Your data matches 11 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001298
Mp00014: Binary trees to 132-avoiding permutationPermutations
Mp00223: Permutations runsortPermutations
Mp00175: Permutations inverse Foata bijectionPermutations
St001298: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1] => 0
[.,[.,.]]
=> [2,1] => [1,2] => [1,2] => 1
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 2
[.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => [1,2,3] => 2
[[.,.],[.,.]]
=> [3,1,2] => [1,2,3] => [1,2,3] => 2
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => [3,1,2] => 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 2
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 3
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 3
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 3
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,4,3] => [4,1,2,3] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 3
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 3
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 3
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,3,2,4] => [3,1,2,4] => 2
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 2
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,2,4,3] => [4,1,2,3] => 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => [3,4,1,2] => 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 3
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => 3
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,2,4,3,5] => [4,1,2,3,5] => 3
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,2,5,3,4] => [1,5,2,3,4] => 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,2,5,3,4] => [1,5,2,3,4] => 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,2,3,5,4] => [5,1,2,3,4] => 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,2,4,5,3] => [4,5,1,2,3] => 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,3,2,4,5] => [3,1,2,4,5] => 3
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,3,2,4,5] => [3,1,2,4,5] => 3
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => 3
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => 3
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,2,4,3,5] => [4,1,2,3,5] => 3
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,3,4,2,5] => [3,4,1,2,5] => 3
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => 4
Description
The number of repeated entries in the Lehmer code of a permutation. The Lehmer code of a permutation $\pi$ is the sequence $(v_1,\dots,v_n)$, with $v_i=|\{j > i: \pi(j) < \pi(i)\}$. This statistic counts the number of distinct elements in this sequence.
Mp00014: Binary trees to 132-avoiding permutationPermutations
Mp00223: Permutations runsortPermutations
Mp00126: Permutations cactus evacuationPermutations
St000863: Permutations ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1] => 1 = 0 + 1
[.,[.,.]]
=> [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 2 = 1 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[[.,.],[.,.]]
=> [3,1,2] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,4,3] => [4,1,2,3] => 3 = 2 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 3 = 2 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 3 = 2 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 3 = 2 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,2,4,3] => [4,1,2,3] => 3 = 2 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => [3,1,2,4] => 3 = 2 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,3,5,4] => [5,1,2,3,4] => 4 = 3 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,2,4,3,5] => [1,4,2,3,5] => 4 = 3 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,2,5,3,4] => [1,5,2,3,4] => 4 = 3 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,2,5,3,4] => [1,5,2,3,4] => 4 = 3 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,2,3,5,4] => [5,1,2,3,4] => 4 = 3 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,2,4,5,3] => [4,1,2,3,5] => 4 = 3 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,2,3,5,4] => [5,1,2,3,4] => 4 = 3 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,3,2,4,5] => [1,3,4,2,5] => 4 = 3 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,3,2,4,5] => [1,3,4,2,5] => 4 = 3 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,4,2,3,5] => [1,2,4,3,5] => 4 = 3 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,4,2,3,5] => [1,2,4,3,5] => 4 = 3 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,2,4,3,5] => [1,4,2,3,5] => 4 = 3 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,3,4,2,5] => [1,3,2,4,5] => 4 = 3 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => 5 = 4 + 1
[[[.,[.,[[.,[.,.]],.]]],.],.]
=> [4,3,5,2,1,6,7] => [1,6,7,2,3,5,4] => [6,1,5,7,2,3,4] => ? = 4 + 1
[[[.,[[.,[.,.]],[.,.]]],.],.]
=> [5,3,2,4,1,6,7] => [1,6,7,2,4,3,5] => [1,6,4,7,2,3,5] => ? = 4 + 1
[[[.,[[[.,.],[.,.]],.]],.],.]
=> [4,2,3,5,1,6,7] => [1,6,7,2,3,5,4] => [6,1,5,7,2,3,4] => ? = 4 + 1
Description
The length of the first row of the shifted shape of a permutation. The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled. This statistic records the length of the first row of $P$ and $Q$.
Matching statistic: St001820
Mp00018: Binary trees left border symmetryBinary trees
Mp00013: Binary trees to posetPosets
Mp00195: Posets order idealsLattices
St001820: Lattices ⟶ ℤResult quality: 50% values known / values provided: 50%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [.,.]
=> ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,.]]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[[.,.],.]
=> [[.,.],.]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[[.,.],.],.]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 3 = 2 + 1
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3 = 2 + 1
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3 = 2 + 1
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3 = 2 + 1
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 3 = 2 + 1
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3 = 2 + 1
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 4 = 3 + 1
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(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)
=> 4 = 3 + 1
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(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)
=> 4 = 3 + 1
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(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)
=> 4 = 3 + 1
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 4 = 3 + 1
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(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)
=> 4 = 3 + 1
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 4 = 3 + 1
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(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)
=> 4 = 3 + 1
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(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)
=> 4 = 3 + 1
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3 + 1
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3 + 1
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(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)
=> 4 = 3 + 1
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(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)
=> 4 = 3 + 1
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 4 + 1
[[.,[.,[.,[.,.]]]],.]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(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)
=> 4 = 3 + 1
[[.,[.,[[.,.],.]]],.]
=> [[.,.],[.,[[.,.],.]]]
=> ([(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)
=> 4 = 3 + 1
[[.,[[.,[.,.]],.]],.]
=> [[.,.],[[.,.],[.,.]]]
=> ([(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)
=> ? = 2 + 1
[[[.,[.,.]],[.,.]],.]
=> [[[.,.],[.,.]],[.,.]]
=> ([(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)
=> ? = 2 + 1
[[[.,[.,[.,.]]],.],.]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[[[.,[[.,.],.]],.],.]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 1
[.,[[.,[.,.]],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> [.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 4 + 1
[.,[[.,[[.,.],.]],[.,.]]]
=> [.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 4 + 1
[.,[[[.,[.,.]],.],[.,.]]]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 1
[.,[[.,[.,[[.,.],.]]],.]]
=> [.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 1
[.,[[.,[[.,.],[.,.]]],.]]
=> [.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 1
[.,[[.,[[.,[.,.]],.]],.]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 3 + 1
[.,[[.,[[[.,.],.],.]],.]]
=> [.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 1
[.,[[[.,[.,.]],[.,.]],.]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 3 + 1
[.,[[[.,[.,[.,.]]],.],.]]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 3 + 1
[.,[[[.,[[.,.],.]],.],.]]
=> [.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 3 + 1
[.,[[[[.,[.,.]],.],.],.]]
=> [.,[[[[.,.],.],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 1
[[.,[.,.]],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4 + 1
[[.,[.,.]],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4 + 1
[[.,[.,.]],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4 + 1
[[.,[.,.]],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ? = 3 + 1
[[.,[.,.]],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4 + 1
[[.,[.,[.,.]]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 4 + 1
[[.,[.,[.,.]]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 4 + 1
[[.,[[.,.],.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 4 + 1
[[.,[[.,.],.]],[[.,.],.]]
=> [[.,[[.,.],.]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 4 + 1
[[[.,.],[.,.]],[.,[.,.]]]
=> [[[.,[.,[.,.]]],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 1
[[[.,.],[.,.]],[[.,.],.]]
=> [[[.,[[.,.],.]],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 1
[[[.,[.,.]],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4 + 1
[[[.,[.,.]],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4 + 1
[[.,[.,[.,[.,.]]]],[.,.]]
=> [[.,[.,.]],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 4 + 1
[[.,[.,[[.,.],.]]],[.,.]]
=> [[.,[.,.]],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 4 + 1
[[.,[[.,.],[.,.]]],[.,.]]
=> [[.,[.,.]],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 4 + 1
[[.,[[.,[.,.]],.]],[.,.]]
=> [[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ? = 3 + 1
[[.,[[[.,.],.],.]],[.,.]]
=> [[.,[.,.]],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 4 + 1
[[[.,.],[.,[.,.]]],[.,.]]
=> [[[.,[.,.]],[.,[.,.]]],.]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 4 + 1
[[[.,.],[[.,.],.]],[.,.]]
=> [[[.,[.,.]],[[.,.],.]],.]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 4 + 1
[[[.,[.,.]],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 3 + 1
[[[.,[.,[.,.]]],.],[.,.]]
=> [[[.,[.,.]],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 3 + 1
[[[.,[[.,.],.]],.],[.,.]]
=> [[[.,[.,.]],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 3 + 1
[[[[.,.],[.,.]],.],[.,.]]
=> [[[[.,[.,.]],.],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 1
[[[[.,[.,.]],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4 + 1
[[.,[.,[.,[.,[.,.]]]]],.]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4 + 1
[[.,[.,[.,[[.,.],.]]]],.]
=> [[.,.],[.,[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4 + 1
[[.,[.,[[.,.],[.,.]]]],.]
=> [[.,.],[.,[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4 + 1
[[.,[.,[[.,[.,.]],.]]],.]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ? = 3 + 1
[[.,[.,[[[.,.],.],.]]],.]
=> [[.,.],[.,[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4 + 1
Description
The size of the image of the pop stack sorting operator. The pop stack sorting operator is defined by $Pop_L^\downarrow(x) = x\wedge\bigwedge\{y\in L\mid y\lessdot x\}$. This statistic returns the size of $Pop_L^\downarrow(L)\}$.
Matching statistic: St001720
Mp00018: Binary trees left border symmetryBinary trees
Mp00013: Binary trees to posetPosets
Mp00195: Posets order idealsLattices
St001720: Lattices ⟶ ℤResult quality: 50% values known / values provided: 50%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [.,.]
=> ([],1)
=> ([(0,1)],2)
=> 2 = 0 + 2
[.,[.,.]]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[[.,.],.]
