Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001569
(load all 22 compositions to match this statistic)
Mapping
St001569: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 2
[2,1,3,4] => 1
[2,1,4,3] => 1
[2,3,1,4] => 2
[2,3,4,1] => 1
[2,4,1,3] => 2
[2,4,3,1] => 2
[3,1,2,4] => 2
[3,1,4,2] => 2
[3,2,1,4] => 2
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 2
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 1
[4,3,1,2] => 2
[4,3,2,1] => 1
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 2
[1,3,2,4,5] => 1
[1,3,2,5,4] => 1
[1,3,4,2,5] => 2
[1,3,4,5,2] => 2
[1,3,5,2,4] => 2
[1,3,5,4,2] => 2
[1,4,2,3,5] => 2
[1,4,2,5,3] => 2
[1,4,3,2,5] => 2
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
[1,4,5,3,2] => 2
Description
The maximal modular displacement of a permutation. This is $\max_{1\leq i \leq n} \left(\min(\pi(i)-i\pmod n, i-\pi(i)\pmod n)\right)$ for a permutation $\pi$ of $\{1,\dots,n\}$.
Matching statistic: St001875
(load all 2 compositions to match this statistic)
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00205: Posets —maximal antichains⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 33% ●values known / values provided: 39%●distinct values known / distinct values provided: 33%
Values
[1,2] => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 0 + 1
[2,1] => [2,1] => ([],2)
 => ([],1)
 => ? = 1 + 1
[1,2,3] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,1)],2)
 => ? = 0 + 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,1)],2)
 => ? = 1 + 1
[2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ? = 1 + 1
[2,3,1] => [2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? = 1 + 1
[3,1,2] => [3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? = 1 + 1
[3,2,1] => [3,2,1] => ([],3)
 => ([],1)
 => ? = 1 + 1
[1,2,3,4] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,1)],2)
 => ? = 0 + 1
[1,2,4,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,1)],2)
 => ? = 1 + 1
[1,3,2,4] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,1)],2)
 => ? = 1 + 1
[1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,1)],2)
 => ? = 2 + 1
[1,4,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,1)],2)
 => ? = 2 + 1
[1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,1)],2)
 => ? = 2 + 1
[2,1,3,4] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ? = 1 + 1
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ? = 1 + 1
[2,3,1,4] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[2,3,4,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ? = 1 + 1
[2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[2,4,3,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ? = 2 + 1
[3,1,2,4] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? = 2 + 1
[3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? = 2 + 1
[3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 2 + 1
[4,1,2,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ? = 1 + 1
[4,1,3,2] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ? = 2 + 1
[4,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? = 2 + 1
[4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 1 + 1
[4,3,1,2] => [4,3,1,2] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 2 + 1
[4,3,2,1] => [4,3,2,1] => ([],4)
 => ([],1)
 => ? = 1 + 1
[1,2,3,4,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ? = 0 + 1
[1,2,3,5,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ? = 1 + 1
[1,2,4,3,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ? = 1 + 1
[1,2,4,5,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ? = 2 + 1
[1,2,5,3,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ? = 2 + 1
[1,2,5,4,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ? = 2 + 1
[1,3,2,4,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ? = 1 + 1
[1,3,2,5,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ? = 1 + 1
[1,3,4,2,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ? = 2 + 1
[1,3,4,5,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ? = 2 + 1
[1,3,5,2,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ? = 2 + 1
[1,3,5,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ? = 2 + 1
[1,4,2,3,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ? = 2 + 1
[1,4,2,5,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ? = 2 + 1
[1,4,3,2,5] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ? = 2 + 1
[1,4,3,5,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ? = 2 + 1
[1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ? = 2 + 1
[1,4,5,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ? = 2 + 1
[1,5,2,3,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ? = 2 + 1
