- FindStat

Your data matches 11 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001247

Mapping

St001247: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of parts of a partition that are not congruent 2 modulo 3.

Matching statistic: St000873

Mapping

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000873: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 47%

Values

[1]
 => [1,0]
 => [1] => [1] => ? = 1
[2]
 => [1,0,1,0]
 => [1,2] => [1,2] => 2
[1,1]
 => [1,1,0,0]
 => [2,1] => [2,1] => 0
[3]
 => [1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 3
[2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 4
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,4,3] => 2
[2,2]
 => [1,1,1,0,0,0]
 => [3,1,2] => [2,3,1] => 0
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,2,3] => 2
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4,1,2,3] => 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 5
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,5,4] => 3
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [1,3,4,2] => 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,5,3,4] => 3
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [4,2,3,1] => 0
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,5,2,3,4] => 2
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 4
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [1,2,4,5,3] => 2
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => [1,2,3,6,4,5] => 4
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [2,4,1,3] => 2
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [1,5,3,4,2] => 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,4,5,6,3] => [1,2,6,3,4,5] => 3
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [2,3,4,1] => 0
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => [5,2,3,1,4] => 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,2] => [1,6,2,3,4,5] => 2
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 7
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => 5
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,4,5] => [1,2,3,5,6,4] => 3
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => 5
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [1,3,5,2,4] => 3
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,5,3,6,4] => [1,2,6,4,5,3] => 2
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => 4
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => [5,2,4,1,3] => 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [1,3,4,5,2] => 1
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,6,3] => [1,6,3,4,2,5] => 2
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => 3
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => [2,5,3,4,1] => 0
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,4,5,6,2] => [6,2,3,1,4,5] => 1
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => ? ∊ {2,2}
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => ? ∊ {2,2}
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => ? ∊ {0,0,1,1,2,4,5,6,6,8}
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,8,7] => ? ∊ {0,0,1,1,2,4,5,6,6,8}
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => 4
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,5,7,8,6] => [1,2,3,4,5,8,6,7] => ? ∊ {0,0,1,1,2,4,5,6,6,8}
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,5,6,3,4] => [1,2,4,6,3,5] => 4
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,6,4,7,5] => [1,2,3,7,5,6,4] => 3
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,4,6,7,8,5] => [1,2,3,4,8,5,6,7] => ? ∊ {0,0,1,1,2,4,5,6,6,8}
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => [2,5,1,3,4] => 2
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,4,5,2,6,3] => [1,6,3,5,2,4] => 2
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,6,3,4,5] => [1,2,4,5,6,3] => 2
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,5,3,6,7,4] => [1,2,7,4,5,3,6] => 3
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,8,4] => [1,2,3,8,4,5,6,7] => ? ∊ {0,0,1,1,2,4,5,6,6,8}
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => [2,4,5,1,3] => 3
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [3,4,1,5,6,2] => [6,2,4,1,3,5] => 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,4,2,5,6,7,3] => [1,7,3,4,2,5,6] => ? ∊ {0,0,1,1,2,4,5,6,6,8}
[3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,8,3] => [1,2,8,3,4,5,6,7] => ? ∊ {0,0,1,1,2,4,5,6,6,8}
[2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [3,1,4,5,6,7,2] => [7,2,3,1,4,5,6] => ? ∊ {0,0,1,1,2,4,5,6,6,8}
[2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,7,8,2] => [1,8,2,3,4,5,6,7] => ? ∊ {0,0,1,1,2,4,5,6,6,8}
[1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,8,1] => [8,1,2,3,4,5,6,7] => ? ∊ {0,0,1,1,2,4,5,6,6,8}
[9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => ? ∊ {1,1,1,1,1,2,2,2,3,3,4,4,5,5,6,7,7,9}
[8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,9,8] => ? ∊ {1,1,1,1,1,2,2,2,3,3,4,4,5,5,6,7,7,9}
[7,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,5,8,6,7] => [1,2,3,4,5,7,8,6] => ? ∊ {1,1,1,1,1,2,2,2,3,3,4,4,5,5,6,7,7,9}
[7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,5,6,8,9,7] => [1,2,3,4,5,6,9,7,8] => ? ∊ {1,1,1,1,1,2,2,2,3,3,4,4,5,5,6,7,7,9}
[6,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,4,7,5,8,6] => [1,2,3,4,8,6,7,5] => ? ∊ {1,1,1,1,1,2,2,2,3,3,4,4,5,5,6,7,7,9}
[6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,4,5,7,8,9,6] => [1,2,3,4,5,9,6,7,8] => ? ∊ {1,1,1,1,1,2,2,2,3,3,4,4,5,5,6,7,7,9}
[5,2,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,3,6,4,7,8,5] => [1,2,3,8,5,6,4,7] => ? ∊ {1,1,1,1,1,2,2,2,3,3,4,4,5,5,6,7,7,9}
[5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,3,4,6,7,8,9,5] => [1,2,3,4,9,5,6,7,8] => ? ∊ {1,1,1,1,1,2,2,2,3,3,4,4,5,5,6,7,7,9}
[4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,4,5,2,6,7,3] => [1,7,3,5,2,4,6] => ? ∊ {1,1,1,1,1,2,2,2,3,3,4,4,5,5,6,7,7,9}
[4,2,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,2,5,3,6,7,8,4] => [1,2,8,4,5,3,6,7] => ? ∊ {1,1,1,1,1,2,2,2,3,3,4,4,5,5,6,7,7,9}
[4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,8,9,4] => [1,2,3,9,4,5,6,7,8] => ? ∊ {1,1,1,1,1,2,2,2,3,3,4,4,5,5,6,7,7,9}
[3,3,1,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [3,4,1,5,6,7,2] => [7,2,4,1,3,5,6] => ? ∊ {1,1,1,1,1,2,2,2,3,3,4,4,5,5,6,7,7,9}
[3,2,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,4,2,5,6,7,8,3] => [1,8,3,4,2,5,6,7] => ? ∊ {1,1,1,1,1,2,2,2,3,3,4,4,5,5,6,7,7,9}
[3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,8,9,3] => [1,2,9,3,4,5,6,7,8] => ? ∊ {1,1,1,1,1,2,2,2,3,3,4,4,5,5,6,7,7,9}
[2,2,2,1,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [4,1,2,5,6,7,3] => [2,7,3,4,1,5,6] => ? ∊ {1,1,1,1,1,2,2,2,3,3,4,4,5,5,6,7,7,9}
[2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [3,1,4,5,6,7,8,2] => [8,2,3,1,4,5,6,7] => ? ∊ {1,1,1,1,1,2,2,2,3,3,4,4,5,5,6,7,7,9}
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,4,5,6,7,8,9,2] => [1,9,2,3,4,5,6,7,8] => ? ∊ {1,1,1,1,1,2,2,2,3,3,4,4,5,5,6,7,7,9}
[1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,7,8,9,1] => [9,1,2,3,4,5,6,7,8] => ? ∊ {1,1,1,1,1,2,2,2,3,3,4,4,5,5,6,7,7,9}
[10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,7,8,8,10}
[9,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,10,9] => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,7,8,8,10}
[8,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,5,6,9,7,8] => [1,2,3,4,5,6,8,9,7] => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,7,8,8,10}
[8,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,4,5,6,7,9,10,8] => [1,2,3,4,5,6,7,10,8,9] => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,7,8,8,10}
[7,3]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,3,4,7,8,5,6] => [1,2,3,4,6,8,5,7] => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,7,8,8,10}
[7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,4,5,8,6,9,7] => [1,2,3,4,5,9,7,8,6] => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,7,8,8,10}
[7,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,4,5,6,8,9,10,7] => [1,2,3,4,5,6,10,7,8,9] => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,7,8,8,10}
[6,3,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,2,3,6,7,4,8,5] => [1,2,3,8,5,7,4,6] => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,7,8,8,10}
[6,2,2]
 => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,2,3,4,8,5,6,7] => [1,2,3,4,6,7,8,5] => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,7,8,8,10}
[6,2,1,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,3,4,7,5,8,9,6] => [1,2,3,4,9,6,7,5,8] => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,7,8,8,10}
[6,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,3,4,5,7,8,9,10,6] => [1,2,3,4,5,10,6,7,8,9] => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,7,8,8,10}
[5,4,1]
 => [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,4,5,6,2,7,3] => [1,7,3,6,2,4,5] => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,7,8,8,10}
[5,3,1,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,2,5,6,3,7,8,4] => [1,2,8,4,6,3,5,7] => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,7,8,8,10}
[5,2,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,2,3,7,4,5,8,6] => [1,2,3,5,8,6,7,4] => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,7,8,8,10}
[5,2,1,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,2,3,6,4,7,8,9,5] => [1,2,3,9,5,6,4,7,8] => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,7,8,8,10}
[5,1,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,3,4,6,7,8,9,10,5] => [1,2,3,4,10,5,6,7,8,9] => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,7,8,8,10}
[4,4,1,1]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [3,4,5,1,6,7,2] => [7,2,5,1,3,4,6] => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,7,8,8,10}
[4,3,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,4,5,2,6,7,8,3] => [1,8,3,5,2,4,6,7] => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,7,8,8,10}
[4,2,2,1,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,2,6,3,4,7,8,5] => [1,2,4,8,5,6,3,7] => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,4,5,5,6,6,6,7,8,8,10}

Description

The aix statistic of a permutation. According to [1], this statistic on finite strings

$\pi$ of integers is given as follows: let

$m$ be the leftmost occurrence of the minimal entry and let

$\pi = \alpha\ m\ \beta$ . Then

$\operatorname{aix}\pi = \begin{cases} \operatorname{aix}\alpha & \text{ if } \alpha,\beta \neq \emptyset \\ 1 + \operatorname{aix}\beta & \text{ if } \alpha = \emptyset \\ 0 & \text{ if } \beta = \emptyset \end{cases}\ .$

Matching statistic: St001017

Mapping

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001017: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 41%

Values

[1]
 => [1,0]
 => [1,0]
 => [1,0]
 => 1
[2]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 3
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 4
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 4
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 2
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 2
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 0
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {1,2,3,3,5,5,7}
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? ∊ {1,2,3,3,5,5,7}
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 4
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? ∊ {1,2,3,3,5,5,7}
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 2
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? ∊ {1,2,3,3,5,5,7}
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? ∊ {1,2,3,3,5,5,7}
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 0
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? ∊ {1,2,3,3,5,5,7}
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? ∊ {1,2,3,3,5,5,7}
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,1,1,3,3,3,4,4,5,6,6,8}
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,1,1,3,3,3,4,4,5,6,6,8}
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? ∊ {0,0,1,1,3,3,3,4,4,5,6,6,8}
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,1,1,3,3,3,4,4,5,6,6,8}
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 4
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => ? ∊ {0,0,1,1,3,3,3,4,4,5,6,6,8}
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? ∊ {0,0,1,1,3,3,3,4,4,5,6,6,8}
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 3
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => ? ∊ {0,0,1,1,3,3,3,4,4,5,6,6,8}
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? ∊ {0,0,1,1,3,3,3,4,4,5,6,6,8}
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? ∊ {0,0,1,1,3,3,3,4,4,5,6,6,8}
[3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? ∊ {0,0,1,1,3,3,3,4,4,5,6,6,8}
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 2
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 1
[2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? ∊ {0,0,1,1,3,3,3,4,4,5,6,6,8}
[2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? ∊ {0,0,1,1,3,3,3,4,4,5,6,6,8}
[1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? ∊ {0,0,1,1,3,3,3,4,4,5,6,6,8}
[9]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[8,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[7,2]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? ∊ {0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[7,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? ∊ {0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[6,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? ∊ {0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
 => ? ∊ {0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[5,4]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 2
[5,3,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => ? ∊ {0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? ∊ {0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[5,2,1,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? ∊ {0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
 => ? ∊ {0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[4,4,1]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 2
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 3
[4,3,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => ? ∊ {0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => ? ∊ {0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[4,2,1,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? ∊ {0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? ∊ {0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[3,3,1,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? ∊ {0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[3,2,2,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[3,2,1,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? ∊ {0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? ∊ {0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[2,2,2,1,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? ∊ {0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? ∊ {0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? ∊ {0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? ∊ {0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[10]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,7,8,8,10}
[9,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,7,8,8,10}
[8,2]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,7,8,8,10}
[8,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,7,8,8,10}
[7,3]
 => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,7,8,8,10}
[7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,7,8,8,10}
[7,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0]
 => ? ∊ {0,0,0,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,6,6,6,7,8,8,10}

