1
[4,2,3,1] => ([(2,3)],4)
 => [4,4,4]
 => [3,3,3,3]
 => ? = 0 + 1
[4,3,1,2] => ([(2,3)],4)
 => [4,4,4]
 => [3,3,3,3]
 => ? = 0 + 1
[4,3,2,1] => ([],4)
 => [4,4,4,4,4,4]
 => [6,6,6,6]
 => ? = 0 + 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => [1]
 => [1]
 => 1 = 0 + 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => [2]
 => [1,1]
 => 1 = 0 + 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [2]
 => [1,1]
 => 1 = 0 + 1
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,1,1]
 => 1 = 0 + 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => [3]
 => [1,1,1]
 => 1 = 0 + 1
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => [3,3]
 => [2,2,2]
 => 1 = 0 + 1
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [2]
 => [1,1]
 => 1 = 0 + 1
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2]
 => [2,2]
 => 1 = 0 + 1
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,1,1]
 => 1 = 0 + 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => 1 = 0 + 1
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => ? = 0 + 1
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [3]
 => [1,1,1]
 => 1 = 0 + 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => 1 = 0 + 1
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [2,2,2]
 => 1 = 0 + 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => ? = 1 + 1
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,2]
 => [2,2,1,1]
 => 1 = 0 + 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [3,3,3,3]
 => ? = 0 + 1
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => [4]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => ? = 0 + 1
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => ? = 0 + 1
[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [3,3,3,3]
 => ? = 0 + 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => [4,4,4]
 => [3,3,3,3]
 => ? = 0 + 1
[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => [4,4,4,4,4,4]
 => [6,6,6,6]
 => ? = 0 + 1
[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => [2]
 => [1,1]
 => 1 = 0 + 1
[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => [2,2]
 => [2,2]
 => 1 = 0 + 1
[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [2,2]
 => [2,2]
 => 1 = 0 + 1
[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [6]
 => [1,1,1,1,1,1]
 => 1 = 0 + 1
[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [6]
 => [1,1,1,1,1,1]
 => 1 = 0 + 1
[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [6,6]
 => [2,2,2,2,2,2]
 => ? = 0 + 1
[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => [3]
 => [1,1,1]
 => 1 = 0 + 1
[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => 1 = 0 + 1
[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => 1 = 0 + 1
[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => [5]
 => [1,1,1,1,1]
 => 1 = 0 + 1
[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => 1 = 0 + 1
[2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
 => [5,5]
 => [2,2,2,2,2]
 => ? = 0 + 1
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [5,3]
 => [2,2,2,1,1]
 => ? = 0 + 1
[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 + 1
[2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
 => [5,5]
 => [2,2,2,2,2]
 => ? = 1 + 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => [5,4]
 => [2,2,2,2,1]
 => ? = 0 + 1
[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0 + 1
[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]
 => ? = 0 + 1
[2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5)
 => [4,4,3]
 => [3,3,3,2]
 => ? = 0 + 1
[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 0 + 1
[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]
 => ? = 0 + 1
[2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => [15,15]
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => ? = 0 + 1
[3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => [5,3]
 => [2,2,2,1,1]
 => ? = 0 + 1
[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]
 => ? = 0 + 1
[3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6,6]
 => [2,2,2,2,2,2]
 => ? = 0 + 1
[3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => ? = 1 + 1
[3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => [5,5]
 => [2,2,2,2,2]
 => ? = 2 + 1
[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]
 => ? = 1 + 1
[3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [5,5,5,5]
 => [4,4,4,4,4]
 => ? = 2 + 1
[3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => [5,4]
 => [2,2,2,2,1]
 => ? = 0 + 1
[3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [4,4,4]
 => [3,3,3,3]
 => ? = 0 + 1
[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]
 => ? = 2 + 1
[3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => [5,5]
 => [2,2,2,2,2]
 => ? = 0 + 1
[3,4,5,2,1] => ([(2,3),(3,4)],5)
 => [5,5,5,5]
 => [4,4,4,4,4]
 => ? = 0 + 1
[3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => [5,4]
 => [2,2,2,2,1]
 => ? = 0 + 1
[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]
 => ? = 0 + 1
[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]
 => ? = 0 + 1
[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]
 => ? = 1 + 1
[3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
 => [10,10]
 => [2,2,2,2,2,2,2,2,2,2]
 => ? = 0 + 1
[3,5,4,2,1] => ([(2,3),(2,4)],5)
 => [10,10,10,10]
 => [4,4,4,4,4,4,4,4,4,4]
 => ? = 0 + 1
[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 + 1
[4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
 => [4,4,3]
 => [3,3,3,2]
 => ? = 1 + 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => [5,4]
 => [2,2,2,2,1]
 => ? = 0 + 1
[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]
 => ? = 0 + 1

Description

The number of monotone factorisations of genus zero of a permutation of given cycle type. A monotone factorisation of genus zero of a permutation

$\pi\in\mathfrak S_n$ with

$\ell$ cycles, including fixed points, is a tuple of

$r = n - \ell$ transpositions

$(a_1, b_1),\dots,(a_r, b_r)$ with

$b_1 \leq \dots \leq b_r$ and

$a_i < b_i$ for all

$i$ , whose product, in this order, is

$\pi$ . For example, the cycle

$(2,3,1)$ has the two factorizations

$(2,3)(1,3)$ and

$(1,2)(2,3)$ .

