Processing math: 100%

Your data matches 58 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001527
Mapping
St001527: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1
[2]
 => 2
[1,1]
 => 1
[3]
 => 3
[2,1]
 => 1
[1,1,1]
 => 3
[4]
 => 4
[3,1]
 => 1
[2,2]
 => 4
[2,1,1]
 => 1
[1,1,1,1]
 => 2
[5]
 => 5
[4,1]
 => 1
[3,2]
 => 5
[3,1,1]
 => 5
[2,2,1]
 => 5
[2,1,1,1]
 => 1
[1,1,1,1,1]
 => 5
[6]
 => 6
[5,1]
 => 1
[4,2]
 => 6
[4,1,1]
 => 3
[3,3]
 => 3
[3,2,1]
 => 2
[3,1,1,1]
 => 3
[2,2,2]
 => 6
[2,2,1,1]
 => 3
[2,1,1,1,1]
 => 2
[1,1,1,1,1,1]
 => 3
[7]
 => 7
[6,1]
 => 1
[5,2]
 => 7
[5,1,1]
 => 7
[4,3]
 => 7
[4,2,1]
 => 7
[4,1,1,1]
 => 1
[3,3,1]
 => 7
[3,2,2]
 => 7
[3,2,1,1]
 => 7
[3,1,1,1,1]
 => 7
[2,2,2,1]
 => 7
[2,2,1,1,1]
 => 7
[2,1,1,1,1,1]
 => 1
[1,1,1,1,1,1,1]
 => 7
Description
The cyclic permutation representation number of an integer partition. This is the size of the largest cyclic group C such that the fake degree is the character of a permutation representation of C.
Matching statistic: St001614
(load all 2 compositions to match this statistic)
Mapping
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001614: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1],[]]
 => 1
[2]
 => [[2],[]]
 => 2
[1,1]
 => [[1,1],[]]
 => 1
[3]
 => [[3],[]]
 => 3
[2,1]
 => [[2,1],[]]
 => 1
[1,1,1]
 => [[1,1,1],[]]
 => 3
[4]
 => [[4],[]]
 => 4
[3,1]
 => [[3,1],[]]
 => 1
[2,2]
 => [[2,2],[]]
 => 4
[2,1,1]
 => [[2,1,1],[]]
 => 1
[1,1,1,1]
 => [[1,1,1,1],[]]
 => 2
[5]
 => [[5],[]]
 => 5
[4,1]
 => [[4,1],[]]
 => 1
[3,2]
 => [[3,2],[]]
 => 5
[3,1,1]
 => [[3,1,1],[]]
 => 5
[2,2,1]
 => [[2,2,1],[]]
 => 5
[2,1,1,1]
 => [[2,1,1,1],[]]
 => 1
[1,1,1,1,1]
 => [[1,1,1,1,1],[]]
 => 5
[6]
 => [[6],[]]
 => 6
[5,1]
 => [[5,1],[]]
 => 1
[4,2]
 => [[4,2],[]]
 => 6
[4,1,1]
 => [[4,1,1],[]]
 => 3
[3,3]
 => [[3,3],[]]
 => 3
[3,2,1]
 => [[3,2,1],[]]
 => 2
[3,1,1,1]
 => [[3,1,1,1],[]]
 => 3
[2,2,2]
 => [[2,2,2],[]]
 => 6
[2,2,1,1]
 => [[2,2,1,1],[]]
 => 3
[2,1,1,1,1]
 => [[2,1,1,1,1],[]]
 => 2
[1,1,1,1,1,1]
 => [[1,1,1,1,1,1],[]]
 => 3
[7]
 => [[7],[]]
 => 7
[6,1]
 => [[6,1],[]]
 => 1
[5,2]
 => [[5,2],[]]
 => 7
[5,1,1]
 => [[5,1,1],[]]
 => 7
[4,3]
 => [[4,3],[]]
 => 7
[4,2,1]
 => [[4,2,1],[]]
 => 7
[4,1,1,1]
 => [[4,1,1,1],[]]
 => 1
[3,3,1]
 => [[3,3,1],[]]
 => 7
[3,2,2]
 => [[3,2,2],[]]
 => 7
[3,2,1,1]
 => [[3,2,1,1],[]]
 => 7
[3,1,1,1,1]
 => [[3,1,1,1,1],[]]
 => 7
[2,2,2,1]
 => [[2,2,2,1],[]]
 => 7
[2,2,1,1,1]
 => [[2,2,1,1,1],[]]
 => 7
[2,1,1,1,1,1]
 => [[2,1,1,1,1,1],[]]
 => 1
[1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1],[]]
 => 7
Description
The cyclic permutation representation number of a skew partition. This is the size of the largest cyclic group C such that the fake degree is the character of a permutation representation of C. See [[St001527]] for the restriction of this statistic to integer partitions.
Matching statistic: St001232
(load all 10 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0]
 => [1,0]
 => [1,0]
 => 0 = 1 - 1
[2]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[1,1]
 => [1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 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]
 => 0 = 1 - 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? ∊ {1,2} - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? ∊ {1,2} - 1
[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
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 5 - 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? ∊ {1,5,5,5} - 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 5 - 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? ∊ {1,5,5,5} - 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? ∊ {1,5,5,5} - 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]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? ∊ {1,5,5,5} - 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]
 => 0 = 1 - 1
[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]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 6 - 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? ∊ {2,2,3,3,3,3,3,6} - 1
[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]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 6 - 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? ∊ {2,2,3,3,3,3,3,6} - 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? ∊ {2,2,3,3,3,3,3,6} - 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]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ? ∊ {2,2,3,3,3,3,3,6} - 1
[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]
 => ? ∊ {2,2,3,3,3,3,3,6} - 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]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? ∊ {2,2,3,3,3,3,3,6} - 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]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ? ∊ {2,2,3,3,3,3,3,6} - 1
[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]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => ? ∊ {2,2,3,3,3,3,3,6} - 1
[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
[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]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 6 = 7 - 1
[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]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => ? ∊ {1,1,7,7,7,7,7,7,7,7,7,7} - 1
[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]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 6 = 7 - 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ? ∊ {1,1,7,7,7,7,7,7,7,7,7,7} - 1
[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]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ? ∊ {1,1,7,7,7,7,7,7,7,7,7,7} - 1
[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]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => ? ∊ {1,1,7,7,7,7,7,7,7,7,7,7} - 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? ∊ {1,1,7,7,7,7,7,7,7,7,7,7} - 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? ∊ {1,1,7,7,7,7,7,7,7,7,7,7} - 1
[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]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => ? ∊ {1,1,7,7,7,7,7,7,7,7,7,7} - 1
[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]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => ? ∊ {1,1,7,7,7,7,7,7,7,7,7,7} - 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? ∊ {1,1,7,7,7,7,7,7,7,7,7,7} - 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]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => ? ∊ {1,1,7,7,7,7,7,7,7,7,7,7} - 1
[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]
 => [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
 => ? ∊ {1,1,7,7,7,7,7,7,7,7,7,7} - 1
[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]
 => [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
 => ? ∊ {1,1,7,7,7,7,7,7,7,7,7,7} - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000454
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 71%
Values
[1]
 => 10 => [1,2] => ([(1,2)],3)
 => 1
[2]
 => 100 => [1,3] => ([(2,3)],4)
 => 1
[1,1]
 => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[3]
 => 1000 => [1,4] => ([(3,4)],5)
 => 1
[2,1]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
[1,1,1]
 => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[4]
 => 10000 => [1,5] => ([(4,5)],6)
 => 1
[3,1]
 => 10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,4}
[2,2]
 => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2
[2,1,1]
 => 10110 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,4}
[1,1,1,1]
 => 11110 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[5]
 => 100000 => [1,6] => ([(5,6)],7)
 => 1
[4,1]
 => 100010 => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,5,5,5,5}
[3,2]
 => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,5,5,5,5}
[3,1,1]
 => 100110 => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,5,5,5,5}
[2,2,1]
 => 11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,5,5,5,5}
[2,1,1,1]
 => 101110 => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,5,5,5,5}
[1,1,1,1,1]
 => 111110 => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[6]
 => 1000000 => [1,7] => ([(6,7)],8)
 => ? ∊ {1,2,3,3,3,3,6,6,6}
[5,1]
 => 1000010 => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,2,3,3,3,3,6,6,6}
[4,2]
 => 100100 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,3,3,3,3,6,6,6}
[4,1,1]
 => 1000110 => [1,4,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,2,3,3,3,3,6,6,6}
[3,3]
 => 11000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 2
[3,2,1]