[1,5,2,4,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ? = 2 + 1
[1,5,3,2,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ? = 2 + 1
[1,5,3,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ? = 2 + 1
[1,5,4,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1)],2)
 => ? = 2 + 1
[2,3,1,4,5] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[2,3,1,5,4] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[2,3,4,1,5] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[2,3,5,1,4] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[2,4,1,3,5] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[2,4,1,5,3] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[2,4,3,1,5] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[2,4,5,1,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[2,5,1,3,4] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[2,5,1,4,3] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[2,5,3,1,4] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[2,5,4,1,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[3,1,2,4,5] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[3,1,2,5,4] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[3,1,4,2,5] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[3,1,4,5,2] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[3,1,5,2,4] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[3,1,5,4,2] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[3,2,4,1,5] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[3,2,5,1,4] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[3,4,1,2,5] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,4,1,5,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,4,2,1,5] => [3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[3,4,2,5,1] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[3,4,5,1,2] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,5,1,2,4] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,5,1,4,2] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,5,2,1,4] => [3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[3,5,2,4,1] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[3,5,4,1,2] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,1,2,3,5] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[4,1,2,5,3] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[4,1,3,2,5] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[4,1,3,5,2] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[4,1,5,2,3] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[4,1,5,3,2] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[4,2,1,3,5] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[4,2,1,5,3] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[4,2,3,1,5] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,2,3,5,1] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[4,2,5,1,3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,2,5,3,1] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[4,3,1,2,5] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[4,3,1,5,2] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3 = 2 + 1
[4,3,5,1,2] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St001604
Mapping
Mp00239: Permutations —Corteel⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 33%
Values
[1,2] => [1,2] => [2]
 => []
 => ? = 0 - 2
[2,1] => [2,1] => [1,1]
 => [1]
 => ? = 1 - 2
[1,2,3] => [1,2,3] => [3]
 => []
 => ? = 0 - 2
[1,3,2] => [1,3,2] => [2,1]
 => [1]
 => ? = 1 - 2
[2,1,3] => [2,1,3] => [2,1]
 => [1]
 => ? = 1 - 2
[2,3,1] => [3,2,1] => [1,1,1]
 => [1,1]
 => ? = 1 - 2
[3,1,2] => [3,1,2] => [2,1]
 => [1]
 => ? = 1 - 2
[3,2,1] => [2,3,1] => [2,1]
 => [1]
 => ? = 1 - 2
[1,2,3,4] => [1,2,3,4] => [4]
 => []
 => ? = 0 - 2
[1,2,4,3] => [1,2,4,3] => [3,1]
 => [1]
 => ? = 1 - 2
[1,3,2,4] => [1,3,2,4] => [3,1]
 => [1]
 => ? = 1 - 2
[1,3,4,2] => [1,4,3,2] => [2,1,1]
 => [1,1]
 => ? = 2 - 2
[1,4,2,3] => [1,4,2,3] => [3,1]
 => [1]
 => ? = 2 - 2
[1,4,3,2] => [1,3,4,2] => [3,1]
 => [1]
 => ? = 2 - 2
[2,1,3,4] => [2,1,3,4] => [3,1]
 => [1]
 => ? = 1 - 2
[2,1,4,3] => [2,1,4,3] => [2,2]
 => [2]
 => ? = 1 - 2
[2,3,1,4] => [3,2,1,4] => [2,1,1]
 => [1,1]
 => ? = 2 - 2
[2,3,4,1] => [4,2,3,1] => [2,1,1]
 => [1,1]
 => ? = 1 - 2
[2,4,1,3] => [4,2,1,3] => [2,1,1]
 => [1,1]
 => ? = 2 - 2
[2,4,3,1] => [3,2,4,1] => [2,1,1]
 => [1,1]
 => ? = 2 - 2
[3,1,2,4] => [3,1,2,4] => [3,1]
 => [1]
 => ? = 2 - 2
[3,1,4,2] => [4,1,3,2] => [2,1,1]
 => [1,1]
 => ? = 2 - 2
[3,2,1,4] => [2,3,1,4] => [3,1]
 => [1]
 => ? = 2 - 2
[3,2,4,1] => [2,4,3,1] => [2,1,1]
 => [1,1]
 => ? = 2 - 2
[3,4,1,2] => [4,3,2,1] => [1,1,1,1]
 => [1,1,1]
 => 0 = 2 - 2
[3,4,2,1] => [4,3,1,2] => [2,1,1]
 => [1,1]
 => ? = 2 - 2
[4,1,2,3] => [4,1,2,3] => [3,1]
 => [1]
 => ? = 1 - 2
[4,1,3,2] => [3,1,4,2] => [2,2]
 => [2]
 => ? = 2 - 2
[4,2,1,3] => [2,4,1,3] => [2,2]
 => [2]
 => ? = 2 - 2
[4,2,3,1] => [2,3,4,1] => [3,1]
 => [1]
 => ? = 1 - 2
[4,3,1,2] => [3,4,2,1] => [2,1,1]
 => [1,1]
 => ? = 2 - 2
[4,3,2,1] => [3,4,1,2] => [2,2]
 => [2]
 => ? = 1 - 2
[1,2,3,4,5] => [1,2,3,4,5] => [5]
 => []
 => ? = 0 - 2
[1,2,3,5,4] => [1,2,3,5,4] => [4,1]
 => [1]
 => ? = 1 - 2
[1,2,4,3,5] => [1,2,4,3,5] => [4,1]
 => [1]
 => ? = 1 - 2
[1,2,4,5,3] => [1,2,5,4,3] => [3,1,1]
 => [1,1]
 => ? = 2 - 2
[1,2,5,3,4] => [1,2,5,3,4] => [4,1]
 => [1]
 => ? = 2 - 2
[1,2,5,4,3] => [1,2,4,5,3] => [4,1]
 => [1]
 => ? = 2 - 2
[1,3,2,4,5] => [1,3,2,4,5] => [4,1]
 => [1]
 => ? = 1 - 2
[1,3,2,5,4] => [1,3,2,5,4] => [3,2]
 => [2]
 => ? = 1 - 2
[1,3,4,2,5] => [1,4,3,2,5] => [3,1,1]
 => [1,1]
 => ? = 2 - 2
[1,3,4,5,2] => [1,5,3,4,2] => [3,1,1]
 => [1,1]
 => ? = 2 - 2
[1,3,5,2,4] => [1,5,3,2,4] => [3,1,1]
 => [1,1]
 => ? = 2 - 2
[1,3,5,4,2] => [1,4,3,5,2] => [3,1,1]
 => [1,1]
 => ? = 2 - 2
[1,4,2,3,5] => [1,4,2,3,5] => [4,1]
 => [1]
 => ? = 2 - 2
[1,4,2,5,3] => [1,5,2,4,3] => [3,1,1]
 => [1,1]
 => ? = 2 - 2
[1,4,3,2,5] => [1,3,4,2,5] => [4,1]
 => [1]
 => ? = 2 - 2
[1,4,3,5,2] => [1,3,5,4,2] => [3,1,1]
 => [1,1]
 => ? = 2 - 2
[1,4,5,2,3] => [1,5,4,3,2] => [2,1,1,1]
 => [1,1,1]
 => 0 = 2 - 2
[1,4,5,3,2] => [1,5,4,2,3] => [3,1,1]
 => [1,1]
 => ? = 2 - 2
[1,5,2,3,4] => [1,5,2,3,4] => [4,1]
 => [1]
 => ? = 2 - 2
[1,5,2,4,3] => [1,4,2,5,3] => [3,2]
 => [2]
 => ? = 2 - 2
[2,1,4,5,3] => [2,1,5,4,3] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
[2,3,1,5,4] => [3,2,1,5,4] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
[2,4,1,5,3] => [5,2,1,4,3] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
[2,4,3,5,1] => [3,2,5,4,1] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
[2,4,5,1,3] => [5,2,4,3,1] => [2,1,1,1]
 => [1,1,1]
 => 0 = 2 - 2
[2,4,5,3,1] => [5,2,4,1,3] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
[2,5,1,4,3] => [4,2,1,5,3] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
[2,5,3,1,4] => [3,2,5,1,4] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
[2,5,4,1,3] => [4,2,5,3,1] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
[2,5,4,3,1] => [4,2,5,1,3] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
[3,4,1,2,5] => [4,3,2,1,5] => [2,1,1,1]
 => [1,1,1]
 => 0 = 2 - 2
[3,4,1,5,2] => [5,3,2,4,1] => [2,1,1,1]
 => [1,1,1]
 => 0 = 2 - 2
[3,4,2,5,1] => [5,3,1,4,2] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
[3,4,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1]
 => [1,1,1,1]
 => 0 = 2 - 2
[3,4,5,2,1] => [5,4,3,1,2] => [2,1,1,1]
 => [1,1,1]
 => 0 = 2 - 2
[3,5,1,2,4] => [5,3,2,1,4] => [2,1,1,1]
 => [1,1,1]
 => 0 = 2 - 2
[3,5,1,4,2] => [4,3,2,5,1] => [2,1,1,1]
 => [1,1,1]
 => 0 = 2 - 2
[3,5,2,4,1] => [4,3,1,5,2] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
[3,5,4,1,2] => [5,3,4,2,1] => [2,1,1,1]
 => [1,1,1]
 => 0 = 2 - 2
[3,5,4,2,1] => [5,3,4,1,2] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
[4,1,3,5,2] => [3,1,5,4,2] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
[4,1,5,2,3] => [5,1,4,3,2] => [2,1,1,1]
 => [1,1,1]
 => 0 = 2 - 2
[4,2,1,5,3] => [2,5,1,4,3] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
[4,2,5,1,3] => [2,5,4,3,1] => [2,1,1,1]
 => [1,1,1]
 => 0 = 2 - 2
[4,2,5,3,1] => [2,5,4,1,3] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
[4,3,1,5,2] => [3,5,2,4,1] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
[4,3,2,5,1] => [3,5,1,4,2] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
[4,3,5,1,2] => [4,5,3,2,1] => [2,1,1,1]
 => [1,1,1]
 => 0 = 2 - 2
[4,3,5,2,1] => [4,5,3,1,2] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
[4,5,1,2,3] => [5,4,2,3,1] => [2,1,1,1]
 => [1,1,1]
 => 0 = 2 - 2
[4,5,1,3,2] => [5,4,2,1,3] => [2,1,1,1]
 => [1,1,1]
 => 0 = 2 - 2
[4,5,2,1,3] => [5,4,1,3,2] => [2,1,1,1]
 => [1,1,1]
 => 0 = 2 - 2
[4,5,3,1,2] => [4,3,5,2,1] => [2,1,1,1]
 => [1,1,1]
 => 0 = 2 - 2
[4,5,3,2,1] => [4,3,5,1,2] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
[5,1,4,2,3] => [4,1,5,3,2] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
[5,3,1,2,4] => [3,5,2,1,4] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
[5,3,4,1,2] => [3,5,4,2,1] => [2,1,1,1]
 => [1,1,1]
 => 0 = 2 - 2
[5,3,4,2,1] => [3,5,4,1,2] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
[5,4,1,2,3] => [4,5,2,3,1] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
[5,4,1,3,2] => [4,5,2,1,3] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
[5,4,2,1,3] => [4,5,1,3,2] => [2,2,1]
 => [2,1]
 => 0 = 2 - 2
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000264
