Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001811
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St001811: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
 => [2,1] => [1,2] => [1,2] => 0
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [1,2,3] => [1,2,3] => 0
{{1,2},{3}}
 => [2,1,3] => [1,2,3] => [1,2,3] => 0
{{1,3},{2}}
 => [3,2,1] => [1,3,2] => [1,3,2] => 1
{{1},{2,3}}
 => [1,3,2] => [1,2,3] => [1,2,3] => 0
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 2
{{1,2},{3,4}}
 => [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,4,2,3] => [1,4,2,3] => 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 3
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => 4
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,2,4,3,5] => [1,2,4,3,5] => 2
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,3,4] => 2
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,3,4] => 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 3
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,3,4,5,2] => [1,5,4,3,2] => 3
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,3,4,2,5] => [1,4,3,2,5] => 2
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [1,3,4,2,5] => [1,4,3,2,5] => 2
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,3,5,2,4] => [1,5,3,2,4] => 2
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,3,5,2,4] => [1,5,3,2,4] => 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [1,3,2,5,4] => [1,3,2,5,4] => 4
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,4,5,2,3] => [1,5,2,4,3] => 3
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,4,2,3,5] => [1,4,2,3,5] => 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1
Description
The Castelnuovo-Mumford regularity of a permutation. The ''Castelnuovo-Mumford regularity'' of a permutation $\sigma$ is the ''Castelnuovo-Mumford regularity'' of the ''matrix Schubert variety'' $X_\sigma$. Equivalently, it is the difference between the degrees of the ''Grothendieck polynomial'' and the ''Schubert polynomial'' for $\sigma$. It can be computed by subtracting the ''Coxeter length'' [[St000018]] from the ''Rajchgot index'' [[St001759]].
Matching statistic: St001857
(load all 19 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001857: Signed permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 40%
Values
{{1,2}}
 => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [2,3,1] => 0
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => 0
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [2,3,4,1] => ? = 0
{{1,2,3},{4}}
 => [2,3,1,4] => [2,3,1,4] => ? = 0
{{1,2,4},{3}}
 => [2,4,3,1] => [2,4,3,1] => ? = 2
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => ? = 0
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => ? = 0
{{1,3,4},{2}}
 => [3,2,4,1] => [3,2,4,1] => ? = 2
{{1,3},{2,4}}
 => [3,4,1,2] => [3,4,1,2] => ? = 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => ? = 1
{{1,4},{2,3}}
 => [4,3,2,1] => [4,3,2,1] => ? = 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,3,4,2] => ? = 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => ? = 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [4,2,3,1] => ? = 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,4,3,2] => ? = 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => ? = 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => ? = 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [2,3,4,5,1] => ? = 0
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [2,3,4,1,5] => ? = 0
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [2,3,5,4,1] => ? = 3
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [2,3,1,5,4] => ? = 0
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [2,3,1,4,5] => ? = 0
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [2,4,3,5,1] => ? = 4
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [2,4,5,1,3] => ? = 2
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [2,4,3,1,5] => ? = 2
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [2,5,4,3,1] => ? = 2
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,4,5,3] => ? = 0
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [2,5,3,4,1] => ? = 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [2,1,5,4,3] => ? = 3
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [3,2,4,5,1] => ? = 3
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [3,5,4,1,2] => ? = 2
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [3,2,4,1,5] => ? = 2
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [3,4,5,2,1] => ? = 2
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [3,4,1,5,2] => ? = 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [3,4,1,2,5] => ? = 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [3,2,5,4,1] => ? = 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [3,5,1,4,2] => ? = 4
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [3,2,1,5,4] => ? = 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [3,2,1,4,5] => ? = 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [4,3,2,5,1] => ? = 3
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [4,3,5,1,2] => ? = 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [5,3,4,2,1] => ? = 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,3,4,5,2] => ? = 0
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,3,4,2,5] => ? = 0
{{1,5},{2,3},{4}}
 => [5,3,2,4,1] => [5,3,2,4,1] => ? = 1
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [1,3,5,4,2] => ? = 3
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [1,3,2,5,4] => ? = 0
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,3,2,4,5] => ? = 0
Description
The number of edges in the reduced word graph of a signed permutation. The reduced word graph of a signed permutation $\pi$ has the reduced words of $\pi$ as vertices and an edge between two reduced words if they differ by exactly one braid move.
Matching statistic: St001623
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001623: Lattices ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 40%
Values
{{1,2}}
 => [2,1] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
{{1},{2}}
 => [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
{{1,2,3}}
 => [2,3,1] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 2 = 0 + 2
{{1,2},{3}}
 => [2,1,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 2 = 0 + 2
{{1,3},{2}}
 => [3,2,1] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 3 = 1 + 2
{{1},{2,3}}
 => [1,3,2] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 2 = 0 + 2
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 2 = 0 + 2
{{1,2,3,4}}
 => [2,3,4,1] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 0 + 2
{{1,2,3},{4}}
 => [2,3,1,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 0 + 2
{{1,2,4},{3}}
 => [2,4,3,1] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ? = 2 + 2
{{1,2},{3,4}}
 => [2,1,4,3] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 0 + 2
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 0 + 2
{{1,3,4},{2}}
 => [3,2,4,1] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 2 + 2
{{1,3},{2,4}}
 => [3,4,1,2] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 1 + 2
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 1 + 2
{{1,4},{2,3}}
 => [4,3,2,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 1 + 2
{{1},{2,3,4}}
 => [1,3,4,2] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 0 + 2
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 0 + 2
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 1 + 2
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ? = 2 + 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 0 + 2
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 0 + 2
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
 => ? = 0 + 2
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
 => ? = 0 + 2
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ? = 3 + 2
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
 => ? = 0 + 2
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
 => ? = 0 + 2
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 4 + 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 2 + 2
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 2 + 2
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,2,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 2 + 2
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
 => ? = 0 + 2
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
 => ? = 0 + 2
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,2,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 2 + 2
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ? = 3 + 2
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
 => ? = 0 + 2
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
 => ? = 0 + 2
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 3 + 2
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 2 + 2
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 2 + 2
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 2 + 2
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 1 + 2
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 1 + 2
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ? = 2 + 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 4 + 2
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 1 + 2
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 1 + 2
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
 => ? = 3 + 2
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 1 + 2
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 1 + 2
{{1,5},{2,3,4}}
 => [5,3,4,2,1] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
 => ? = 0 + 2
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
 => ? = 0 + 2
{{1,5},{2,3},{4}}
 => [5,3,2,4,1] => [1,5,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 2
{{1},{2,3,5},{4}}
 => [1,3,5,4,2] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ? = 3 + 2
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
 => ? = 0 + 2
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
 => ? = 0 + 2
Description
The number of doubly irreducible elements of a lattice. An element $d$ of a lattice $L$ is '''doubly irreducible''' if it is both join and meet irreducible. That means, $d$ is neither the least nor the greatest element of $L$ and if $d=x\vee y$ or $d=x\wedge y$, then $d\in\{x,y\}$ for all $x,y\in L$. In a finite lattice, the doubly irreducible elements are those which cover and are covered by a unique element.