 => 101010 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,3,3,3,3,6,6,6}
[3,1,1,1]
 => 1001110 => [1,3,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,2,3,3,3,3,6,6,6}
[2,2,2]
 => 11100 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[2,2,1,1]
 => 110110 => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,3,3,3,3,6,6,6}
[2,1,1,1,1]
 => 1011110 => [1,2,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,2,3,3,3,3,6,6,6}
[1,1,1,1,1,1]
 => 1111110 => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,2,3,3,3,3,6,6,6}
[7]
 => 10000000 => [1,8] => ([(7,8)],9)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[6,1]
 => 10000010 => [1,6,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[5,2]
 => 1000100 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[5,1,1]
 => 10000110 => [1,5,1,2] => ([(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[4,3]
 => 101000 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[4,2,1]
 => 1001010 => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[4,1,1,1]
 => 10001110 => [1,4,1,1,2] => ([(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[3,3,1]
 => 110010 => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[3,2,2]
 => 101100 => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[3,2,1,1]
 => 1010110 => [1,2,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[3,1,1,1,1]
 => 10011110 => [1,3,1,1,1,2] => ([(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[2,2,2,1]
 => 111010 => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[2,2,1,1,1]
 => 1101110 => [1,1,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[2,1,1,1,1,1]
 => 10111110 => [1,2,1,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,1,1,1,1,1]
 => 11111110 => [1,1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
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: St001645
(load all 3 compositions to match this statistic)
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1]]
 => [1] => ([],1)
 => 1
[2]
 => [[1,2]]
 => [1,2] => ([],2)
 => ? = 1
[1,1]
 => [[1],[2]]
 => [2,1] => ([(0,1)],2)
 => 2
[3]
 => [[1,2,3]]
 => [1,2,3] => ([],3)
 => ? ∊ {1,3}
[2,1]
 => [[1,3],[2]]
 => [2,1,3] => ([(1,2)],3)
 => ? ∊ {1,3}
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => ([],4)
 => ? ∊ {1,1,2}
[3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {1,1,2}
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
[2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,2}
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => ([],5)
 => ? ∊ {1,1,5,5,5,5}
[4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ? ∊ {1,1,5,5,5,5}
[3,2]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {1,1,5,5,5,5}
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,5,5,5,5}
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,5,5,5,5}
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,5,5,5,5}
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => ([],6)
 => ? ∊ {1,2,2,3,3,3,3,3}
[5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => ([(4,5)],6)
 => ? ∊ {1,2,2,3,3,3,3,3}
[4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {1,2,2,3,3,3,3,3}
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => ([(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,2,3,3,3,3,3}
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => 6
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [4,2,5,1,3,6] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,2,3,3,3,3,3}
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,2,3,3,3,3,3}
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 6
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [5,3,2,6,1,4] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,2,3,3,3,3,3}
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [5,4,3,2,1,6] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,2,2,3,3,3,3,3}
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[7]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => ([],7)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7}
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => ([(5,6)],7)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7}
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => ([(3,5),(3,6),(4,5),(4,6)],7)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7}
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => ([(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7}
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7}
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => ([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7}
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => [4,3,2,1,5,6,7] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7}
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [5,2,6,7,1,3,4] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(4,5),(5,6)],7)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7}
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [5,6,3,4,1,2,7] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7}
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [5,3,2,6,1,4,7] => ([(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7}
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7}
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3] => ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5] => ([(0,1),(0,6),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7}
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1,7] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7}
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7
Description
The pebbling number of a connected graph.
Matching statistic: St001330
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 71%
Values
[1]
 => 10 => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[2]
 => 100 => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[1,1]
 => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3]
 => 1000 => [1,4] => ([(3,4)],5)
 => 2 = 1 + 1
[2,1]
 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,1,1]
 => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4]
 => 10000 => [1,5] => ([(4,5)],6)
 => 2 = 1 + 1
[3,1]
 => 10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,4} + 1
[2,2]
 => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[2,1,1]
 => 10110 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,4} + 1
[1,1,1,1]
 => 11110 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
[5]
 => 100000 => [1,6] => ([(5,6)],7)
 => ? ∊ {1,1,5,5,5,5} + 1
[4,1]
 => 100010 => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,5,5,5,5} + 1
[3,2]
 => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,5,5,5,5} + 1
[3,1,1]
 => 100110 => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,5,5,5,5} + 1
[2,2,1]
 => 11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,5,5,5,5} + 1
[2,1,1,1]
 => 101110 => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,5,5,5,5} + 1
[1,1,1,1,1]
 => 111110 => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 6 = 5 + 1
[6]
 => 1000000 => [1,7] => ([(6,7)],8)
 => ? ∊ {1,2,3,3,3,3,6,6,6} + 1
[5,1]
 => 1000010 => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,2,3,3,3,3,6,6,6} + 1
[4,2]
 => 100100 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,3,3,3,3,6,6,6} + 1
[4,1,1]
 => 1000110 => [1,4,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,2,3,3,3,3,6,6,6} + 1
[3,3]
 => 11000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[3,2,1]
 => 101010 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,3,3,3,3,6,6,6} + 1
[3,1,1,1]
 => 1001110 => [1,3,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,2,3,3,3,3,6,6,6} + 1
[2,2,2]
 => 11100 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
[2,2,1,1]
 => 110110 => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,2,3,3,3,3,6,6,6} + 1
[2,1,1,1,1]
 => 1011110 => [1,2,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,2,3,3,3,3,6,6,6} + 1
[1,1,1,1,1,1]
 => 1111110 => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,2,3,3,3,3,6,6,6} + 1
[7]
 => 10000000 => [1,8] => ([(7,8)],9)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[6,1]
 => 10000010 => [1,6,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[5,2]
 => 1000100 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[5,1,1]
 => 10000110 => [1,5,1,2] => ([(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[4,3]
 => 101000 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[4,2,1]
 => 1001010 => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[4,1,1,1]
 => 10001110 => [1,4,1,1,2] => ([(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[3,3,1]
 => 110010 => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[3,2,2]
 => 101100 => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[3,2,1,1]
 => 1010110 => [1,2,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[3,1,1,1,1]
 => 10011110 => [1,3,1,1,1,2] => ([(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[2,2,2,1]
 => 111010 => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[2,2,1,1,1]
 => 1101110 => [1,1,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[2,1,1,1,1,1]
 => 10111110 => [1,2,1,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[1,1,1,1,1,1,1]
 => 11111110 => [1,1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
Description
The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of q possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number HG(G) of a graph G is the largest integer q such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of q possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000663
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000663: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 43%
Values
[1]
 => [1,0]
 => [(1,2)]
 => [2,1] => 0 = 1 - 1
[2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 1 = 2 - 1
[1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => [3,4,2,1] => 0 = 1 - 1
[3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 2 = 3 - 1
[2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => 0 = 1 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => 2 = 3 - 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => ? ∊ {1,2,4,4} - 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,7,8,6,5] => ? ∊ {1,2,4,4} - 1
[2,2]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => 0 = 1 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => ? ∊ {1,2,4,4} - 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? ∊ {1,2,4,4} - 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => ? ∊ {1,1,5,5,5,5,5} - 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,9,10,8,7] => ? ∊ {1,1,5,5,5,5,5} - 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,6,7,8,5,4,3] => ? ∊ {1,1,5,5,5,5,5} - 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,7,9,6,10,8,5] => ? ∊ {1,1,5,5,5,5,5} - 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => ? ∊ {1,1,5,5,5,5,5} - 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9)]
 => [2,1,5,7,4,9,6,10,8,3] => ? ∊ {1,1,5,5,5,5,5} - 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [3,5,2,7,4,9,6,10,8,1] => ? ∊ {1,1,5,5,5,5,5} - 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} - 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
 => [2,1,4,3,6,5,8,7,11,12,10,9] => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} - 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => [2,1,4,3,8,9,10,7,6,5] => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} - 1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
 => [2,1,4,3,6,5,9,11,8,12,10,7] => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} - 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} - 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,6,7,9,5,4,10,8,3] => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} - 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
 => [2,1,4,3,7,9,6,11,8,12,10,5] => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} - 1
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} - 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [4,5,7,3,2,9,6,10,8,1] => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} - 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [2,1,5,7,4,9,6,11,8,12,10,3] => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} - 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
 => [3,5,2,7,4,9,6,11,8,12,10,1] => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} - 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
 => [2,1,4,3,6,5,8,7,10,9,12,11,14,13] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
 => [2,1,4,3,6,5,8,7,10,9,13,14,12,11] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
 => [2,1,4,3,6,5,10,11,12,9,8,7] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,14),(10,11),(12,13)]
 => [2,1,4,3,6,5,8,7,11,13,10,14,12,9] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => [2,1,6,8,9,5,10,7,4,3] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
 => [2,1,4,3,8,9,11,7,6,12,10,5] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,14),(8,9),(10,11),(12,13)]
 => [2,1,4,3,6,5,9,11,8,13,10,14,12,7] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => [4,6,7,3,9,5,2,10,8,1] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [2,1,7,8,9,10,6,5,4,3] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
 => [2,1,6,7,9,5,4,11,8,12,10,3] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,14),(6,7),(8,9),(10,11),(12,13)]
 => [2,1,4,3,7,9,6,11,8,13,10,14,12,5] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [5,6,7,9,4,3,2,10,8,1] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
 => [4,5,7,3,2,9,6,11,8,12,10,1] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => [2,1,5,7,4,9,6,11,8,13,10,14,12,3] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => [3,5,2,7,4,9,6,11,8,13,10,14,12,1] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} - 1
Description
The number of right floats of a permutation. Let π be a permutation of length n. A raft of π is a non-empty maximal sequence of consecutive small ascents, [[St000441]], and a right float is a large ascent not consecutive to any raft of π. See Definition 3.10 and Example 3.11 in [1].
Matching statistic: St001000
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001000: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 43%
Values
[1]
 => 10 => [1,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
[2]
 => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1]
 => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[3]
 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[2,1]
 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,1]
 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[4]
 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {1,1,4,4} + 1
[3,1]
 => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => ? ∊ {1,1,4,4} + 1
[2,2]
 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[2,1,1]
 => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ? ∊ {1,1,4,4} + 1
[1,1,1,1]
 => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,1,4,4} + 1
[5]
 => 100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {1,1,5,5,5,5,5} + 1
[4,1]
 => 100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? ∊ {1,1,5,5,5,5,5} + 1
[3,2]
 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => ? ∊ {1,1,5,5,5,5,5} + 1
[3,1,1]
 => 100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => ? ∊ {1,1,5,5,5,5,5} + 1
[2,2,1]
 => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => ? ∊ {1,1,5,5,5,5,5} + 1
[2,1,1,1]
 => 101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,1,5,5,5,5,5} + 1
[1,1,1,1,1]
 => 111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,1,5,5,5,5,5} + 1
[6]
 => 1000000 => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} + 1
[5,1]
 => 1000010 => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} + 1
[4,2]
 => 100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} + 1
[4,1,1]
 => 1000110 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} + 1
[3,3]
 => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} + 1
[3,2,1]
 => 101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} + 1
[3,1,1,1]
 => 1001110 => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} + 1
[2,2,2]
 => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} + 1
[2,2,1,1]
 => 110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} + 1
[2,1,1,1,1]
 => 1011110 => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} + 1
[1,1,1,1,1,1]
 => 1111110 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} + 1
[7]
 => 10000000 => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[6,1]
 => 10000010 => [1,6,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[5,2]