(load all 2 compositions to match this statistic)
Mapping
Mp00223: Permutations —runsort⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 33%
Values
[1,2] => [1,2] => [1,2] => ([],2)
 => ? = 0 + 1
[2,1] => [1,2] => [1,2] => ([],2)
 => ? = 1 + 1
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
 => ? = 0 + 1
[1,3,2] => [1,3,2] => [1,2,3] => ([],3)
 => ? = 1 + 1
[2,1,3] => [1,3,2] => [1,2,3] => ([],3)
 => ? = 1 + 1
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
 => ? = 1 + 1
[3,1,2] => [1,2,3] => [1,2,3] => ([],3)
 => ? = 1 + 1
[3,2,1] => [1,2,3] => [1,2,3] => ([],3)
 => ? = 1 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? = 0 + 1
[1,2,4,3] => [1,2,4,3] => [1,2,3,4] => ([],4)
 => ? = 1 + 1
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => ([],4)
 => ? = 1 + 1
[1,3,4,2] => [1,3,4,2] => [1,2,3,4] => ([],4)
 => ? = 2 + 1
[1,4,2,3] => [1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => ? = 2 + 1
[1,4,3,2] => [1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => ? = 2 + 1
[2,1,3,4] => [1,3,4,2] => [1,2,3,4] => ([],4)
 => ? = 1 + 1
[2,1,4,3] => [1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => ? = 1 + 1
[2,3,1,4] => [1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => ? = 2 + 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? = 1 + 1
[2,4,1,3] => [1,3,2,4] => [1,2,3,4] => ([],4)
 => ? = 2 + 1
[2,4,3,1] => [1,2,4,3] => [1,2,3,4] => ([],4)
 => ? = 2 + 1
[3,1,2,4] => [1,2,4,3] => [1,2,3,4] => ([],4)
 => ? = 2 + 1
[3,1,4,2] => [1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => ? = 2 + 1
[3,2,1,4] => [1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
 => ? = 2 + 1
[3,2,4,1] => [1,2,4,3] => [1,2,3,4] => ([],4)
 => ? = 2 + 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? = 2 + 1
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? = 2 + 1
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? = 1 + 1
[4,1,3,2] => [1,3,2,4] => [1,2,3,4] => ([],4)
 => ? = 2 + 1
[4,2,1,3] => [1,3,2,4] => [1,2,3,4] => ([],4)
 => ? = 2 + 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? = 1 + 1
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? = 2 + 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4,5] => ([],5)
 => ? = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => ([],5)
 => ? = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,3,4,5] => ([],5)
 => ? = 2 + 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ? = 2 + 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ? = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => ([],5)
 => ? = 1 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [1,2,3,4,5] => ([],5)
 => ? = 2 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [1,2,3,4,5] => ([],5)
 => ? = 2 + 1
[1,3,5,2,4] => [1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ? = 2 + 1
[1,3,5,4,2] => [1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ? = 2 + 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ? = 2 + 1
[1,4,2,5,3] => [1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? = 2 + 1
[1,4,3,2,5] => [1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? = 2 + 1
[1,4,3,5,2] => [1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ? = 2 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ? = 2 + 1
[1,4,5,3,2] => [1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
 => ? = 2 + 1
[1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,5,3,4,2] => [1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,5,4,2,3] => [1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,5,4,3,2] => [1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[2,1,5,3,4] => [1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[2,1,5,4,3] => [1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[2,3,1,5,4] => [1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[2,3,4,1,5] => [1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[3,1,5,4,2] => [1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[3,2,1,5,4] => [1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[3,4,1,5,2] => [1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[3,4,2,1,5] => [1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[4,1,5,2,3] => [1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[4,1,5,3,2] => [1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[4,2,1,5,3] => [1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[4,2,3,1,5] => [1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[4,3,1,5,2] => [1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[4,3,2,1,5] => [1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
Description
The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.
Matching statistic: St001624
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Values
[1,2] => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[2,1] => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2 = 1 + 1
[2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2 = 1 + 1
[2,3,1] => [3,2,1] => ([],3)
 => ([(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 + 1
[3,1,2] => [3,2,1] => ([],3)
 => ([(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 + 1
[3,2,1] => [3,2,1] => ([],3)
 => ([(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 + 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 2 = 1 + 1
[1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => ? = 2 + 1
[1,4,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => ? = 2 + 1
[1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => ? = 2 + 1
[2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 2 = 1 + 1
[2,3,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? = 2 + 1
[2,3,4,1] => [4,2,3,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 1 + 1
[2,4,1,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(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)
 => ? = 2 + 1
[2,4,3,1] => [4,3,2,1] => ([],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),(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)
 => ? = 2 + 1
[3,1,2,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? = 2 + 1
[3,1,4,2] => [4,2,3,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 2 + 1
[3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? = 2 + 1
[3,2,4,1] => [4,2,3,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 2 + 1
[3,4,1,2] => [4,3,2,1] => ([],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),(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)
 => ? = 2 + 1
[3,4,2,1] => [4,3,2,1] => ([],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),(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)
 => ? = 2 + 1
[4,1,2,3] => [4,2,3,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 1 + 1
[4,1,3,2] => [4,2,3,1] => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 2 + 1
[4,2,1,3] => [4,3,2,1] => ([],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),(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)
 => ? = 2 + 1
[4,2,3,1] => [4,3,2,1] => ([],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),(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)
 => ? = 1 + 1
[4,3,1,2] => [4,3,2,1] => ([],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),(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)
 => ? = 2 + 1
[4,3,2,1] => [4,3,2,1] => ([],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),(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)
 => ? = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 2 = 1 + 1
[1,2,4,5,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 2 + 1
[1,2,5,3,4] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 2 + 1
[1,2,5,4,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ? = 1 + 1
[1,3,4,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
 => ? = 2 + 1
[1,3,4,5,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
 => ? = 2 + 1
[1,3,5,2,4] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 2 + 1
[1,3,5,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 2 + 1
[1,4,2,3,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
 => ? = 2 + 1
[1,4,2,5,3] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
 => ? = 2 + 1
[1,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
 => ? = 2 + 1
[1,4,3,5,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
 => ? = 2 + 1
[1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 2 + 1
[1,4,5,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 2 + 1
[1,5,2,3,4] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
 => ? = 2 + 1
[1,5,2,4,3] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
 => ? = 2 + 1
[1,5,3,2,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 2 + 1
[1,5,3,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 2 + 1
[1,5,4,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 2 + 1
[1,5,4,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 2 + 1
[2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 2 = 1 + 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => ? = 1 + 1
[2,1,4,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
 => ? = 1 + 1
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
 => ? = 2 + 1
[2,1,5,3,4] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
 => ? = 2 + 1
[2,1,5,4,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
 => ? = 2 + 1
[2,3,1,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
 => ? = 2 + 1
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
 => ? = 2 + 1
[2,3,4,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
 => ? = 2 + 1
Description
The breadth of a lattice. The '''breadth''' of a lattice is the least integer $b$ such that any join $x_1\vee x_2\vee\cdots\vee x_n$, with $n > b$, can be expressed as a join over a proper subset of $\{x_1,x_2,\ldots,x_n\}$.