Description

Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001491

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 12%

Values

[1]
 => []
 =>  => ? => ? = 1
[2]
 => []
 =>  => ? => ? ∊ {0,2}
[1,1]
 => [1]
 => 10 => 00 => ? ∊ {0,2}
[3]
 => []
 =>  => ? => ? ∊ {1,3}
[2,1]
 => [1]
 => 10 => 00 => ? ∊ {1,3}
[1,1,1]
 => [1,1]
 => 110 => 010 => 1
[4]
 => []
 =>  => ? => ? ∊ {0,2,4}
[3,1]
 => [1]
 => 10 => 00 => ? ∊ {0,2,4}
[2,2]
 => [2]
 => 100 => 000 => ? ∊ {0,2,4}
[2,1,1]
 => [1,1]
 => 110 => 010 => 1
[1,1,1,1]
 => [1,1,1]
 => 1110 => 0110 => 2
[5]
 => []
 =>  => ? => ? ∊ {0,3,3,5}
[4,1]
 => [1]
 => 10 => 00 => ? ∊ {0,3,3,5}
[3,2]
 => [2]
 => 100 => 000 => ? ∊ {0,3,3,5}
[3,1,1]
 => [1,1]
 => 110 => 010 => 1
[2,2,1]
 => [2,1]
 => 1010 => 0010 => 1
[2,1,1,1]
 => [1,1,1]
 => 1110 => 0110 => 2
[1,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 01110 => ? ∊ {0,3,3,5}
[6]
 => []
 =>  => ? => ? ∊ {0,2,2,3,4,4,6}
[5,1]
 => [1]
 => 10 => 00 => ? ∊ {0,2,2,3,4,4,6}
[4,2]
 => [2]
 => 100 => 000 => ? ∊ {0,2,2,3,4,4,6}
[4,1,1]
 => [1,1]
 => 110 => 010 => 1
[3,3]
 => [3]
 => 1000 => 0000 => ? ∊ {0,2,2,3,4,4,6}
[3,2,1]
 => [2,1]
 => 1010 => 0010 => 1
[3,1,1,1]
 => [1,1,1]
 => 1110 => 0110 => 2
[2,2,2]
 => [2,2]
 => 1100 => 0100 => 1
[2,2,1,1]
 => [2,1,1]
 => 10110 => 00110 => ? ∊ {0,2,2,3,4,4,6}
[2,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 01110 => ? ∊ {0,2,2,3,4,4,6}
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => 011110 => ? ∊ {0,2,2,3,4,4,6}
[7]
 => []
 =>  => ? => ? ∊ {0,2,2,2,3,3,3,4,5,5,7}
[6,1]
 => [1]
 => 10 => 00 => ? ∊ {0,2,2,2,3,3,3,4,5,5,7}
[5,2]
 => [2]
 => 100 => 000 => ? ∊ {0,2,2,2,3,3,3,4,5,5,7}
[5,1,1]
 => [1,1]
 => 110 => 010 => 1
[4,3]
 => [3]
 => 1000 => 0000 => ? ∊ {0,2,2,2,3,3,3,4,5,5,7}
[4,2,1]
 => [2,1]
 => 1010 => 0010 => 1
[4,1,1,1]
 => [1,1,1]
 => 1110 => 0110 => 2
[3,3,1]
 => [3,1]
 => 10010 => 00010 => ? ∊ {0,2,2,2,3,3,3,4,5,5,7}
[3,2,2]
 => [2,2]
 => 1100 => 0100 => 1
[3,2,1,1]
 => [2,1,1]
 => 10110 => 00110 => ? ∊ {0,2,2,2,3,3,3,4,5,5,7}
[3,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 01110 => ? ∊ {0,2,2,2,3,3,3,4,5,5,7}
[2,2,2,1]
 => [2,2,1]
 => 11010 => 01010 => ? ∊ {0,2,2,2,3,3,3,4,5,5,7}
[2,2,1,1,1]
 => [2,1,1,1]
 => 101110 => 001110 => ? ∊ {0,2,2,2,3,3,3,4,5,5,7}
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => 011110 => ? ∊ {0,2,2,2,3,3,3,4,5,5,7}
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1111110 => 0111110 => ? ∊ {0,2,2,2,3,3,3,4,5,5,7}
[8]
 => []
 =>  => ? => ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[7,1]
 => [1]
 => 10 => 00 => ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[6,2]
 => [2]
 => 100 => 000 => ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[6,1,1]
 => [1,1]
 => 110 => 010 => 1
[5,3]
 => [3]
 => 1000 => 0000 => ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[5,2,1]
 => [2,1]
 => 1010 => 0010 => 1
[5,1,1,1]
 => [1,1,1]
 => 1110 => 0110 => 2
[4,4]
 => [4]
 => 10000 => 00000 => ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[4,3,1]
 => [3,1]
 => 10010 => 00010 => ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[4,2,2]
 => [2,2]
 => 1100 => 0100 => 1
[4,2,1,1]
 => [2,1,1]
 => 10110 => 00110 => ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[4,1,1,1,1]
 => [1,1,1,1]
 => 11110 => 01110 => ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[3,3,2]
 => [3,2]
 => 10100 => 00100 => ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[3,3,1,1]
 => [3,1,1]
 => 100110 => 000110 => ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[3,2,2,1]
 => [2,2,1]
 => 11010 => 01010 => ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[3,2,1,1,1]
 => [2,1,1,1]
 => 101110 => 001110 => ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => 111110 => 011110 => ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[2,2,2,2]
 => [2,2,2]
 => 11100 => 01100 => ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[2,2,2,1,1]
 => [2,2,1,1]
 => 110110 => 010110 => ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => 1011110 => 0011110 => ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1111110 => 0111110 => ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 11111110 => 01111110 => ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[9]
 => []
 =>  => ? => ? ∊ {0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[8,1]
 => [1]
 => 10 => 00 => ? ∊ {0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9}
[7,1,1]
 => [1,1]
 => 110 => 010 => 1
[6,2,1]
 => [2,1]
 => 1010 => 0010 => 1
[6,1,1,1]
 => [1,1,1]
 => 1110 => 0110 => 2
[5,2,2]
 => [2,2]
 => 1100 => 0100 => 1
[8,1,1]
 => [1,1]
 => 110 => 010 => 1
[7,2,1]
 => [2,1]
 => 1010 => 0010 => 1
[7,1,1,1]
 => [1,1,1]
 => 1110 => 0110 => 2
[6,2,2]
 => [2,2]
 => 1100 => 0100 => 1
[9,1,1]
 => [1,1]
 => 110 => 010 => 1
[8,2,1]
 => [2,1]
 => 1010 => 0010 => 1
[8,1,1,1]
 => [1,1,1]
 => 1110 => 0110 => 2
[7,2,2]
 => [2,2]
 => 1100 => 0100 => 1
[10,1,1]
 => [1,1]
 => 110 => 010 => 1
[9,2,1]
 => [2,1]
 => 1010 => 0010 => 1
[9,1,1,1]
 => [1,1,1]
 => 1110 => 0110 => 2
[8,2,2]
 => [2,2]
 => 1100 => 0100 => 1
[11,1,1]
 => [1,1]
 => 110 => 010 => 1
[10,2,1]
 => [2,1]
 => 1010 => 0010 => 1
[10,1,1,1]
 => [1,1,1]
 => 1110 => 0110 => 2
[9,2,2]
 => [2,2]
 => 1100 => 0100 => 1
[12,1,1]
 => [1,1]
 => 110 => 010 => 1
[11,2,1]
 => [2,1]
 => 1010 => 0010 => 1
[11,1,1,1]
 => [1,1,1]
 => 1110 => 0110 => 2
[10,2,2]
 => [2,2]
 => 1100 => 0100 => 1
[13,1,1]
 => [1,1]
 => 110 => 010 => 1
[12,2,1]
 => [2,1]
 => 1010 => 0010 => 1
[12,1,1,1]
 => [1,1,1]
 => 1110 => 0110 => 2
[11,2,2]
 => [2,2]
 => 1100 => 0100 => 1
[14,1,1]
 => [1,1]
 => 110 => 010 => 1
[13,2,1]
 => [2,1]
 => 1010 => 0010 => 1
[13,1,1,1]
 => [1,1,1]
 => 1110 => 0110 => 2
[12,2,2]
 => [2,2]
 => 1100 => 0100 => 1

Description

The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let

$A_n=K[x]/(x^n)$ . We associate to a nonempty subset S of an (n-1)-set the module

$M_S$ , which is the direct sum of

$A_n$ -modules with indecomposable non-projective direct summands of dimension

$i$ when

$i$ is in

$S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of

$M_S$ . We decode the subset as a binary word so that for example the subset

$S=\{1,3 \}$ of

$\{1,2,3 \}$ is decoded as 101.