 => 1000100 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[5,1,1]
 => 10000110 => [1,5,1,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[4,3]
 => 101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[4,2,1]
 => 1001010 => [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[4,1,1,1]
 => 10001110 => [1,4,1,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[3,3,1]
 => 110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[3,2,2]
 => 101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[3,2,1,1]
 => 1010110 => [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[3,1,1,1,1]
 => 10011110 => [1,3,1,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[2,2,2,1]
 => 111010 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[2,2,1,1,1]
 => 1101110 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[2,1,1,1,1,1]
 => 10111110 => [1,2,1,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[1,1,1,1,1,1,1]
 => 11111110 => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
Description
Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001514
Mapping
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001514: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 43%
Values
[1]
 => 10 => [1,2] => [1,0,1,1,0,0]
 => 2 = 1 + 1
[2]
 => 100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,1]
 => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[3]
 => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[2,1]
 => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,1]
 => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[4]
 => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {1,1,4,4} + 1
[3,1]
 => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => ? ∊ {1,1,4,4} + 1
[2,2]
 => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[2,1,1]
 => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ? ∊ {1,1,4,4} + 1
[1,1,1,1]
 => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,1,4,4} + 1
[5]
 => 100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {1,1,5,5,5,5,5} + 1
[4,1]
 => 100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? ∊ {1,1,5,5,5,5,5} + 1
[3,2]
 => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => ? ∊ {1,1,5,5,5,5,5} + 1
[3,1,1]
 => 100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => ? ∊ {1,1,5,5,5,5,5} + 1
[2,2,1]
 => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => ? ∊ {1,1,5,5,5,5,5} + 1
[2,1,1,1]
 => 101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,1,5,5,5,5,5} + 1
[1,1,1,1,1]
 => 111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,1,5,5,5,5,5} + 1
[6]
 => 1000000 => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} + 1
[5,1]
 => 1000010 => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} + 1
[4,2]
 => 100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} + 1
[4,1,1]
 => 1000110 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} + 1
[3,3]
 => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} + 1
[3,2,1]
 => 101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} + 1
[3,1,1,1]
 => 1001110 => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} + 1
[2,2,2]
 => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} + 1
[2,2,1,1]
 => 110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} + 1
[2,1,1,1,1]
 => 1011110 => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} + 1
[1,1,1,1,1,1]
 => 1111110 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,2,2,3,3,3,3,3,6,6,6} + 1
[7]
 => 10000000 => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[6,1]
 => 10000010 => [1,6,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[5,2]
 => 1000100 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[5,1,1]
 => 10000110 => [1,5,1,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[4,3]
 => 101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[4,2,1]
 => 1001010 => [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[4,1,1,1]
 => 10001110 => [1,4,1,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[3,3,1]
 => 110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[3,2,2]
 => 101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[3,2,1,1]
 => 1010110 => [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[3,1,1,1,1]
 => 10011110 => [1,3,1,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[2,2,2,1]
 => 111010 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[2,2,1,1,1]
 => 1101110 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[2,1,1,1,1,1]
 => 10111110 => [1,2,1,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
[1,1,1,1,1,1,1]
 => 11111110 => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7} + 1
Description
The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule.
Matching statistic: St000352
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000352: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 29%
Values
[1]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 1
[2]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => 2
[1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [4,5,6,3,2,1,8,7] => ? ∊ {3,3}
[2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,6,7,8,5,4,3] => ? ∊ {3,3}
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => [5,6,7,8,4,3,2,1,10,9] => ? ∊ {1,1,4,4}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => [3,5,2,6,4,1,8,7] => ? ∊ {1,1,4,4}
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [3,4,2,1,7,8,6,5] => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => ? ∊ {1,1,4,4}
