- FindStat

Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000019

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The cardinality of the support of a permutation. A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$. The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product. See [2], Definition 1 and Proposition 10. The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$. Thus, the connectivity set is the complement of the support.

Matching statistic: St001207
(load all 4 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 20%

Values

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

Description

The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.

Matching statistic: St001095

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001095: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 20%

Values

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

Description

The number of non-isomorphic posets with precisely one further covering relation.

Matching statistic: St001964

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 10%

Values

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

Description

The interval resolution global dimension of a poset. This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.

Matching statistic: St001683

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001683: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 30%

Values

[1,0]
 => [1,1,0,0]
 => [2,3,1] => [3,1,2] => 0
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => [4,2,3,1] => 0
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => 0
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [4,2,5,3,1] => 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [5,2,3,1,4] => 0
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [5,3,4,1,2] => 0
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [5,2,3,4,1] => 0
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => 0
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [5,3,1,6,4,2] => 2
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,2,6,3,1,5] => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [6,4,5,2,3,1] => 0
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [5,2,4,6,3,1] => 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [6,2,3,1,4,5] => 0
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => [5,3,6,4,1,2] => 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,5,4,1,6,3] => [6,3,4,1,2,5] => 0
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => [5,2,3,6,4,1] => 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => [5,2,6,3,4,1] => 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5,3,4,1,6,2] => [6,2,3,4,1,5] => 0
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [2,3,6,5,1,4] => [6,4,5,1,2,3] => 0
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2,6,4,5,1,3] => [6,3,4,5,1,2] => 0
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [6,3,4,5,1,2] => [6,2,3,4,5,1] => 0
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => [6,4,2,7,5,3,1] => ? = 3
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,1,2,3,7,4] => [5,3,1,7,4,2,6] => ? = 2
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => [4,2,7,5,6,3,1] => ? = 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,7,1,2,6,3,4] => [6,3,1,5,7,4,2] => ? = 3
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [5,4,1,2,6,7,3] => [4,2,7,3,1,5,6] => ? = 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => [6,4,7,5,2,3,1] => ? = 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [6,3,1,5,2,7,4] => [7,4,5,2,3,1,6] => ? = 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [7,4,1,6,2,3,5] => [7,5,2,4,6,3,1] => ? = 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [6,7,1,5,2,3,4] => [6,3,1,7,4,5,2] => ? = 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => [5,2,4,7,3,1,6] => ? = 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [4,3,1,7,6,2,5] => [7,5,6,2,3,1,4] => ? = 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [7,3,1,5,6,2,4] => [7,4,5,6,2,3,1] => ? = 0
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [7,4,1,5,6,2,3] => [6,2,4,5,7,3,1] => ? = 3
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [4,3,1,5,6,7,2] => [7,2,3,1,4,5,6] => ? = 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => [6,4,1,2,7,5,3] => ? = 2
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => [5,3,7,4,1,2,6] => ? = 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [2,7,4,1,6,3,5] => [7,5,6,3,4,1,2] => ? = 0
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [2,7,5,1,6,3,4] => [6,3,5,7,4,1,2] => ? = 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,5,4,1,6,7,3] => [7,3,4,1,2,5,6] => ? = 0
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [6,3,7,1,2,4,5] => [6,4,1,7,5,2,3] => ? = 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [6,3,5,1,2,7,4] => [5,2,3,7,4,1,6] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [6,7,4,1,2,3,5] => [6,3,4,1,7,5,2] => ? = 3
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [6,7,5,1,2,3,4] => [7,4,1,6,3,5,2] => ? = 2
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,5,4,1,2,7,3] => [5,2,7,3,4,1,6] => ? = 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [7,3,4,1,6,2,5] => [7,5,6,2,3,4,1] => ? = 0
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [7,3,5,1,6,2,4] => [6,2,3,5,7,4,1] => ? = 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [7,5,4,1,6,2,3] => [6,2,5,7,3,4,1] => ? = 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [5,3,4,1,6,7,2] => [7,2,3,4,1,5,6] => ? = 0
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [2,3,7,6,1,4,5] => [6,4,7,5,1,2,3] => ? = 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [2,3,6,5,1,7,4] => [7,4,5,1,2,3,6] => ? = 0
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,7,4,6,1,3,5] => [6,3,4,7,5,1,2] => ? = 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,7,6,5,1,3,4] => [6,3,7,4,5,1,2] => ? = 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,6,4,5,1,7,3] => [7,3,4,5,1,2,6] => ? = 0
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [7,3,4,6,1,2,5] => [6,2,3,4,7,5,1] => ? = 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [7,3,6,5,1,2,4] => [6,2,3,7,4,5,1] => ? = 2
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [7,6,4,5,1,2,3] => [6,2,7,3,4,5,1] => ? = 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [6,3,4,5,1,7,2] => [7,2,3,4,5,1,6] => ? = 0
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [2,3,4,7,6,1,5] => [7,5,6,1,2,3,4] => ? = 0
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [2,3,7,5,6,1,4] => [7,4,5,6,1,2,3] => ? = 0
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [2,7,4,5,6,1,3] => [7,3,4,5,6,1,2] => ? = 0
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [7,3,4,5,6,1,2] => [7,2,3,4,5,6,1] => ? = 0
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [8,7,1,2,3,4,5,6] => [8,6,4,2,7,5,3,1] => ? = 4
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [7,6,1,2,3,4,8,5] => [6,4,2,8,5,3,1,7] => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [5,8,1,2,3,7,4,6] => [5,3,1,8,6,7,4,2] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [8,6,1,2,3,7,4,5] => [7,4,2,6,8,5,3,1] => ? = 4
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [5,6,1,2,3,7,8,4] => [5,3,1,8,4,2,6,7] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [8,4,1,2,7,3,5,6] => [4,2,7,5,8,6,3,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [7,4,1,2,6,3,8,5] => [4,2,8,5,6,3,1,7] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [5,8,1,2,7,3,4,6] => [8,6,3,1,5,7,4,2] => ? = 3

Description

The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation.