Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001958
(load all 4 compositions to match this statistic)
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001958: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => 0
[1,0,1,0]
 => [1,2] => 1
[1,1,0,0]
 => [2,1] => 1
[1,0,1,0,1,0]
 => [1,2,3] => 1
[1,0,1,1,0,0]
 => [1,3,2] => 2
[1,1,0,0,1,0]
 => [2,1,3] => 2
[1,1,0,1,0,0]
 => [2,3,1] => 2
[1,1,1,0,0,0]
 => [3,2,1] => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 3
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 3
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 3
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 3
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 3
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 3
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 3
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 3
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 3
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 3
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 3
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 4
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 4
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 4
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 4
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 4
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 4
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 4
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 4
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 4
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => 4
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 4
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 3
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 4
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 4
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 4
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 4
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 4
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 4
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 4
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 4
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 4
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 4
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => 4
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 4
Description
The degree of the polynomial interpolating the values of a permutation. Given a permutation $\pi\in\mathfrak S_n$ there is a polynomial $p$ of minimal degree such that $p(n)=\pi(n)$ for $n\in\{1,\dots,n\}$. This statistic records the degree of $p$.
Matching statistic: St001232
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 86%
Values
[1,0]
 => [1] => [1] => [1,0]
 => 0
[1,0,1,0]
 => [2,1] => [1,2] => [1,0,1,0]
 => 1
[1,1,0,0]
 => [1,2] => [1,2] => [1,0,1,0]
 => 1
[1,0,1,0,1,0]
 => [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => ? = 1
[1,0,1,1,0,0]
 => [2,3,1] => [1,2,3] => [1,0,1,0,1,0]
 => ? = 2
[1,1,0,0,1,0]
 => [3,1,2] => [1,2,3] => [1,0,1,0,1,0]
 => ? = 2
[1,1,0,1,0,0]
 => [2,1,3] => [1,3,2] => [1,0,1,1,0,0]
 => 2
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => ? = 3
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => ? = 3
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => ? = 3
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => ? = 3
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => ? = 3
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => ? = 3
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 3
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 3
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 3
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => ? = 3
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => ? = 3
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => ? = 3
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => ? = 3
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => ? = 4
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 3
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => ? = 4
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => ? = 4
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 3
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => ? = 4
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => ? = 4
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => ? = 4
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => 4
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => 4
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => ? = 4
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => ? = 3
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => ? = 4
[1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => ? = 3
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => ? = 4
[1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => ? = 4
[1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => ? = 3
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => ? = 4
[1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => ? = 4
[1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => ? = 4
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,1,5] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,1,6] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [4,5,3,2,1,6] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[1,1,0,1,0,1,1,0,0,0,1,0]
 => [6,3,4,2,1,5] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [5,3,4,2,1,6] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [4,3,5,2,1,6] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [3,4,5,2,1,6] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [6,4,2,3,1,5] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [5,4,2,3,1,6] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [4,5,2,3,1,6] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [6,3,2,4,1,5] => [1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [5,3,2,4,1,6] => [1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,1,6] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,1,6] => [1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => [1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [5,2,3,4,1,6] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,1,6] => [1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,1,6] => [1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,1,6] => [1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001645
Mapping
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => ([],1)
 => 1 = 0 + 1
[1,0,1,0]
 => [2,1] => [2,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,1,0,0]
 => [1,2] => [1,2] => ([],2)
 => ? = 1 + 1
[1,0,1,0,1,0]
 => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => ? = 1 + 1
[1,0,1,1,0,0]
 => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,1,0,0,1,0]
 => [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,1,0,1,0,0]
 => [1,3,2] => [1,3,2] => ([(1,2)],3)
 => ? = 2 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => ([],3)
 => ? = 1 + 1
[1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? = 1 + 1
[1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4 = 3 + 1
[1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => ? = 3 + 1
[1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ? = 3 + 1
[1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ? = 3 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 4 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => ? = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 4 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => ? = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ? = 4 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,4,6,1,3,5] => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => 6 = 5 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,4,5,6,1,3] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 6 = 5 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [2,5,6,1,3,4] => [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
 => [3,4,6,1,2,5] => [5,4,6,2,1,3] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,1,2] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [3,5,1,6,2,4] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 6 = 5 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
 => [3,5,6,1,2,4] => [5,6,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,1,1,0,0,0,1,1,0,1,0,0]
 => [4,5,1,6,2,3] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [4,5,6,1,2,3] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,1,1,0,1,0,0,0,1,1,0,0]
 => [4,6,1,2,3,5] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 6 = 5 + 1
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,6,1,2,3,4] => [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,4,6,1,7,3,5] => [4,6,7,1,5,2,3] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 7 = 6 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,4,6,7,1,3,5] => [5,6,7,4,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,6),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,4,5,7,1,3,6] => [5,6,3,7,1,2,4] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 7 = 6 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,4,5,6,7,1,3] => [6,7,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 7 = 6 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [2,5,6,1,7,3,4] => [4,7,6,1,5,3,2] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,5,6,7,1,3,4] => [5,7,6,4,1,3,2] => ([(0,1),(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [2,5,7,1,3,4,6] => [4,6,7,1,5,2,3] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 7 = 6 + 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,6,7,1,3,4,5] => [4,7,6,1,5,3,2] => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
Description
The pebbling number of a connected graph.