Matching statistic: St001880
(load all 6 compositions to match this statistic)

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 41%

Values

[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => ([(0,1)],2)
 => ? = 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => ? ∊ {0,2}
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => ([(1,2)],3)
 => ? ∊ {0,2}
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => ? ∊ {1,1}
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 3
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => ? ∊ {1,1}
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ? ∊ {0,1,2,2}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => ? ∊ {0,1,2,2}
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,1,2,2}
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => ([(3,4)],5)
 => ? ∊ {0,1,2,2}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
 => ? ∊ {0,1,2,3,3,5}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => ? ∊ {0,1,2,3,3,5}
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => ? ∊ {0,1,2,3,3,5}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ? ∊ {0,1,2,3,3,5}
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,1,2,3,3,5}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(4,5)],6)
 => ? ∊ {0,1,2,3,3,5}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6)],7)
 => ? ∊ {0,1,1,2,2,2,3,4,6}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {0,1,1,2,2,2,3,4,6}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,2,2,2,3,4,6}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => ([(1,2),(1,3),(1,4)],5)
 => ? ∊ {0,1,1,2,2,2,3,4,6}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,2,2,2,3,4,6}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => ([(2,3),(2,4)],5)
 => ? ∊ {0,1,1,2,2,2,3,4,6}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ? ∊ {0,1,1,2,2,2,3,4,6}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,1,1,2,2,2,3,4,6}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ([(5,6)],7)
 => ? ∊ {0,1,1,2,2,2,3,4,6}
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7)],8)
 => ? ∊ {0,1,1,2,2,2,2,3,3,3,4,5,7}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {0,1,1,2,2,2,2,3,3,3,4,5,7}
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
 => ? ∊ {0,1,1,2,2,2,2,3,3,3,4,5,7}
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ? ∊ {0,1,1,2,2,2,2,3,3,3,4,5,7}
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 5
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ? ∊ {0,1,1,2,2,2,2,3,3,3,4,5,7}
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ? ∊ {0,1,1,2,2,2,2,3,3,3,4,5,7}
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ? ∊ {0,1,1,2,2,2,2,3,3,3,4,5,7}
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {0,1,1,2,2,2,2,3,3,3,4,5,7}
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ? ∊ {0,1,1,2,2,2,2,3,3,3,4,5,7}
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ([(1,4),(2,3),(2,4)],5)
 => ? ∊ {0,1,1,2,2,2,2,3,3,3,4,5,7}
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
 => ? ∊ {0,1,1,2,2,2,2,3,3,3,4,5,7}
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? ∊ {0,1,1,2,2,2,2,3,3,3,4,5,7}
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ([(6,7)],8)
 => ? ∊ {0,1,1,2,2,2,2,3,3,3,4,5,7}
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8)],9)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,6,6,8}
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,6,6,8}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,6,6,8}
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,6,6,8}
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,6,6,8}
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => 2
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,6,6,8}
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => ([(1,2),(1,3),(1,4),(1,5)],6)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,6,6,8}
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 5
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,6,6,8}
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => ([(0,5),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,6,6,8}
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,6,6,8}
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,6,6,8}
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,6,6,8}
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,3,3,3,3,4,4,6,6,8}
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
 => 2
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2
[6,2,1,1]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => 2
[5,4,1]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => 2
[5,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => 1
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[6,2,1,1,1]
 => [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 1
[5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 6
[5,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
 => 2
[6,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
 => 2
[6,3,2,1]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
 => 3
[6,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => 1
[5,4,2,1]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 6
[5,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => 5
[6,3,2,1,1]
 => [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
 => 3
[6,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => 1
[5,4,3,1]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 6
[5,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => 5
[5,3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 6
[6,5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => 3
[6,4,3,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 7
[6,4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
 => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
 => 2
[6,3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
 => 2
[6,5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
 => 3
[6,4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => ([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
 => 3
[6,3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
 => 2
[5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[6,5,4,1]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
 => 3
[6,5,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => 2
[6,4,3,2,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 7
[6,4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
 => 3
[6,3,2,2,2,1]
 => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
 => 1

Description

The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.