=> [[.,.],.]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 1 + 2
[[[.,.],.],.]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 4 = 2 + 2
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 2 + 2
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 2 + 2
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 2 + 2
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 4 = 2 + 2
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 2 + 2
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 5 = 3 + 2
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(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)
=> 5 = 3 + 2
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(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)
=> 5 = 3 + 2
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(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)
=> 5 = 3 + 2
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 5 = 3 + 2
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(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)
=> 5 = 3 + 2
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 5 = 3 + 2
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(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)
=> 5 = 3 + 2
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(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)
=> 5 = 3 + 2
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3 + 2
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3 + 2
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(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)
=> 5 = 3 + 2
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(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)
=> 5 = 3 + 2
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[[.,[.,[.,[.,.]]]],.]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(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)
=> 5 = 3 + 2
[[.,[.,[[.,.],.]]],.]
=> [[.,.],[.,[[.,.],.]]]
=> ([(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)
=> 5 = 3 + 2
[[.,[[.,[.,.]],.]],.]
=> [[.,.],[[.,.],[.,.]]]
=> ([(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)
=> ? = 2 + 2
[[[.,[.,.]],[.,.]],.]
=> [[[.,.],[.,.]],[.,.]]
=> ([(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)
=> ? = 2 + 2
[[[.,[.,[.,.]]],.],.]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 2
[[[.,[[.,.],.]],.],.]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 2
[.,[[.,[.,.]],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 2
[.,[[.,[.,.]],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 2
[.,[[.,[.,[.,.]]],[.,.]]]
=> [.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 4 + 2
[.,[[.,[[.,.],.]],[.,.]]]
=> [.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 4 + 2
[.,[[[.,[.,.]],.],[.,.]]]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 2
[.,[[.,[.,[.,[.,.]]]],.]]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 2
[.,[[.,[.,[[.,.],.]]],.]]
=> [.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 2
[.,[[.,[[.,.],[.,.]]],.]]
=> [.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 2
[.,[[.,[[.,[.,.]],.]],.]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 3 + 2
[.,[[.,[[[.,.],.],.]],.]]
=> [.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 2
[.,[[[.,[.,.]],[.,.]],.]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 3 + 2
[.,[[[.,[.,[.,.]]],.],.]]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 3 + 2
[.,[[[.,[[.,.],.]],.],.]]
=> [.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 3 + 2
[.,[[[[.,[.,.]],.],.],.]]
=> [.,[[[[.,.],.],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 2
[[.,[.,.]],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4 + 2
[[.,[.,.]],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4 + 2
[[.,[.,.]],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4 + 2
[[.,[.,.]],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ? = 3 + 2
[[.,[.,.]],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4 + 2
[[.,[.,[.,.]]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 4 + 2
[[.,[.,[.,.]]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 4 + 2
[[.,[[.,.],.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 4 + 2
[[.,[[.,.],.]],[[.,.],.]]
=> [[.,[[.,.],.]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 4 + 2
[[[.,.],[.,.]],[.,[.,.]]]
=> [[[.,[.,[.,.]]],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 2
[[[.,.],[.,.]],[[.,.],.]]
=> [[[.,[[.,.],.]],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 2
[[[.,[.,.]],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4 + 2
[[[.,[.,.]],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4 + 2
[[.,[.,[.,[.,.]]]],[.,.]]
=> [[.,[.,.]],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 4 + 2
[[.,[.,[[.,.],.]]],[.,.]]
=> [[.,[.,.]],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 4 + 2
[[.,[[.,.],[.,.]]],[.,.]]
=> [[.,[.,.]],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 4 + 2
[[.,[[.,[.,.]],.]],[.,.]]
=> [[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(2,11),(3,7),(3,8),(4,10),(4,13),(5,10),(5,12),(6,3),(6,12),(6,13),(7,15),(8,15),(9,1),(10,2),(10,14),(11,9),(12,7),(12,14),(13,8),(13,14),(14,11),(14,15),(15,9)],16)
=> ? = 3 + 2
[[.,[[[.,.],.],.]],[.,.]]
=> [[.,[.,.]],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 4 + 2
[[[.,.],[.,[.,.]]],[.,.]]
=> [[[.,[.,.]],[.,[.,.]]],.]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 4 + 2
[[[.,.],[[.,.],.]],[.,.]]
=> [[[.,[.,.]],[[.,.],.]],.]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 4 + 2
[[[.,[.,.]],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,4),(0,5),(0,6),(1,11),(3,10),(3,12),(4,7),(4,8),(5,7),(5,9),(6,3),(6,8),(6,9),(7,14),(8,12),(8,14),(9,10),(9,14),(10,13),(11,2),(12,1),(12,13),(13,11),(14,13)],15)
=> ? = 3 + 2
[[[.,[.,[.,.]]],.],[.,.]]
=> [[[.,[.,.]],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 3 + 2
[[[.,[[.,.],.]],.],[.,.]]
=> [[[.,[.,.]],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,2),(4,10),(5,3),(5,7),(6,4),(6,7),(7,8),(7,10),(8,11),(9,12),(10,9),(10,11),(11,12),(12,1)],13)
=> ? = 3 + 2
[[[[.,.],[.,.]],.],[.,.]]
=> [[[[.,[.,.]],.],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 2
[[[[.,[.,.]],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4 + 2
[[.,[.,[.,[.,[.,.]]]]],.]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4 + 2
[[.,[.,[.,[[.,.],.]]]],.]
=> [[.,.],[.,[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4 + 2
[[.,[.,[[.,.],[.,.]]]],.]
=> [[.,.],[.,[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4 + 2
[[.,[.,[[.,[.,.]],.]]],.]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(0,5),(2,11),(3,7),(3,8),(4,8),(4,9),(5,7),(5,9),(6,2),(6,10),(7,12),(8,12),(9,6),(9,12),(10,11),(11,1),(12,10)],13)
=> ? = 3 + 2
[[.,[.,[[[.,.],.],.]]],.]
=> [[.,.],[.,[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,3),(0,6),(2,10),(3,7),(4,5),(4,9),(5,2),(5,8),(6,4),(6,7),(7,9),(8,10),(9,8),(10,1)],11)
=> ? = 4 + 2
Description
The minimal length of a chain of small intervals in a lattice. An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.
Mp00018: Binary trees left border symmetryBinary trees
Mp00013: Binary trees to posetPosets
St001879: Posets ⟶ ℤResult quality: 35% values known / values provided: 35%distinct values known / distinct values provided: 71%
Values
[.,.]
=> [.,.]
=> ([],1)
=> ? = 0
[.,[.,.]]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ? = 1
[[.,.],.]
=> [[.,.],.]
=> ([(0,1)],2)
=> ? = 1
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 2
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 2
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 1
[[[.,.],.],.]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 2
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 2
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 2
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 3
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 3
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 3
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[[.,[.,[.,[.,.]]]],.]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3
[[.,[.,[[.,.],.]]],.]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3
[[.,[[.,.],[.,.]]],.]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3
[[.,[[.,[.,.]],.]],.]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 2
[[.,[[[.,.],.],.]],.]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3
[[[.,.],[.,[.,.]]],.]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3
[[[.,.],[[.,.],.]],.]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3
[[[.,[.,.]],[.,.]],.]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 2
[[[[.,.],.],[.,.]],.]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 3
[[[.,[.,[.,.]]],.],.]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2
[[[.,[[.,.],.]],.],.]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2
[[[[.,.],[.,.]],.],.]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3
[[[[.,[.,.]],.],.],.]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3
[[[[[.,.],.],.],.],.]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[.,[.,[[.,.],.]]]]]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[.,[[.,.],[.,.]]]]]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[.,[[.,[.,.]],.]]]]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? = 4
[.,[.,[.,[[[.,.],.],.]]]]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[[.,.],[.,[.,.]]]]]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[[.,.],[[.,.],.]]]]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[[.,[.,.]],[.,.]]]]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 4
[.,[.,[[[.,.],.],[.,.]]]]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[.,[[.,[.,[.,.]]],.]]]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 4
[.,[.,[[.,[[.,.],.]],.]]]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 4
[.,[.,[[[.,.],[.,.]],.]]]
=> [.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? = 4
[.,[.,[[[.,[.,.]],.],.]]]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 4
[.,[.,[[[[.,.],.],.],.]]]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[.,.],[.,[.,[.,.]]]]]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[.,.],[.,[[.,.],.]]]]
=> [.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[.,.],[[.,.],[.,.]]]]
=> [.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[.,.],[[.,[.,.]],.]]]
=> [.,[[.,[[.,.],[.,.]]],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? = 4
[.,[[.,.],[[[.,.],.],.]]]
=> [.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[.,[.,.]],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4
[.,[[.,[.,.]],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4
[.,[[[.,.],.],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[[.,.],.],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[.,[.,[.,.]]],[.,.]]]
=> [.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 4
[.,[[.,[[.,.],.]],[.,.]]]
=> [.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 4
[.,[[[.,.],[.,.]],[.,.]]]
=> [.,[[[.,[.,.]],[.,.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 4
[.,[[[.,[.,.]],.],[.,.]]]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4
[.,[[[[.,.],.],.],[.,.]]]
=> [.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[.,[[.,[.,[.,[.,.]]]],.]]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4
[.,[[[[[.,.],.],.],.],.]]
=> [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[.,.],[.,[.,[.,[.,.]]]]]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[.,.],[.,[.,[[.,.],.]]]]
=> [[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[.,.],[.,[[.,.],[.,.]]]]
=> [[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[.,.],[.,[[[.,.],.],.]]]
=> [[.,[.,[[[.,.],.],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[.,.],[[.,.],[.,[.,.]]]]
=> [[.,[[.,[.,[.,.]]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[[.,.],[[.,.],[[.,.],.]]]