Matching statistic: St000848
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000848: Posets ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 43%
Values
[1,0]
 => [1] => ([],1)
 => ([],1)
 => ? = 0 - 1
[1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 0 = 1 - 1
[1,1,0,0]
 => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [3,2,1] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3 - 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3 - 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3 - 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3 - 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3 - 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
 => ([(0,2),(0,3),(1,5),(1,8),(2,6),(2,7),(3,1),(3,6),(3,7),(4,9),(5,9),(6,4),(6,8),(7,4),(7,5),(7,8),(8,9)],10)
 => ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ([(0,3),(0,4),(1,8),(1,9),(2,10),(2,11),(3,1),(3,5),(3,6),(3,7),(4,2),(4,5),(4,6),(4,7),(5,9),(5,11),(6,9),(6,10),(7,8),(7,10),(7,11),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
 => ([(0,2),(0,3),(1,4),(1,5),(2,7),(2,8),(3,1),(3,7),(3,8),(4,9),(5,9),(6,9),(7,5),(7,6),(8,4),(8,6)],10)
 => ? = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
 => ([(0,2),(0,3),(1,5),(1,8),(2,6),(2,7),(3,1),(3,6),(3,7),(4,9),(5,9),(6,4),(6,8),(7,4),(7,5),(7,8),(8,9)],10)
 => ? = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ([(0,2),(0,3),(1,5),(1,9),(2,6),(2,7),(2,8),(3,1),(3,6),(3,7),(3,8),(4,10),(5,10),(6,9),(7,4),(7,9),(8,4),(8,5),(9,10)],11)
 => ? = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ([(0,2),(0,3),(1,5),(1,9),(2,6),(2,7),(2,8),(3,1),(3,6),(3,7),(3,8),(4,10),(5,10),(6,9),(7,4),(7,9),(8,4),(8,5),(9,10)],11)
 => ? = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ([(0,3),(0,4),(1,8),(1,9),(2,10),(2,11),(3,1),(3,5),(3,6),(3,7),(4,2),(4,5),(4,6),(4,7),(5,9),(5,11),(6,9),(6,10),(7,8),(7,10),(7,11),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ([(0,3),(0,4),(1,8),(1,9),(2,10),(2,11),(3,1),(3,5),(3,6),(3,7),(4,2),(4,5),(4,6),(4,7),(5,9),(5,11),(6,9),(6,10),(7,8),(7,10),(7,11),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 4 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 4 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 4 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 3 - 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
 => ([(0,2),(0,3),(1,8),(1,9),(2,5),(2,6),(2,7),(3,1),(3,5),(3,6),(3,7),(4,10),(5,8),(5,9),(6,4),(6,9),(7,4),(7,8),(8,10),(9,10)],11)
 => ? = 4 - 1
[1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 4 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 4 - 1
[1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 4 - 1
[1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 4 - 1
[1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,2,1] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ([(0,2),(0,5),(1,11),(2,6),(2,7),(3,4),(3,9),(3,12),(4,1),(4,8),(4,10),(5,3),(5,6),(5,7),(6,12),(7,9),(7,12),(8,11),(9,8),(9,10),(10,11),(12,10)],13)
 => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,6,3,2,1] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ([(0,2),(0,5),(1,11),(2,6),(2,7),(3,4),(3,9),(3,12),(4,1),(4,8),(4,10),(5,3),(5,6),(5,7),(6,12),(7,9),(7,12),(8,11),(9,8),(9,10),(10,11),(12,10)],13)
 => ? = 5 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 5 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
Description
The balance constant multiplied with the number of linear extensions of a poset. A pair of elements $x,y$ of a poset is $\alpha$-balanced if the proportion $P(x,y)$ of linear extensions where $x$ comes before $y$ is between $\alpha$ and $1-\alpha$. The balance constant of a poset is $\max\min(P(x,y), P(y,x)).$ Kislitsyn [1] conjectured that every poset which is not a chain is $1/3$-balanced. Brightwell, Felsner and Trotter [2] show that it is at least $(1-\sqrt 5)/10$-balanced. Olson and Sagan [3] exhibit various posets that are $1/2$-balanced.