Matching statistic: St001233

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001233: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 29%

Values

[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 2 = 1 + 1
[2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3 = 2 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4 = 3 + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 3 = 2 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 3 = 2 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 5 = 4 + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => ? ∊ {3,5} + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 2 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 4 = 3 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {3,5} + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
 => ? ∊ {1,4,4,6} + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
 => ? ∊ {1,4,4,6} + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 1 = 0 + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 3 = 2 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2 = 1 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4 = 3 + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 3 = 2 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 3 = 2 + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? ∊ {1,4,4,6} + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {1,4,4,6} + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0]
 => ? ∊ {2,2,3,3,4,5,5,7} + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0]
 => ? ∊ {2,2,3,3,4,5,5,7} + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
 => ? ∊ {2,2,3,3,4,5,5,7} + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
 => ? ∊ {2,2,3,3,4,5,5,7} + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 4 = 3 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 3 = 2 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? ∊ {2,2,3,3,4,5,5,7} + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 3 = 2 + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? ∊ {2,2,3,3,4,5,5,7} + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? ∊ {2,2,3,3,4,5,5,7} + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,2,3,3,4,5,5,7} + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,0]
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,5,6,6,8} + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0,0]
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,5,6,6,8} + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0]
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,5,6,6,8} + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,5,6,6,8} + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,5,6,6,8} + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,5,6,6,8} + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,5,6,6,8} + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,5,6,6,8} + 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 3 = 2 + 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,5,6,6,8} + 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 3 = 2 + 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,5,6,6,8} + 1
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,5,6,6,8} + 1
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,5,6,6,8} + 1
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,5,6,6,8} + 1
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,5,6,6,8} + 1
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,5,6,6,8} + 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => ? ∊ {1,1,2,2,2,3,3,3,3,4,4,4,5,6,6,8} + 1
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0,0]
 => ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9} + 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,1,0,0,0]
 => ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9} + 1
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0,0]
 => ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9} + 1
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0,0]
 => ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9} + 1
[6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
 => ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9} + 1
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
 => ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9} + 1
[6,1,1,1]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,1,0,0,0]
 => ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9} + 1
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
 => ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9} + 1
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
 => ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9} + 1
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9} + 1
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9} + 1
[5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9} + 1
[4,4,1]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9} + 1
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 1 = 0 + 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 3 = 2 + 1
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9} + 1
[4,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9} + 1
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9} + 1
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9} + 1
[3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9} + 1
[3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9} + 1
[3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => ? ∊ {1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,6,7,7,9} + 1
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 2 = 1 + 1

Description

The number of indecomposable 2-dimensional modules with projective dimension one.

Matching statistic: St001875

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 12%

Values

[1]
 => [1,0,1,0]
 => [[1,1],[]]
 => ([],1)
 => ? = 1
[2]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,2}
[1,1]
 => [1,0,1,1,0,0]
 => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,2}
[3]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => ([],1)
 => ? ∊ {1,1,3}
[2,1]
 => [1,0,1,0,1,0]
 => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {1,1,3}
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => ([],1)
 => ? ∊ {1,1,3}
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => ([],1)
 => ? ∊ {0,1,2,2,4}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,1,2,2,4}
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,1,2,2,4}
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => ([],1)
 => ? ∊ {0,1,2,2,4}
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => ([],1)
 => ? ∊ {0,1,2,2,4}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[3,3,3,3],[2]]
 => ([],1)
 => ? ∊ {0,1,1,2,3,3,5}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => ([],1)
 => ? ∊ {0,1,1,2,3,3,5}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,1,1,2,3,3,5}
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,1,1,2,3,3,5}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {0,1,1,2,3,3,5}
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => ([],1)
 => ? ∊ {0,1,1,2,3,3,5}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3,1],[]]
 => ([],1)
 => ? ∊ {0,1,1,2,3,3,5}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[4,4,4,4],[3]]
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,3,4,4,6}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[4,4,4],[3]]
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,3,4,4,6}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,1,1,1,2,2,2,3,4,4,6}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,3,4,4,6}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,1,1,1,2,2,2,3,4,4,6}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,3,4,4,6}
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,3,4,4,6}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,1,1,1,2,2,2,3,4,4,6}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,1,1,1,2,2,2,3,4,4,6}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[4,4,1],[]]
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,3,4,4,6}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [[4,4,4,1],[]]
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,3,4,4,6}
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [[4,4,4,4,4],[3]]
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[3,3,3,3,3],[2]]
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[3,3,3,3],[2,1]]
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [[3,3,3,2],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [[3,3,3,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1],[]]
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2,1],[]]
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [[3,3,3,3,1],[]]
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[4,4,4,4,1],[]]
 => ([],1)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [[5,5,5,5,5],[4]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [[5,5,5,5],[4]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [[4,4,4,3],[3]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [[4,4,4,4],[3,1]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[4,4,3],[3]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [[2,2,2,2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,3,3,4,4,4,5,6,6,8}
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 3
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [[3,3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 3
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[3,3,2,2],[1,1]]
 => ([(0,2),(2,1)],3)
 => 3
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [[4,4,4,3],[3,1]]
 => ([(0,2),(2,1)],3)
 => 3
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [[4,4,3],[3,1]]
 => ([(0,2),(2,1)],3)
 => 3
[4,4,2]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [[4,3,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [[4,3,3],[2,1]]
 => ([(0,2),(2,1)],3)
 => 3
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [[4,3,2],[1,1]]
 => ([(0,2),(2,1)],3)
 => 3
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [[4,3,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [[4,4,2],[2,1]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[3,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [[4,4,3,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [[3,3,2,2],[2,1]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[5,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2,1],[2]]
 => ([(0,2),(2,1)],3)
 => 3
[5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [[3,3,3,2],[2,1,1]]
 => ([(0,2),(2,1)],3)
 => 3
[5,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3,1],[2,1]]
 => ([(0,2),(2,1)],3)
 => 3
[4,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 3
[3,3,3,1,1]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2,1],[1,1]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[5,3,3,1]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 3
[5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 4
[4,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[4,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 3
[5,3,3,2]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[5,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 4
[4,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[5,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 3
[5,4,4,2,1]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [[6,4],[3]]
 => ([(0,2),(2,1)],3)
 => 3
[5,4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [[6,3],[2]]
 => ([(0,2),(2,1)],3)
 => 3

Description

The number of simple modules with projective dimension at most 1.

Matching statistic: St000454

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 12%

Values

[1]
 => [1,0,1,0]
 => [.,[.,.]]
 => ([(0,1)],2)
 => 1
[2]
 => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => ([(0,2),(1,2)],3)
 => ? ∊ {0,2}
[1,1]
 => [1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => ([(0,2),(1,2)],3)
 => ? ∊ {0,2}
[3]
 => [1,1,1,0,0,0,1,0]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,1,3}
[2,1]
 => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => ([(0,2),(1,2)],3)
 => ? ∊ {1,1,3}
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,3}
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {0,1,2,2,4}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,1,2,2,4}
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,1,2,2,4}
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,1,2,2,4}
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,1,2,2,4}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? ∊ {0,1,1,3,3,5}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {0,1,1,3,3,5}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[.,[.,[.,.]]],.]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,1,1,3,3,5}
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [.,[[.,[.,.]],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,1,1,3,3,5}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,1,1,3,3,5}
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,1,1,3,3,5}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],[[.,.],.]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ? ∊ {0,1,1,1,2,2,2,3,4,4,6}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[[[.,.],.],.],[.,[.,.]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? ∊ {0,1,1,1,2,2,2,3,4,4,6}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {0,1,1,1,2,2,2,3,4,4,6}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,1,1,1,2,2,2,3,4,4,6}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {0,1,1,1,2,2,2,3,4,4,6}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,1,1,1,2,2,2,3,4,4,6}
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,1,1,1,2,2,2,3,4,4,6}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,1,1,1,2,2,2,3,4,4,6}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,1,1,1,2,2,2,3,4,4,6}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [.,[[[[.,.],.],.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {0,1,1,1,2,2,2,3,4,4,6}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[[.,.],.]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ? ∊ {0,1,1,1,2,2,2,3,4,4,6}
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [[[.,.],[.,.]],[[.,.],[.,[.,.]]]]
 => ([(0,7),(1,6),(2,6),(3,4),(4,7),(5,6),(5,7)],8)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],[.,[.,.]]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[[.,[.,.]],.],[.,[.,.]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [[.,[[.,.],.]],[.,[.,.]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [.,[[[.,[.,.]],.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [.,[[.,[[.,.],.]],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[.,[.,.]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,7),(5,7),(6,7)],8)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,3,4,5,5,7}
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [[[[.,.],.],[.,.]],[[.,.],[.,[.,.]]]]
 => ([(0,8),(1,7),(2,4),(3,5),(4,7),(5,8),(6,7),(6,8)],9)
 => ? ∊ {0,0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],[[[.,.],.],.]]
 => ([(0,6),(1,5),(2,7),(3,4),(3,5),(4,7),(6,7)],8)
 => ? ∊ {0,0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [[[[.,.],.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [[[[.,.],[.,[.,.]]],.],[.,.]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ? ∊ {0,0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[[.,.],.],[[.,[.,.]],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? ∊ {0,0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],[.,[.,.]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? ∊ {0,0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [[[[.,.],.],[.,[.,.]]],.]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[[.,.],.],[.,[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? ∊ {0,0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? ∊ {0,0,1,1,1,1,2,2,2,3,3,3,3,4,4,4,5,6,6,8}
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [.,[[[[.,.],[.,.]],.],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2
[6,3]
 => [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
 => [[[.,[.,.]],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2
[2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [.,[[.,[[.,.],[.,.]]],[.,.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2
[6,2,2,2]
 => [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
 => [[.,.],[[[.,.],.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2
[6,3,2,2]
 => [1,1,1,0,0,1,1,0,1,0,0,0,1,0]
 => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2
[6,2,2,2,1]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [[[[[.,.],.],[.,[.,.]]],.],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2
[5,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
 => [[.,[[[.,.],.],[[.,.],.]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2
[5,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [.,[[[[.,.],.],[[.,.],.]],.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2
[6,3,2,2,1]
 => [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
 => [[[[.,[.,.]],[.,[.,.]]],.],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2
[6,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [[.,[[[.,.],.],[.,[.,.]]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2
[6,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
 => [.,[[[[.,.],.],[.,[.,.]]],.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2
[5,3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,1,0,0]
 => [[.,[[.,[.,.]],[[.,.],.]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2
[5,3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [.,[[[.,[.,.]],[[.,.],.]],.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2
[5,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [.,[.,[[[.,.],.],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2
[4,4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
 => [[[.,.],[.,[[.,.],[.,.]]]],.]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2
[4,4,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => [.,[[.,.],[[[.,.],[.,.]],.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2
[6,3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,0,0,0,1,0]
 => [[.,[[.,[.,.]],[.,[.,.]]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2
[6,3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [.,[[[.,[.,.]],[.,[.,.]]],.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2
[6,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [.,[.,[[[.,.],.],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2
[5,3,2,2,2,1]
 => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2
[4,4,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [.,[[.,.],[.,[[.,.],[.,.]]]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2
[6,3,2,2,2,1]
 => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2

Description

The largest eigenvalue of a graph if it is integral. If a graph is

$d$ -regular, then its largest eigenvalue equals

$d$ . One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.