=> [[.,[[.,[[.,.],.]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Mp00018: Binary trees left border symmetryBinary trees
Mp00013: Binary trees to posetPosets
St001880: Posets ⟶ ℤResult quality: 35% values known / values provided: 35%distinct values known / distinct values provided: 71%
Values
[.,.]
=> [.,.]
=> ([],1)
=> ? = 0 + 1
[.,[.,.]]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ? = 1 + 1
[[.,.],.]
=> [[.,.],.]
=> ([(0,1)],2)
=> ? = 1 + 1
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 1 + 1
[[[.,.],.],.]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 2 + 1
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 2 + 1
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 3 + 1
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 + 1
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 + 1
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 + 1
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 3 + 1
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 + 1
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 3 + 1
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 1
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 1
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3 + 1
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3 + 1
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 + 1
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 1
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[[.,[.,[.,[.,.]]]],.]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 1
[[.,[.,[[.,.],.]]],.]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 1
[[.,[[.,.],[.,.]]],.]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 1
[[.,[[.,[.,.]],.]],.]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 2 + 1
[[.,[[[.,.],.],.]],.]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 1
[[[.,.],[.,[.,.]]],.]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 + 1
[[[.,.],[[.,.],.]],.]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 + 1
[[[.,[.,.]],[.,.]],.]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 2 + 1
[[[[.,.],.],[.,.]],.]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 3 + 1
[[[.,[.,[.,.]]],.],.]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2 + 1
[[[.,[[.,.],.]],.],.]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 2 + 1
[[[[.,.],[.,.]],.],.]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 + 1
[[[[.,[.,.]],.],.],.]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 1
[[[[[.,.],.],.],.],.]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? = 4 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[.,[.,[[.,.],[[.,.],.]]]]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 4 + 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 4 + 1
[.,[.,[[.,[[.,.],.]],.]]]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 4 + 1
[.,[.,[[[.,.],[.,.]],.]]]
=> [.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? = 4 + 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 4 + 1
[.,[.,[[[[.,.],.],.],.]]]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[.,[[.,.],[.,[[.,.],.]]]]
=> [.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[.,[[.,.],[[.,.],[.,.]]]]
=> [.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[.,[[.,.],[[.,[.,.]],.]]]
=> [.,[[.,[[.,.],[.,.]]],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? = 4 + 1
[.,[[.,.],[[[.,.],.],.]]]
=> [.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4 + 1
[.,[[.,[.,.]],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4 + 1
[.,[[[.,.],.],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[.,[[[.,.],.],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> [.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 4 + 1
[.,[[.,[[.,.],.]],[.,.]]]
=> [.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 4 + 1
[.,[[[.,.],[.,.]],[.,.]]]
=> [.,[[[.,[.,.]],[.,.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 4 + 1
[.,[[[.,[.,.]],.],[.,.]]]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4 + 1
[.,[[[[.,.],.],.],[.,.]]]
=> [.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4 + 1
[.,[[[[[.,.],.],.],.],.]]
=> [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[[.,.],[.,[.,[.,[.,.]]]]]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[[.,.],[.,[.,[[.,.],.]]]]
=> [[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[[.,.],[.,[[.,.],[.,.]]]]
=> [[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[[.,.],[.,[[[.,.],.],.]]]
=> [[.,[.,[[[.,.],.],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[[.,.],[[.,.],[.,[.,.]]]]
=> [[.,[[.,[.,[.,.]]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[[.,.],[[.,.],[[.,.],.]]]
=> [[.,[[.,[[.,.],.]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001626
Mp00018: Binary trees left border symmetryBinary trees
Mp00013: Binary trees to posetPosets
Mp00195: Posets order idealsLattices
St001626: Lattices ⟶ ℤResult quality: 21% values known / values provided: 21%distinct values known / distinct values provided: 86%
Values
[.,.]
=> [.,.]
=> ([],1)
=> ([(0,1)],2)
=> 2 = 0 + 2
[.,[.,.]]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[[.,.],.]
=> [[.,.],.]
=> ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 1 + 2
[[[.,.],.],.]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 2 + 2
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 4 = 2 + 2
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 2 + 2
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 2 + 2
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 2 + 2
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 4 = 2 + 2
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 4 = 2 + 2
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 3 + 2
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 5 = 3 + 2
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(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)
=> ? = 3 + 2
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(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)
=> ? = 3 + 2
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(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)
=> ? = 3 + 2
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 5 = 3 + 2
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(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)
=> ? = 3 + 2
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 5 = 3 + 2
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(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)
=> ? = 3 + 2
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(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)
=> ? = 3 + 2
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3 + 2
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 3 + 2
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(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)
=> ? = 3 + 2
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(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)
=> ? = 3 + 2
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[[.,[.,[.,[.,.]]]],.]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(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)
=> ? = 3 + 2
[[.,[.,[[.,.],.]]],.]
=> [[.,.],[.,[[.,.],.]]]
=> ([(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)
=> ? = 3 + 2
[[.,[[.,.],[.,.]]],.]
=> [[.,.],[[.,[.,.]],.]]
=> ([(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)
=> ? = 3 + 2
[[.,[[.,[.,.]],.]],.]
=> [[.,.],[[.,.],[.,.]]]
=> ([(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)
=> ? = 2 + 2
[[.,[[[.,.],.],.]],.]
=> [[.,.],[[[.,.],.],.]]
=> ([(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)
=> ? = 3 + 2
[[[.,.],[.,[.,.]]],.]
=> [[[.,.],[.,[.,.]]],.]
=> ([(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)
=> ? = 3 + 2
[[[.,.],[[.,.],.]],.]
=> [[[.,.],[[.,.],.]],.]
=> ([(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)
=> ? = 3 + 2
[[[.,[.,.]],[.,.]],.]
=> [[[.,.],[.,.]],[.,.]]
=> ([(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)
=> ? = 2 + 2
[[[[.,.],.],[.,.]],.]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 5 = 3 + 2
[[[.,[.,[.,.]]],.],.]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 2
[[[.,[[.,.],.]],.],.]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 2
[[[[.,.],[.,.]],.],.]
=> [[[[.,.],.],[.,.]],.]
=> ([(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)
=> ? = 3 + 2
[[[[.,[.,.]],.],.],.]
=> [[[[.,.],.],.],[.,.]]
=> ([(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)
=> ? = 3 + 2
[[[[[.,.],.],.],.],.]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 4 + 2
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 5 + 2
[.,[.,[.,[.,[[.,.],.]]]]]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 5 + 2
[.,[.,[.,[[.,.],[.,.]]]]]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 5 + 2
[.,[.,[.,[[.,[.,.]],.]]]]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 4 + 2
[.,[.,[.,[[[.,.],.],.]]]]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 5 + 2
[.,[.,[[.,.],[.,[.,.]]]]]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 5 + 2
[.,[.,[[.,.],[[.,.],.]]]]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 5 + 2
[.,[.,[[.,[.,.]],[.,.]]]]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 4 + 2
[.,[.,[[[.,.],.],[.,.]]]]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 5 + 2
[.,[.,[[.,[.,[.,.]]],.]]]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 4 + 2
[.,[.,[[.,[[.,.],.]],.]]]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 4 + 2
[.,[.,[[[.,.],[.,.]],.]]]
=> [.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 4 + 2
[.,[.,[[[.,[.,.]],.],.]]]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 4 + 2
[.,[.,[[[[.,.],.],.],.]]]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 5 + 2
[.,[[.,.],[[.,[.,.]],.]]]
=> [.,[[.,[[.,.],[.,.]]],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 4 + 2
[.,[[.,[.,.]],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 2
[.,[[.,[.,.]],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 2
[.,[[.,[.,[.,.]]],[.,.]]]
=> [.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 4 + 2
[.,[[.,[[.,.],.]],[.,.]]]
=> [.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 4 + 2
[.,[[[.,.],[.,.]],[.,.]]]
=> [.,[[[.,[.,.]],[.,.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 4 + 2
[.,[[[.,[.,.]],.],[.,.]]]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 2
[.,[[.,[.,[.,[.,.]]]],.]]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 2
[.,[[.,[.,[[.,.],.]]],.]]
=> [.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 2
[.,[[.,[[.,.],[.,.]]],.]]
=> [.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 2
[.,[[.,[[.,[.,.]],.]],.]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 3 + 2
[.,[[.,[[[.,.],.],.]],.]]
=> [.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 2
[.,[[[.,.],[.,[.,.]]],.]]
=> [.,[[[.,.],[.,[.,.]]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 4 + 2
[.,[[[.,.],[[.,.],.]],.]]
=> [.,[[[.,.],[[.,.],.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 4 + 2
[.,[[[.,[.,.]],[.,.]],.]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(0,3),(0,4),(0,5),(2,9),(3,8),(3,10),(4,7),(4,10),(5,7),(5,8),(6,1),(7,11),(8,11),(9,6),(10,2),(10,11),(11,9)],12)
=> ? = 3 + 2
[.,[[[[.,.],.],[.,.]],.]]
=> [.,[[[[.,.],[.,.]],.],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 4 + 2
[.,[[[.,[.,[.,.]]],.],.]]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 3 + 2
[.,[[[.,[[.,.],.]],.],.]]
=> [.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(0,5),(0,6),(2,9),(3,8),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8),(7,9),(8,10),(9,10),(10,4)],11)
=> ? = 3 + 2
[.,[[[[.,.],[.,.]],.],.]]
=> [.,[[[[.,.],.],[.,.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 4 + 2
[.,[[[[.,[.,.]],.],.],.]]
=> [.,[[[[.,.],.],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(0,3),(0,6),(2,8),(3,7),(4,2),(4,9),(5,1),(6,4),(6,7),(7,9),(8,5),(9,8)],10)
=> ? = 4 + 2
[[.,.],[.,[[.,[.,.]],.]]]
=> [[.,[.,[[.,.],[.,.]]]],.]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> ? = 4 + 2
[[.,.],[[.,[.,.]],[.,.]]]
=> [[.,[[.,[.,.]],[.,.]]],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> ? = 4 + 2
Description
The number of maximal proper sublattices of a lattice.
Matching statistic: St001668
Mp00012: Binary trees to Dyck path: up step, left tree, down step, right treeDyck paths
Mp00201: Dyck paths RingelPermutations
Mp00065: Permutations permutation posetPosets
St001668: Posets ⟶ ℤResult quality: 6% values known / values provided: 6%distinct values known / distinct values provided: 57%
Values
[.,.]
=> [1,0]
=> [2,1] => ([],2)
=> 0
[.,[.,.]]
=> [1,0,1,0]
=> [3,1,2] => ([(1,2)],3)
=> 1
[[.,.],.]
=> [1,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 2
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 2
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => ([(2,3)],4)
=> 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 3
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 3
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 3
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> 2
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 3
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 3
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 3
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 3
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 2
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> 2
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> 2
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 2
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 3
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> ? = 4
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ([(0,5),(1,3),(3,4),(4,2),(4,5)],6)
=> ? = 4
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => ([(0,4),(0,5),(1,2),(2,3),(2,5),(3,4)],6)
=> ? = 4
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => ([(1,5),(4,3),(5,2),(5,4)],6)
=> ? = 3
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,4),(4,2),(4,5),(5,3)],6)
=> ? = 4
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6)
=> ? = 4
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> ? = 4
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 3
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
=> ? = 4
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ([(1,3),(1,5),(4,2),(5,4)],6)
=> ? = 3
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ([(0,5),(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> ? = 3
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
=> ? = 3
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ([(1,4),(1,5),(4,3),(5,2)],6)
=> ? = 3
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,3),(1,5),(4,2),(5,4)],6)
=> ? = 4
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,3),(1,5),(4,2),(5,4)],6)
=> ? = 4
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
=> ? = 4
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> ? = 4
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(5,2)],6)
=> ? = 3
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6)
=> ? = 4
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => ([(1,5),(2,3),(3,5),(5,4)],6)
=> ? = 3
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => ([(0,5),(1,4),(1,5),(2,3),(3,4),(3,5)],6)
=> ? = 3
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,4),(4,2),(4,5),(5,3)],6)
=> ? = 4
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => ([(0,4),(0,5),(1,2),(2,3),(2,5),(3,4)],6)
=> ? = 4
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => ([(1,5),(2,3),(3,4),(4,5)],6)
=> ? = 3
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
=> ? = 3
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ([(0,4),(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> ? = 3
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ([(1,4),(2,3),(3,5),(4,5)],6)
=> ? = 3
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,3),(3,4),(4,2),(4,5)],6)
=> ? = 4
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> ? = 3
[[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => ([(0,5),(1,5),(2,3),(3,4),(3,5)],6)
=> ? = 3
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => ([(1,4),(1,5),(2,3),(2,4),(3,5)],6)
=> ? = 3
[[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => ([(1,5),(2,3),(2,5),(3,4)],6)
=> ? = 2
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,5),(5,4)],6)
=> ? = 3
[[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ([(0,5),(1,2),(1,3),(1,5),(5,4)],6)
=> ? = 3
[[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,5),(3,5)],6)
=> ? = 3
[[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => ([(1,4),(2,3),(2,5),(4,5)],6)
=> ? = 2
[[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ([(0,5),(1,4),(4,2),(4,3),(4,5)],6)
=> ? = 3
[[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => ([(3,4),(4,5)],6)
=> ? = 2
[[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => ([(0,5),(1,4),(2,3),(2,5),(4,5)],6)
=> ? = 2
[[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => ([(0,5),(1,3),(1,4),(1,5),(4,2)],6)
=> ? = 3
[[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => ([(1,3),(2,4),(4,5)],6)
=> ? = 3
[[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> ? = 4
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 5
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ([(0,6),(1,4),(3,5),(4,3),(5,2),(5,6)],7)
=> ? = 5
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => ([(0,5),(0,6),(1,3),(2,6),(3,4),(4,2),(4,5)],7)
=> ? = 5
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ([(1,5),(4,3),(5,6),(6,2),(6,4)],7)
=> ? = 4
[.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => ([(0,6),(1,4),(4,5),(5,2),(5,6),(6,3)],7)
=> ? = 5
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => ([(0,5),(0,6),(1,4),(3,6),(4,3),(4,5),(6,2)],7)
=> ? = 5
[.,[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => ([(0,2),(1,5),(1,6),(2,3),(2,6),(3,4),(3,5),(6,4)],7)
=> ? = 5
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => ([(1,5),(2,6),(3,6),(4,3),(5,2),(5,4)],7)
=> ? = 4
Description
The number of points of the poset minus the width of the poset.
Matching statistic: St001965
Mp00012: Binary trees to Dyck path: up step, left tree, down step, right treeDyck paths
Mp00201: Dyck paths RingelPermutations
Mp00063: Permutations to alternating sign matrixAlternating sign matrices
St001965: Alternating sign matrices ⟶ ℤResult quality: 6% values known / values provided: 6%distinct values known / distinct values provided: 57%
Values
[.,.]
=> [1,0]
=> [2,1] => [[0,1],[1,0]]
=> 0
[.,[.,.]]
=> [1,0,1,0]
=> [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
=> 1
[[.,.],.]
=> [1,1,0,0]
=> [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 2
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 2
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 2
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 2
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 3
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0]]
=> 3
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 3
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
=> 2
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 3
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
=> 3
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0]]
=> 3
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0]]
=> 2
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
=> 3
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]
=> 2
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> 2
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0]]
=> 2
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
=> 2
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 3
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? = 4
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 4
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 4
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 3
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 4
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 4
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 4
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? = 3
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 4
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> ? = 3
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 3
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 3
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 3
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 4
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 4
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 4
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 4
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 3
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 4
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? = 3
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 3
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> ? = 4
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 4
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? = 3
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 3
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> ? = 3
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> ? = 3
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0]]
=> ? = 4
[[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> ? = 3
[[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 3
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 3
[[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 2
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3
[[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 3
[[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 3
[[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> ? = 2
[[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 3
[[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> ? = 2
[[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 2
[[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 3
[[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 3
[[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 4
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
=> ? = 5
[.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0]]
=> ? = 5
[.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0]]
=> ? = 5
[.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => [[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[0,0,0,0,1,0,0],[1,0,0,0,0,0,0]]
=> ? = 4
[.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,1,2,3,6,7,4] => [[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 5
[.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [4,1,2,7,3,5,6] => [[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0]]
=> ? = 5
[.,[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,1,2,6,3,7,5] => [[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0]]
=> ? = 5
[.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,1,2,5,3,4,6] => [[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0]]
=> ? = 4
Description
The number of decreasable positions in the corner sum matrix of an alternating sign matrix. A decreasable position in a corner sum matrix is an entry such that the matrix obtained by decreasing it by one yields the corner sum matrix of some alternating sign matrix.
Matching statistic: St001773
Mp00014: Binary trees to 132-avoiding permutationPermutations
Mp00223: Permutations runsortPermutations
Mp00170: Permutations to signed permutationSigned permutations
St001773: Signed permutations ⟶ ℤResult quality: 6% values known / values provided: 6%distinct values known / distinct values provided: 57%
Values
[.,.]
=> [1] => [1] => [1] => 1 = 0 + 1
[.,[.,.]]
=> [2,1] => [1,2] => [1,2] => 2 = 1 + 1
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 2 = 1 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[[.,.],[.,.]]
=> [3,1,2] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 3 = 2 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 3 = 2 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 3 = 2 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => [1,3,4,2] => 3 = 2 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 4 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 4 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 4 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 3 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 4 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 4 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 4 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 3 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 4 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 3 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 3 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 3 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 3 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 4 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 4 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 4 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 4 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 3 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 4 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 3 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => ? = 3 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 4 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 4 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 3 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,4,2,3,5] => [1,4,2,3,5] => ? = 3 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => ? = 3 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,3,4,2,5] => [1,3,4,2,5] => ? = 3 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 4 + 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 3 + 1
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 3 + 1
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 3 + 1
[[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => [1,5,2,4,3] => [1,5,2,4,3] => ? = 2 + 1
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [1,5,2,3,4] => [1,5,2,3,4] => ? = 3 + 1
[[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 3 + 1
[[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => [1,2,5,3,4] => [1,2,5,3,4] => ? = 3 + 1
[[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => [1,3,5,2,4] => [1,3,5,2,4] => ? = 2 + 1
[[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => [1,2,3,5,4] => [1,2,3,5,4] => ? = 3 + 1
[[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 2 + 1
[[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => [1,4,5,2,3] => [1,4,5,2,3] => ? = 2 + 1
[[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => [1,2,4,5,3] => [1,2,4,5,3] => ? = 3 + 1
[[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => [1,3,4,5,2] => [1,3,4,5,2] => ? = 3 + 1
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ? = 4 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 5 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> [5,6,4,3,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 5 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [6,4,5,3,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 5 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> [5,4,6,3,2,1] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => ? = 4 + 1
[.,[.,[.,[[[.,.],.],.]]]]
=> [4,5,6,3,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 5 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [6,5,3,4,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 5 + 1
[.,[.,[[.,.],[[.,.],.]]]]
=> [5,6,3,4,2,1] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 5 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> [6,4,3,5,2,1] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ? = 4 + 1
Description
The number of minimal elements in Bruhat order not less than the signed permutation. The minimal elements in question are biGrassmannian, that is both the element and its inverse have at most one descent. This is the size of the essential set of the signed permutation, see [1].
The following 1 statistic also match your data. Click on any of them to see the details.
St001237The number of simple modules with injective dimension at most one or dominant dimension at least one.