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [2,1,7,8,9,10,6,5,4,3] => ? ∊ {1,1,4,4}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
 => [6,7,8,9,10,5,4,3,2,1,12,11] => ? ∊ {1,1,5,5,5,5,5}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [(1,8),(2,7),(3,4),(5,6),(9,10)]
 => [4,6,7,3,8,5,2,1,10,9] => ? ∊ {1,1,5,5,5,5,5}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [3,4,2,1,6,5,8,7] => ? ∊ {1,1,5,5,5,5,5}
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => [2,1,5,6,4,3,8,7] => ? ∊ {1,1,5,5,5,5,5}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,7,8,6,5] => ? ∊ {1,1,5,5,5,5,5}
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [(1,2),(3,10),(4,9),(5,6),(7,8)]
 => [2,1,6,8,9,5,10,7,4,3] => ? ∊ {1,1,5,5,5,5,5}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
 => [2,1,8,9,10,11,12,7,6,5,4,3] => ? ∊ {1,1,5,5,5,5,5}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
 => [7,8,9,10,11,12,6,5,4,3,2,1,14,13] => ? ∊ {2,2,3,3,3,3,3,6,6,6}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
 => [5,7,8,9,4,10,6,3,2,1,12,11] => ? ∊ {2,2,3,3,3,3,3,6,6,6}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [(1,8),(2,5),(3,4),(6,7),(9,10)]
 => [4,5,7,3,2,8,6,1,10,9] => ? ∊ {2,2,3,3,3,3,3,6,6,6}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [(1,8),(2,3),(4,7),(5,6),(9,10)]
 => [3,6,2,7,8,5,4,1,10,9] => ? ∊ {2,2,3,3,3,3,3,6,6,6}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => [4,5,6,3,2,1,9,10,8,7] => ? ∊ {2,2,3,3,3,3,3,6,6,6}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,6,7,9,5,4,10,8,3] => ? ∊ {2,2,3,3,3,3,3,6,6,6}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => [3,4,2,1,8,9,10,7,6,5] => ? ∊ {2,2,3,3,3,3,3,6,6,6}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [(1,2),(3,10),(4,5),(6,9),(7,8)]
 => [2,1,5,8,4,9,10,7,6,3] => ? ∊ {2,2,3,3,3,3,3,6,6,6}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
 => [2,1,7,9,10,11,6,12,8,5,4,3] => ? ∊ {2,2,3,3,3,3,3,6,6,6}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
 => [2,1,9,10,11,12,13,14,8,7,6,5,4,3] => ? ∊ {2,2,3,3,3,3,3,6,6,6}
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,16)]
 => [8,9,10,11,12,13,14,7,6,5,4,3,2,1,16,15] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
 => [6,8,9,10,11,5,12,7,4,3,2,1,14,13] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
 => [5,6,8,9,4,3,10,7,2,1,12,11] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
 => [4,7,8,3,9,10,6,5,2,1,12,11] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10)]
 => [4,5,6,3,2,1,8,7,10,9] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,10)]
 => [3,5,2,7,4,8,6,1,10,9] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,10)]
 => [2,1,6,7,8,5,4,3,10,9] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [(1,6),(2,3),(4,5),(7,10),(8,9)]
 => [3,5,2,6,4,1,9,10,8,7] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9)]
 => [3,4,2,1,7,9,6,10,8,5] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,10),(4,5),(6,7),(8,9)]
 => [2,1,5,7,4,9,6,10,8,3] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
 => [2,1,7,8,10,11,6,5,12,9,4,3] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => [2,1,4,3,8,9,10,7,6,5] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
 => [2,1,6,9,10,5,11,12,8,7,4,3] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
 => [2,1,8,10,11,12,13,7,14,9,6,5,4,3] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
 => [2,1,10,11,12,13,14,15,16,9,8,7,6,5,4,3] => ? ∊ {1,1,1,7,7,7,7,7,7,7,7,7,7,7,7}
Description
The Elizalde-Pak rank of a permutation. This is the largest k such that π(i)>k for all i≤k. According to [1], the length of the longest increasing subsequence in a 321-avoiding permutation is equidistributed with the rank of a 132-avoiding permutation.
The following 48 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001632The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000007The number of saliances of the permutation. St000054The first entry of the permutation. St001487The number of inner corners of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000090The variation of a composition. St000091The descent variation of a composition. St000492The rob statistic of a set partition. St000498The lcs statistic of a set partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000654The first descent of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001151The number of blocks with odd minimum. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001722The number of minimal chains with small intervals between a binary word and the top element. St001896The number of right descents of a signed permutations. St001904The length of the initial strictly increasing segment of a parking function. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000075The orbit size of a standard tableau under promotion. St000089The absolute variation of a composition. St000365The number of double ascents of a permutation. St000383The last part of an integer composition. St000542The number of left-to-right-minima of a permutation. St000562The number of internal points of a set partition. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000709The number of occurrences of 14-2-3 or 14-3-2. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000839The largest opener of a set partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000230Sum of the minimal elements of the blocks of a set partition. St001375The pancake length of a permutation. St001516The number of cyclic bonds of a permutation. St000735The last entry on the main diagonal of a standard tableau.