Matching statistic: St000068
(load all 23 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 14%

Values

[1,2] => [1,2] => [1,2] => ([(0,1)],2)
 => 1 = 0 + 1
[2,1] => [1,2] => [1,2] => ([(0,1)],2)
 => 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,3,2] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[2,1,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[2,3,1] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[3,1,2] => [1,3,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,2,1] => [1,3,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 0 + 1
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 0 + 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 0 + 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 0 + 1
[3,1,2,4] => [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1 = 0 + 1
[3,1,4,2] => [1,3,4,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 0 + 1
[3,2,1,4] => [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1 = 0 + 1
[3,2,4,1] => [1,3,4,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1 + 1
[3,4,1,2] => [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1 = 0 + 1
[3,4,2,1] => [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1 = 0 + 1
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 0 + 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1 = 0 + 1
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1 = 0 + 1
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1 = 0 + 1
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1 = 0 + 1
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 1 = 0 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 1 = 0 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 1 = 0 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 1 = 0 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 0 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [4,1,2,5,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
 => ? = 0 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 0 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [4,1,2,5,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
 => ? = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 0 + 1
[1,4,5,3,2] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 0 + 1
[1,5,2,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => 1 = 0 + 1
[1,5,2,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 0 + 1
[1,5,3,2,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => 1 = 0 + 1
[1,5,3,4,2] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 0 + 1
[1,5,4,2,3] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 0 + 1
[1,5,4,3,2] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 0 + 1
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[2,1,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 1 = 0 + 1
[2,1,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 1 = 0 + 1
[2,4,1,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 0 + 1
[2,4,1,5,3] => [1,2,4,5,3] => [4,1,2,5,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
 => ? = 0 + 1
[2,4,3,1,5] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 0 + 1
[2,4,3,5,1] => [1,2,4,5,3] => [4,1,2,5,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
 => ? = 1 + 1
[2,4,5,1,3] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 0 + 1
[2,4,5,3,1] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 0 + 1
[2,5,1,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 0 + 1
[2,5,3,4,1] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 0 + 1
[2,5,4,1,3] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 0 + 1
[2,5,4,3,1] => [1,2,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 0 + 1
[3,1,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 0 + 1
[3,1,2,5,4] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 0 + 1
[3,1,4,2,5] => [1,3,4,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 0 + 1
[3,1,4,5,2] => [1,3,4,5,2] => [3,1,4,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
 => ? = 0 + 1
[3,1,5,2,4] => [1,3,5,4,2] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 0 + 1
[3,1,5,4,2] => [1,3,5,2,4] => [3,1,2,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 0 + 1
[3,2,1,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 0 + 1
[3,2,1,5,4] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 0 + 1
[3,2,4,1,5] => [1,3,4,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 1 + 1
[3,2,4,5,1] => [1,3,4,5,2] => [3,1,4,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
 => ? = 2 + 1
[3,2,5,1,4] => [1,3,5,4,2] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 1 + 1
[3,2,5,4,1] => [1,3,5,2,4] => [3,1,2,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 2 + 1
[3,4,1,2,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 0 + 1
[3,4,1,5,2] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 0 + 1
[3,4,2,1,5] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 0 + 1
[3,4,2,5,1] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2 + 1
[3,4,5,1,2] => [1,3,5,2,4] => [3,1,2,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 0 + 1
[3,4,5,2,1] => [1,3,5,2,4] => [3,1,2,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 0 + 1
[3,5,1,2,4] => [1,3,2,5,4] => [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ? = 0 + 1
[3,5,1,4,2] => [1,3,2,5,4] => [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ? = 0 + 1
[3,5,2,1,4] => [1,3,2,5,4] => [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ? = 0 + 1
[3,5,2,4,1] => [1,3,2,5,4] => [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ? = 1 + 1
[3,5,4,1,2] => [1,3,4,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 0 + 1
[3,5,4,2,1] => [1,3,4,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 0 + 1
[4,1,2,3,5] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 0 + 1
[4,1,2,5,3] => [1,4,5,3,2] => [4,5,1,3,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 0 + 1
[4,1,3,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ? = 0 + 1
[4,1,3,5,2] => [1,4,5,2,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 1 + 1

Description

The number of minimal elements in a poset.

Matching statistic: St001846
(load all 8 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001846: Lattices ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 14%

Values

[1,2] => ([],2)
 => ([],1)
 => 0
[2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[1,2,3] => ([],3)
 => ([],1)
 => 0
[1,3,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => 0
[2,1,3] => ([(1,2)],3)
 => ([(0,1)],2)
 => 0
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 0
[1,2,3,4] => ([],4)
 => ([],1)
 => 0
[1,2,4,3] => ([(2,3)],4)
 => ([(0,1)],2)
 => 0
[1,3,2,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => 0
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 0
[2,1,3,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => 0
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[2,3,4,1] => ([(0,3),(1,3),(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
[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
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(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)
 => ? = 0
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[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
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 0
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(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)
 => ? = 1
[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)
 => ? = 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(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)
 => ? = 0
[4,1,2,3] => ([(0,3),(1,3),(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
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(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)
 => ? = 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(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)
 => ? = 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(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)
 => ? = 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(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)
 => ? = 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ? = 0
[1,2,3,4,5] => ([],5)
 => ([],1)
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 0
[1,3,2,4,5] => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[1,3,4,5,2] => ([(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
[1,3,5,2,4] => ([(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
[1,3,5,4,2] => ([(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)
 => ? = 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[1,4,2,5,3] => ([(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
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 0
[1,4,3,5,2] => ([(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)
 => ? = 1
[1,4,5,2,3] => ([(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)
 => ? = 0
[1,4,5,3,2] => ([(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)
 => ? = 0
[1,5,2,3,4] => ([(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
[1,5,2,4,3] => ([(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)
 => ? = 0
[1,5,3,2,4] => ([(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)
 => ? = 0
[1,5,3,4,2] => ([(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)
 => ? = 0
[1,5,4,2,3] => ([(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)
 => ? = 0
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ? = 0
[2,1,3,4,5] => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[2,1,4,5,3] => ([(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
[2,1,5,3,4] => ([(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
[2,1,5,4,3] => ([(0,1),(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)
 => ? = 0
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[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
[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
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(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
[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
[2,3,5,4,1] => ([(0,4),(1,4),(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)
 => ? = 0
[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
[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
[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)
 => ? = 0
[2,4,3,5,1] => ([(0,4),(1,4),(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)
 => ? = 1
[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)
 => ? = 0
[2,4,5,3,1] => ([(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)
 => ? = 0
[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
[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)
 => ? = 0
[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)
 => ? = 0
[2,5,3,4,1] => ([(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)
 => ? = 0
[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)
 => ? = 0
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
 => ? = 0
[3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[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
[3,1,4,2,5] => ([(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
[3,1,4,5,2] => ([(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
[3,1,5,2,4] => ([(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
[3,1,5,4,2] => ([(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)
 => ? = 0
[3,2,1,5,4] => ([(0,1),(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)
 => ? = 0
[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)
 => ? = 1
[3,2,4,5,1] => ([(0,4),(1,4),(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)
 => ? = 2
[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)
 => ? = 1
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
 => ? = 2
[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)
 => ? = 0
[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)
 => ? = 0
[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)
 => ? = 0
[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)
 => ? = 2
[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)
 => ? = 0
[3,4,5,2,1] => ([(0,3),(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),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
 => ? = 0
[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)
 => ? = 0
[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)
 => ? = 0
[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)
 => ? = 0

Description

The number of elements which do not have a complement in the lattice. A complement of an element

$x$ in a lattice is an element

$y$ such that the meet of

$x$ and

$y$ is the bottom element and their join is the top element.

Matching statistic: St001719
(load all 9 compositions to match this statistic)