Matching statistic: St000849
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
Mp00125: Posets —dual poset⟶ Posets
St000849: Posets ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 43%
Values
[1,0]
 => [1] => ([],1)
 => ([],1)
 => ? = 0 - 1
[1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 0 = 1 - 1
[1,1,0,0]
 => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 0 = 1 - 1
[1,0,1,0,1,0]
 => [3,2,1] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[1,0,1,1,0,0]
 => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,1,0,0,1,0]
 => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,1,1,0,0,0]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3 - 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3 - 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3 - 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3 - 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(0,2),(0,3),(1,6),(2,4),(2,5),(3,1),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3 - 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 4 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 4 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 4 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
 => ([(0,2),(0,3),(1,5),(1,8),(2,6),(2,7),(3,1),(3,6),(3,7),(4,9),(5,9),(6,4),(6,8),(7,4),(7,5),(7,8),(8,9)],10)
 => ? = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
 => ([(0,2),(0,4),(1,11),(2,5),(2,6),(2,7),(3,1),(3,8),(3,9),(3,10),(4,3),(4,5),(4,6),(4,7),(5,9),(5,10),(6,8),(6,10),(7,8),(7,9),(8,11),(9,11),(10,11)],12)
 => ? = 4 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 4 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 3 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ([(0,3),(0,4),(1,8),(1,9),(2,10),(2,11),(3,1),(3,5),(3,6),(3,7),(4,2),(4,5),(4,6),(4,7),(5,9),(5,11),(6,9),(6,10),(7,8),(7,10),(7,11),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
 => ([(0,2),(0,3),(1,4),(1,5),(2,7),(2,8),(3,1),(3,7),(3,8),(4,9),(5,9),(6,9),(7,5),(7,6),(8,4),(8,6)],10)
 => ? = 3 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
 => ([(0,2),(0,3),(1,5),(1,8),(2,6),(2,7),(3,1),(3,6),(3,7),(4,9),(5,9),(6,4),(6,8),(7,4),(7,5),(7,8),(8,9)],10)
 => ? = 4 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ([(0,2),(0,3),(1,5),(1,9),(2,6),(2,7),(2,8),(3,1),(3,6),(3,7),(3,8),(4,10),(5,10),(6,9),(7,4),(7,9),(8,4),(8,5),(9,10)],11)
 => ? = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 4 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ([(0,2),(0,3),(1,5),(1,9),(2,6),(2,7),(2,8),(3,1),(3,6),(3,7),(3,8),(4,10),(5,10),(6,9),(7,4),(7,9),(8,4),(8,5),(9,10)],11)
 => ? = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 4 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 4 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ([(0,3),(0,4),(1,8),(1,9),(2,10),(2,11),(3,1),(3,5),(3,6),(3,7),(4,2),(4,5),(4,6),(4,7),(5,9),(5,11),(6,9),(6,10),(7,8),(7,10),(7,11),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 4 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ([(0,3),(0,4),(1,8),(1,9),(2,10),(2,11),(3,1),(3,5),(3,6),(3,7),(4,2),(4,5),(4,6),(4,7),(5,9),(5,11),(6,9),(6,10),(7,8),(7,10),(7,11),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 4 - 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 4 - 1
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 4 - 1
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 4 - 1
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 3 - 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
 => ([(0,2),(0,3),(1,8),(1,9),(2,5),(2,6),(2,7),(3,1),(3,5),(3,6),(3,7),(4,10),(5,8),(5,9),(6,4),(6,9),(7,4),(7,8),(8,10),(9,10)],11)
 => ? = 4 - 1
[1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ([(0,3),(0,4),(1,10),(2,7),(2,8),(3,1),(3,5),(3,6),(4,2),(4,5),(4,6),(5,8),(5,10),(6,7),(6,10),(7,9),(8,9),(10,9)],11)
 => ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ([(0,3),(0,4),(1,8),(1,10),(2,7),(2,9),(3,2),(3,5),(3,6),(4,1),(4,5),(4,6),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,11),(8,11),(9,11),(10,11)],12)
 => ? = 4 - 1
[1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 4 - 1
[1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 3 - 1
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 4 - 1
[1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ([(0,2),(0,4),(1,8),(2,5),(2,6),(3,1),(3,7),(3,9),(4,3),(4,5),(4,6),(5,9),(6,7),(6,9),(7,8),(9,8)],10)
 => ? = 4 - 1
[1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 4 - 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,2,1] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ([(0,2),(0,5),(1,11),(2,6),(2,7),(3,4),(3,9),(3,12),(4,1),(4,8),(4,10),(5,3),(5,6),(5,7),(6,12),(7,9),(7,12),(8,11),(9,8),(9,10),(10,11),(12,10)],13)
 => ? = 5 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,6,3,2,1] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
 => ([(0,2),(0,5),(1,11),(2,6),(2,7),(3,4),(3,9),(3,12),(4,1),(4,8),(4,10),(5,3),(5,6),(5,7),(6,12),(7,9),(7,12),(8,11),(9,8),(9,10),(10,11),(12,10)],13)
 => ? = 5 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 5 - 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
Description
The number of 1/3-balanced pairs in a poset. A pair of elements $x,y$ of a poset is $\alpha$-balanced if the proportion of linear extensions where $x$ comes before $y$ is between $\alpha$ and $1-\alpha$. Kislitsyn [1] conjectured that every poset which is not a chain has a $1/3$-balanced pair. Brightwell, Felsner and Trotter [2] show that at least a $(1-\sqrt 5)/10$-balanced pair exists in posets which are not chains. Olson and Sagan [3] show that posets corresponding to skew Ferrers diagrams have a $1/3$-balanced pair.