- FindStat

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

Matching statistic: St000337

Mapping

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000337: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. For a permutation

$\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines

$\textrm{lec} \, \sigma = \sum_{1 \leq i \leq k} \textrm{inv} \, \tau_{i} \, ,$ where

$\textrm{inv} \, \tau_{i}$ is the number of inversions of

$\tau_{i}$ .

Matching statistic: St001928
(load all 3 compositions to match this statistic)

Mapping

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 57%

Values

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

Description

The number of non-overlapping descents in a permutation. In other words, any maximal descending subsequence

$\pi_i,\pi_{i+1},\dots,\pi_k$ contributes

$\lfloor\frac{k-i+1}{2}\rfloor$ to the total count.

Matching statistic: St001624

Mapping

Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001624: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 29%

Values

[1] => [1] => ([],1)
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,2] => [2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,1] => [1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,2,3] => [2,3,1] => ([(1,2)],3)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,3,2] => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2 = 1 + 1
[2,1,3] => [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
[2,3,1] => [3,1,2] => ([(1,2)],3)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[3,1,2] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2 = 1 + 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,2,3,4] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2 + 1
[1,2,4,3] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2 = 1 + 1
[1,3,2,4] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
 => ? = 1 + 1
[1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 2 + 1
[1,4,2,3] => [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
[1,4,3,2] => [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
[2,1,3,4] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
 => ? = 1 + 1
[2,1,4,3] => [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)
 => ? = 1 + 1
[2,3,1,4] => [3,4,2,1] => ([(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
[2,3,4,1] => [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)
 => ? = 1 + 1
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 2 + 1
[2,4,3,1] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2 = 1 + 1
[3,1,2,4] => [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
[3,1,4,2] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
 => ? = 1 + 1
[3,2,1,4] => [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
[3,2,4,1] => [4,3,1,2] => ([(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
[3,4,1,2] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
 => ? = 2 + 1
[3,4,2,1] => [4,1,2,3] => ([(1,2),(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 + 1
[4,1,2,3] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 2 = 1 + 1
[4,1,3,2] => [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
[4,2,1,3] => [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)
 => ? = 1 + 1
[4,2,3,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 2 = 1 + 1
[4,3,1,2] => [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
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,2,3,4,5] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 2 + 1
[1,2,3,5,4] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 2 + 1
[1,2,4,3,5] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
 => ([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
 => ? = 2 + 1
[1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
 => ? = 2 + 1
[1,2,5,3,4] => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
 => ? = 2 + 1
[1,2,5,4,3] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 1 + 1
[1,3,2,4,5] => [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,9),(1,10),(2,8),(2,10),(3,7),(4,6),(5,1),(5,2),(5,6),(6,8),(6,9),(8,11),(9,11),(10,3),(10,11),(11,7)],12)
 => ? = 2 + 1
[1,3,2,5,4] => [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
 => ? = 1 + 1
[1,3,4,2,5] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
 => ([(0,1),(0,2),(1,11),(2,3),(2,4),(2,11),(3,8),(3,10),(4,5),(4,9),(4,10),(5,6),(5,7),(6,13),(7,13),(8,12),(9,7),(9,12),(10,6),(10,12),(11,8),(11,9),(12,13)],14)
 => ? = 2 + 1
[1,3,4,5,2] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 2 + 1
[1,3,5,2,4] => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? = 2 + 1
[1,3,5,4,2] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? = 2 + 1
[1,4,2,3,5] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => ([(0,1),(0,2),(1,11),(2,3),(2,4),(2,11),(3,8),(3,10),(4,5),(4,9),(4,10),(5,6),(5,7),(6,13),(7,13),(8,12),(9,7),(9,12),(10,6),(10,12),(11,8),(11,9),(12,13)],14)
 => ? = 2 + 1
[1,4,2,5,3] => [2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(0,5),(1,8),(2,7),(2,9),(3,7),(3,10),(4,6),(5,2),(5,3),(5,6),(6,9),(6,10),(7,11),(9,11),(10,1),(10,11),(11,8)],12)
 => ? = 1 + 1
[1,4,3,2,5] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ([(0,1),(0,2),(1,12),(2,3),(2,4),(2,5),(2,12),(3,8),(3,10),(3,11),(4,7),(4,9),(4,11),(5,6),(5,9),(5,10),(6,13),(6,14),(7,13),(7,15),(8,14),(8,15),(9,13),(9,16),(10,14),(10,16),(11,15),(11,16),(12,6),(12,7),(12,8),(13,17),(14,17),(15,17),(16,17)],18)
 => ? = 1 + 1
[1,4,3,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1),(0,2),(1,11),(2,4),(2,5),(2,11),(3,6),(3,7),(4,8),(4,10),(5,8),(5,9),(6,13),(7,13),(8,12),(9,6),(9,12),(10,7),(10,12),(11,3),(11,9),(11,10),(12,13)],14)
 => ? = 2 + 1
[1,4,5,2,3] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12)
 => ? = 2 + 1
[1,4,5,3,2] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ? = 2 + 1
[1,5,2,3,4] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ? = 2 + 1
[1,5,2,4,3] => [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
[1,5,3,2,4] => [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)
 => ? = 1 + 1
[1,5,3,4,2] => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ? = 2 + 1
[1,5,4,2,3] => [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
[1,5,4,3,2] => [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
[2,1,3,4,5] => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => ([(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(3,7),(3,8),(4,6),(4,8),(5,1),(5,9),(6,11),(7,11),(8,5),(8,11),(9,10),(11,9)],12)
 => ? = 2 + 1
[2,1,3,5,4] => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 1 + 1
[2,1,4,3,5] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,8),(2,6),(2,7),(3,10),(3,11),(4,9),(4,11),(5,9),(5,10),(6,12),(7,12),(8,12),(9,13),(10,13),(11,1),(11,2),(11,13),(13,7),(13,8)],14)
 => ? = 1 + 1
[2,1,4,5,3] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
 => ? = 2 + 1
[2,1,5,3,4] => [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)
 => ? = 1 + 1
[2,1,5,4,3] => [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)
 => ? = 1 + 1
[2,3,1,4,5] => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,10),(4,9),(4,11),(5,2),(5,10),(5,11),(6,13),(7,1),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8)],14)
 => ? = 2 + 1
[2,3,1,5,4] => [3,4,2,1,5] => ([(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
[2,3,4,1,5] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,1),(2,9),(2,10),(3,8),(3,12),(4,8),(4,11),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,6),(9,15),(10,7),(10,15),(11,9),(11,13),(12,10),(12,13),(13,15),(15,14)],16)
 => ? = 2 + 1
[2,3,4,5,1] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 2 + 1
[2,3,5,1,4] => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 2 + 1
[5,1,4,3,2] => [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
[5,4,1,3,2] => [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
[5,4,3,1,2] => [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
[5,4,3,2,1] => [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
[6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 0 + 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

$\{x_1,x_2,\ldots,x_n\}$ .