Mapping

Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 14%

Values

[1,2] => ([],2)
 => ([],1)
 => 1 = 0 + 1
[2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,2,3] => ([],3)
 => ([],1)
 => 1 = 0 + 1
[1,3,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 0 + 1
[2,1,3] => ([(1,2)],3)
 => ([(0,1)],2)
 => 1 = 0 + 1
[2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,2,3,4] => ([],4)
 => ([],1)
 => 1 = 0 + 1
[1,2,4,3] => ([(2,3)],4)
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,3,2,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[2,1,3,4] => ([(2,3)],4)
 => ([(0,1)],2)
 => 1 = 0 + 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,3,4,1] => ([(0,3),(1,3),(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)
 => 1 = 0 + 1
[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)
 => 1 = 0 + 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(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)
 => ? = 0 + 1
[3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[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)
 => 1 = 0 + 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(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)
 => ? = 1 + 1
[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)
 => ? = 0 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(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)
 => ? = 0 + 1
[4,1,2,3] => ([(0,3),(1,3),(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)
 => 1 = 0 + 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(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)
 => ? = 0 + 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(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)
 => ? = 0 + 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(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)
 => ? = 0 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(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)
 => ? = 0 + 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ? = 0 + 1
[1,2,3,4,5] => ([],5)
 => ([],1)
 => 1 = 0 + 1
[1,2,3,5,4] => ([(3,4)],5)
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,2,4,3,5] => ([(3,4)],5)
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,3,2,4,5] => ([(3,4)],5)
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,3,4,5,2] => ([(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)
 => 1 = 0 + 1
[1,3,5,2,4] => ([(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)
 => 1 = 0 + 1
[1,3,5,4,2] => ([(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)
 => ? = 0 + 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[1,4,2,5,3] => ([(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)
 => 1 = 0 + 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[1,4,3,5,2] => ([(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)
 => ? = 1 + 1
[1,4,5,2,3] => ([(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)
 => ? = 0 + 1
[1,4,5,3,2] => ([(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)
 => ? = 0 + 1
[1,5,2,3,4] => ([(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)
 => 1 = 0 + 1
[1,5,2,4,3] => ([(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)
 => ? = 0 + 1
[1,5,3,2,4] => ([(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)
 => ? = 0 + 1
[1,5,3,4,2] => ([(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)
 => ? = 0 + 1
[1,5,4,2,3] => ([(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)
 => ? = 0 + 1
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ? = 0 + 1
[2,1,3,4,5] => ([(3,4)],5)
 => ([(0,1)],2)
 => 1 = 0 + 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[2,1,4,5,3] => ([(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)
 => 1 = 0 + 1
[2,1,5,3,4] => ([(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)
 => 1 = 0 + 1
[2,1,5,4,3] => ([(0,1),(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)
 => ? = 0 + 1
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[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)
 => 1 = 0 + 1
[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)
 => 1 = 0 + 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(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 + 1
[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 + 1
[2,3,5,4,1] => ([(0,4),(1,4),(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)
 => ? = 0 + 1
[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)
 => 1 = 0 + 1
[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 + 1
[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)
 => ? = 0 + 1
[2,4,3,5,1] => ([(0,4),(1,4),(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)
 => ? = 1 + 1
[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)
 => ? = 0 + 1
[2,4,5,3,1] => ([(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)
 => ? = 0 + 1
[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 + 1
[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)
 => ? = 0 + 1
[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)
 => ? = 0 + 1
[2,5,3,4,1] => ([(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)
 => ? = 0 + 1
[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)
 => ? = 0 + 1
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
 => ? = 0 + 1
[3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[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)
 => 1 = 0 + 1
[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)
 => 1 = 0 + 1
[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 + 1
[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 + 1
[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)
 => ? = 0 + 1
[3,2,1,5,4] => ([(0,1),(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)
 => ? = 0 + 1
[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)
 => ? = 1 + 1
[3,2,4,5,1] => ([(0,4),(1,4),(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)
 => ? = 2 + 1
[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)
 => ? = 1 + 1
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
 => ? = 2 + 1
[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)
 => ? = 0 + 1
[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)
 => ? = 0 + 1
[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)
 => ? = 0 + 1
[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)
 => ? = 2 + 1
[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)
 => ? = 0 + 1
[3,4,5,2,1] => ([(0,3),(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),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
 => ? = 0 + 1
[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)
 => ? = 0 + 1
[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)
 => ? = 0 + 1
[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)
 => ? = 0 + 1

Description

The number of shortest chains of small intervals from the bottom to the top in a lattice. An interval

$[a, b]$ in a lattice is small if

$b$ is a join of elements covering

$a$ .

The following 95 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001820The size of the image of the pop stack sorting operator. 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. 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. St001330The hat guessing number of a graph. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001845The number of join irreducibles minus the rank of a lattice. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001964The interval resolution global dimension of a poset. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000181The number of connected components of the Hasse diagram for the poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001490The number of connected components of a skew partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St001890The maximum magnitude of the Möbius function of a poset. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000699The toughness times the least common multiple of 1,. St000948The chromatic discriminant of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001281The normalized isoperimetric number of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001395The number of strictly unfriendly partitions of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001797The number of overfull subgraphs of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001316The domatic number of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St000636The hull number of a graph. St001029The size of the core of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001654The monophonic hull number of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000322The skewness of a graph. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. 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. 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. St000449The number of pairs of vertices of a graph with distance 4. 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. St001578The minimal number of edges to add or remove to make a graph a line graph. 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. St001871The number of triconnected components of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001765The number of connected components of the friends and strangers graph. St000264The girth of a graph, which is not a tree. St001624The breadth of a lattice. St001570The minimal number of edges to add to make a graph Hamiltonian.