Matching statistic: St000259

Mapping

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 41%

Values

[1]
 => [1,0]
 => [1] => ([],1)
 => 0 = 1 - 1
[2]
 => [1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => 1 = 2 - 1
[1,1]
 => [1,1,0,0]
 => [1,2] => ([],2)
 => ? = 0 - 1
[3]
 => [1,0,1,0,1,0]
 => [2,1,3] => ([(1,2)],3)
 => ? ∊ {1,1} - 1
[2,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,3,2] => ([(1,2)],3)
 => ? ∊ {1,1} - 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {0,1,2,2} - 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[2,2]
 => [1,1,1,0,0,0]
 => [1,2,3] => ([],3)
 => ? ∊ {0,1,2,2} - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,2,2} - 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {0,1,2,2} - 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? ∊ {0,1,1,2,3} - 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,1,1,2,3} - 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 3 - 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 4 = 5 - 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,1,2,3} - 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,2,3} - 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {0,1,1,2,3} - 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
 => ? ∊ {0,1,1,1,2,2,2,3,4} - 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,1,1,1,2,2,2,3,4} - 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 5 = 6 - 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {0,1,1,1,2,2,2,3,4} - 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,2,2,2,3,4} - 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? ∊ {0,1,1,1,2,2,2,3,4} - 1
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([],4)
 => ? ∊ {0,1,1,1,2,2,2,3,4} - 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,1,1,1,2,2,2,3,4} - 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,3,1,5,4,6] => ([(1,2),(3,5),(4,5)],6)
 => ? ∊ {0,1,1,1,2,2,2,3,4} - 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
 => ? ∊ {0,1,1,1,2,2,2,3,4} - 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,6,5,7] => ([(1,6),(2,5),(3,4)],7)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,5} - 1
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,6,5,7] => ([(1,2),(3,6),(4,5),(5,6)],7)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,5} - 1
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,6,3,5,7] => ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,5} - 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,5} - 1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 4 = 5 - 1
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,4,1,6,3,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 6 = 7 - 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,5} - 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [2,3,1,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,5} - 1
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,4,1,5,3,7,6] => ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,5} - 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,5} - 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,5} - 1
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,1,5,4,7,6] => ([(0,3),(1,2),(4,6),(5,6)],7)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,5} - 1
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4,7,6] => ([(1,6),(2,5),(3,4)],7)
 => ? ∊ {0,1,1,1,2,2,2,2,3,3,5} - 1
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,6,5,8,7] => ([(0,7),(1,6),(2,5),(3,4)],8)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,6,8} - 1
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,6,5,8,7] => ([(0,3),(1,2),(4,7),(5,6),(6,7)],8)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,6,8} - 1
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,4,6,1,3,5,7] => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,6,8} - 1
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,6,3,5,8,7] => ([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,6,8} - 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => 4 = 5 - 1
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [2,4,1,6,7,3,5] => ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
 => 5 = 6 - 1
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,4,1,6,3,8,5,7] => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,6,8} - 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,6,8} - 1
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [2,3,1,4,6,5] => ([(1,2),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,6,8} - 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 3 = 4 - 1
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [2,4,1,5,3,6,7] => ([(2,6),(3,5),(4,5),(4,6)],7)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,6,8} - 1
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,4,1,6,3,7,5,8] => ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,6,8} - 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,6,8} - 1
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,6,8} - 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,6,8} - 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,3,1,6,4,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,6,8} - 1
[3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,4,1,5,3,7,6,8] => ([(1,2),(3,6),(4,5),(5,7),(6,7)],8)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,6,8} - 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,6,8} - 1
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,6,8} - 1
[2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,4,2,5,3,7,6] => ([(1,2),(3,6),(4,5),(5,6)],7)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,6,8} - 1
[2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,1,5,4,7,6,8] => ([(1,4),(2,3),(5,7),(6,7)],8)
 => ? ∊ {0,0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4,6,8} - 1
[6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,4,6,1,7,3,5] => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => 4 = 5 - 1
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,4,6,7,1,3,5] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => 3 = 4 - 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 3 = 4 - 1
[4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [2,4,1,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 4 = 5 - 1
[4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 3 - 1
[4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,4,5,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 4 = 5 - 1
[5,3,3]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,4,5,6,7,1,3] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 3 = 4 - 1
[4,3,2,2]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,3,4,6,1,7,5] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 4 = 5 - 1
[5,4,3]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [2,3,4,6,7,1,5] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 3 = 4 - 1
[4,3,3,3]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 3 - 1

Description

The diameter of a connected graph. This is the greatest distance between any pair of vertices.

Matching statistic: St001198
(load all 7 compositions to match this statistic)

Mapping

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 6%

Values

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

Description

The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ .

The following 1 statistic also match your data. Click on any of them to see the details.

St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ .