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Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St000707
St000707: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> 2
[1,1]
=> 1
[3]
=> 6
[2,1]
=> 2
[1,1,1]
=> 1
[4]
=> 24
[3,1]
=> 6
[2,2]
=> 4
[2,1,1]
=> 2
[1,1,1,1]
=> 1
[5]
=> 120
[4,1]
=> 24
[3,2]
=> 12
[3,1,1]
=> 6
[2,2,1]
=> 4
[2,1,1,1]
=> 2
[1,1,1,1,1]
=> 1
[6]
=> 720
[5,1]
=> 120
[4,2]
=> 48
[4,1,1]
=> 24
[3,3]
=> 36
[3,2,1]
=> 12
[3,1,1,1]
=> 6
[2,2,2]
=> 8
[2,2,1,1]
=> 4
[2,1,1,1,1]
=> 2
[1,1,1,1,1,1]
=> 1
[7]
=> 5040
[6,1]
=> 720
[5,2]
=> 240
[5,1,1]
=> 120
[4,3]
=> 144
[4,2,1]
=> 48
[4,1,1,1]
=> 24
[3,3,1]
=> 36
[3,2,2]
=> 24
[3,2,1,1]
=> 12
[3,1,1,1,1]
=> 6
[2,2,2,1]
=> 8
[2,2,1,1,1]
=> 4
[2,1,1,1,1,1]
=> 2
[1,1,1,1,1,1,1]
=> 1
[8]
=> 40320
[7,1]
=> 5040
[6,2]
=> 1440
[6,1,1]
=> 720
[5,3]
=> 720
[5,2,1]
=> 240
[5,1,1,1]
=> 120
Description
The product of the factorials of the parts.
Matching statistic: St000110
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 25%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 25%
Values
[2]
=> [[1,2]]
=> [1,2] => [2,1] => 2
[1,1]
=> [[1],[2]]
=> [2,1] => [1,2] => 1
[3]
=> [[1,2,3]]
=> [1,2,3] => [3,2,1] => 6
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => [2,1,3] => 2
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [1,2,3] => 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [4,3,2,1] => 24
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [3,2,1,4] => 6
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => 4
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [2,1,3,4] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [5,4,3,2,1] => 120
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [4,3,2,1,5] => 24
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [3,2,1,5,4] => 12
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,2,1,4,5] => 6
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [2,1,4,3,5] => 4
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [2,1,3,4,5] => 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [6,5,4,3,2,1] => 720
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [5,4,3,2,1,6] => 120
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [4,3,2,1,6,5] => 48
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [4,3,2,1,5,6] => 24
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => 36
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [3,2,1,5,4,6] => 12
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [3,2,1,4,5,6] => 6
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [2,1,4,3,6,5] => 8
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [2,1,4,3,5,6] => 4
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [2,1,3,4,5,6] => 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => 5040
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => [6,5,4,3,2,1,7] => 720
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [5,4,3,2,1,7,6] => 240
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => [5,4,3,2,1,6,7] => 120
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [4,3,2,1,7,6,5] => ? ∊ {4,8,12,24,36,48,144}
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => [4,3,2,1,6,5,7] => ? ∊ {4,8,12,24,36,48,144}
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [4,3,2,1,5,6,7] => 24
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => [3,2,1,6,5,4,7] => ? ∊ {4,8,12,24,36,48,144}
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [3,2,1,5,4,7,6] => ? ∊ {4,8,12,24,36,48,144}
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => [3,2,1,5,4,6,7] => ? ∊ {4,8,12,24,36,48,144}
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => [3,2,1,4,5,6,7] => 6
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => [2,1,4,3,6,5,7] => ? ∊ {4,8,12,24,36,48,144}
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => [2,1,4,3,5,6,7] => ? ∊ {4,8,12,24,36,48,144}
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => [2,1,3,4,5,6,7] => 2
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => 40320
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,8] => 5040
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7] => 1440
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => [6,5,4,3,2,1,7,8] => 720
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6] => 720
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8] => ? ∊ {4,8,12,16,24,36,48,72,96,144,240}
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [8,7,6,1,2,3,4,5] => [5,4,3,2,1,6,7,8] => 120
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => 576
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8] => ? ∊ {4,8,12,16,24,36,48,72,96,144,240}
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7] => ? ∊ {4,8,12,16,24,36,48,72,96,144,240}
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => [4,3,2,1,6,5,7,8] => ? ∊ {4,8,12,16,24,36,48,72,96,144,240}
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => [4,3,2,1,5,6,7,8] => 24
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7] => ? ∊ {4,8,12,16,24,36,48,72,96,144,240}
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8] => ? ∊ {4,8,12,16,24,36,48,72,96,144,240}
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,8] => ? ∊ {4,8,12,16,24,36,48,72,96,144,240}
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => [3,2,1,5,4,6,7,8] => ? ∊ {4,8,12,16,24,36,48,72,96,144,240}
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => [3,2,1,4,5,6,7,8] => 6
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7] => ? ∊ {4,8,12,16,24,36,48,72,96,144,240}
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => [2,1,4,3,6,5,7,8] => ? ∊ {4,8,12,16,24,36,48,72,96,144,240}
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => [2,1,4,3,5,6,7,8] => ? ∊ {4,8,12,16,24,36,48,72,96,144,240}
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => [2,1,3,4,5,6,7,8] => 2
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => 1
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,1] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [9,1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [8,9,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,9,8] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [9,8,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,8,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [7,8,9,1,2,3,4,5,6] => [6,5,4,3,2,1,9,8,7] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [9,7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [9,8,7,1,2,3,4,5,6] => [6,5,4,3,2,1,7,8,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [6,7,8,9,1,2,3,4,5] => [5,4,3,2,1,9,8,7,6] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [9,6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [8,9,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,9,8] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [9,8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [9,8,7,6,1,2,3,4,5] => [5,4,3,2,1,6,7,8,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [9,5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [8,9,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,9,8] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [9,8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,1,2,3,4] => [4,3,2,1,6,5,7,8,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,1,2,3,4] => [4,3,2,1,5,6,7,8,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [3,2,1,6,5,4,9,8,7] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [9,7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [9,8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [8,9,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,9,8] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [9,8,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,8,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [9,8,7,6,4,5,1,2,3] => [3,2,1,5,4,6,7,8,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,1,2,3] => [3,2,1,4,5,6,7,8,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,3,4,1,2] => [2,1,4,3,6,5,8,7,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,3,4,1,2] => [2,1,4,3,6,5,7,8,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,3,4,1,2] => [2,1,4,3,5,6,7,8,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,1,2] => [2,1,3,4,5,6,7,8,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [1,2,3,4,5,6,7,8,9,10] => [10,9,8,7,6,5,4,3,2,1] => ? ∊ {1,2,4,6,8,12,16,24,24,32,36,48,48,72,96,120,144,144,192,216,240,288,480,576,720,720,864,1152,1440,1440,2880,2880,4320,5040,10080,14400,17280,30240,40320,80640,362880,3628800}
[9,1]
=> [[1,2,3,4,5,6,7,8,9],[10]]
=> [10,1,2,3,4,5,6,7,8,9] => [9,8,7,6,5,4,3,2,1,10] => ? ∊ {1,2,4,6,8,12,16,24,24,32,36,48,48,72,96,120,144,144,192,216,240,288,480,576,720,720,864,1152,1440,1440,2880,2880,4320,5040,10080,14400,17280,30240,40320,80640,362880,3628800}
Description
The number of permutations less than or equal to a permutation in left weak order.
This is the same as the number of permutations less than or equal to the given permutation in right weak order.
Matching statistic: St000040
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000040: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 22%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000040: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 22%
Values
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 1
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 2
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 1
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => [1,3,2] => 2
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 6
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 1
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => 2
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [1,3,4,2] => 4
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => 6
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 24
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,5,4] => 2
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,2,4,5,3] => 4
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => 6
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,3,5,4,2] => 12
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => 24
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 120
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => 2
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [1,2,3,5,6,4] => 4
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [1,2,3,6,5,4] => 6
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,2,4,5,6,3] => 8
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [1,2,4,6,5,3] => 12
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [1,2,6,5,4,3] => 24
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [1,3,5,6,4,2] => 36
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [1,3,6,5,4,2] => 48
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [1,6,5,4,3,2] => 120
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 720
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 2
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [1,2,3,4,6,7,5] => 4
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => [1,2,3,4,7,6,5] => 6
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [1,2,3,5,6,7,4] => 8
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => [1,2,3,5,7,6,4] => 12
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => 24
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => [1,2,4,5,7,6,3] => 24
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [1,2,4,6,7,5,3] => 36
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => [1,2,4,7,6,5,3] => 48
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => [1,2,7,6,5,4,3] => 120
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => [1,3,5,7,6,4,2] => 144
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => [1,3,7,6,5,4,2] => 240
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => [1,7,6,5,4,3,2] => 720
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => 5040
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,8,7] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [1,2,3,4,5,7,8,6] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => [1,2,3,4,5,8,7,6] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => [1,2,3,4,6,7,8,5] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => [1,2,3,4,6,8,7,5] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [8,7,6,1,2,3,4,5] => [1,2,3,4,8,7,6,5] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [1,2,3,5,6,7,8,4] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => [1,2,3,5,6,8,7,4] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => [1,2,3,5,7,8,6,4] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => [1,2,3,5,8,7,6,4] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => [1,2,3,8,7,6,5,4] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => [1,2,4,5,7,8,6,3] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => [1,2,4,5,8,7,6,3] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => [1,2,4,6,8,7,5,3] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => [1,2,4,8,7,6,5,3] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => [1,2,8,7,6,5,4,3] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [1,3,5,7,8,6,4,2] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => [1,3,5,8,7,6,4,2] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => [1,3,8,7,6,5,4,2] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => [1,8,7,6,5,4,3,2] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,9,8] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [8,9,1,2,3,4,5,6,7] => [1,2,3,4,5,6,8,9,7] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [9,8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,9,8,7] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [7,8,9,1,2,3,4,5,6] => [1,2,3,4,5,7,8,9,6] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [9,7,8,1,2,3,4,5,6] => [1,2,3,4,5,7,9,8,6] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [9,8,7,1,2,3,4,5,6] => [1,2,3,4,5,9,8,7,6] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [6,7,8,9,1,2,3,4,5] => [1,2,3,4,6,7,8,9,5] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [9,6,7,8,1,2,3,4,5] => [1,2,3,4,6,7,9,8,5] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [8,9,6,7,1,2,3,4,5] => [1,2,3,4,6,8,9,7,5] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [9,8,6,7,1,2,3,4,5] => [1,2,3,4,6,9,8,7,5] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [9,8,7,6,1,2,3,4,5] => [1,2,3,4,9,8,7,6,5] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [9,5,6,7,8,1,2,3,4] => [1,2,3,5,6,7,9,8,4] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [8,9,5,6,7,1,2,3,4] => [1,2,3,5,6,8,9,7,4] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [9,8,5,6,7,1,2,3,4] => [1,2,3,5,6,9,8,7,4] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => [1,2,3,5,7,9,8,6,4] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,1,2,3,4] => [1,2,3,5,9,8,7,6,4] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,1,2,3,4] => [1,2,3,9,8,7,6,5,4] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [1,2,4,5,7,8,9,6,3] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [9,7,8,4,5,6,1,2,3] => [1,2,4,5,7,9,8,6,3] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [9,8,7,4,5,6,1,2,3] => [1,2,4,5,9,8,7,6,3] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [8,9,6,7,4,5,1,2,3] => [1,2,4,6,8,9,7,5,3] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [9,8,6,7,4,5,1,2,3] => [1,2,4,6,9,8,7,5,3] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [9,8,7,6,4,5,1,2,3] => [1,2,4,9,8,7,6,5,3] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,1,2,3] => [1,2,9,8,7,6,5,4,3] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,3,4,1,2] => [1,3,5,7,9,8,6,4,2] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,3,4,1,2] => [1,3,5,9,8,7,6,4,2] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,3,4,1,2] => [1,3,9,8,7,6,5,4,2] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
Description
The number of regions of the inversion arrangement of a permutation.
The inversion arrangement Aw consists of the hyperplanes xi−xj=0 such that (i,j) is an inversion of w.
Postnikov [4] conjectured that the number of regions in Aw equals the number of permutations in the interval [id,w] in the strong Bruhat order if and only if w avoids 4231, 35142, 42513, 351624. This conjecture was proved by Hultman-Linusson-Shareshian-Sjöstrand [1].
Oh-Postnikov-Yoo [3] showed that the number of regions of Aw is |χGw(−1)| where χGw is the chromatic polynomial of the inversion graph Gw. This is the graph with vertices 1,2,…,n and edges (i,j) for i⪇ w_i\gneq w_j.
For a permutation w=w_1\cdots w_n, Lewis-Morales [2] and Hultman (see appendix in [2]) showed that this number equals the number of placements of n non-attacking rooks on the south-west Rothe diagram of w.
Matching statistic: St000269
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000269: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 22%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000269: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 22%
Values
[2]
=> [[1,2]]
=> [2] => ([],2)
=> 1
[1,1]
=> [[1],[2]]
=> [1,1] => ([(0,1)],2)
=> 2
[3]
=> [[1,2,3]]
=> [3] => ([],3)
=> 1
[2,1]
=> [[1,3],[2]]
=> [1,2] => ([(1,2)],3)
=> 2
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 6
[4]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> 1
[3,1]
=> [[1,3,4],[2]]
=> [1,3] => ([(2,3)],4)
=> 2
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 4
[2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 6
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 24
[5]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> 1
[4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => ([(3,4)],5)
=> 2
[3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> 4
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 6
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 12
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 24
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 120
[6]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => ([(4,5)],6)
=> 2
[4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => ([(3,5),(4,5)],6)
=> 4
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 6
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 8
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 12
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 24
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 36
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 48
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [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)
=> 120
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => ([(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)
=> 720
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => ([],7)
=> 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [1,6] => ([(5,6)],7)
=> 2
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [2,5] => ([(4,6),(5,6)],7)
=> 4
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 6
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 8
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> 12
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 24
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 24
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 36
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 48
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 120
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 144
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [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)
=> 240
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [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)
=> 720
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => ([(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)
=> 5040
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [8] => ([],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [1,7] => ([(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [2,6] => ([(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [1,1,6] => ([(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [3,5] => ([(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [1,1,1,5] => ([(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [1,1,2,4] => ([(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [1,1,1,1,4] => ([(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [2,3,3] => ([(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [1,1,3,3] => ([(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [1,1,1,2,3] => ([(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,3] => ([(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [1,1,2,2,2] => ([(1,7),(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [1,1,1,1,2,2] => ([(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [9] => ([],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [1,8] => ([(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [2,7] => ([(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [1,1,7] => ([(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [3,6] => ([(5,8),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [1,2,6] => ([(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [1,1,1,6] => ([(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [4,5] => ([(4,8),(5,8),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [1,3,5] => ([(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [2,2,5] => ([(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [1,1,2,5] => ([(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [1,1,1,1,5] => ([(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [1,4,4] => ([(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [2,3,4] => ([(3,8),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [1,1,3,4] => ([(3,8),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [1,2,2,4] => ([(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> [1,1,1,2,4] => ([(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,4] => ([(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [3,3,3] => ([(2,8),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [1,2,3,3] => ([(2,8),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [1,1,1,3,3] => ([(2,8),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [2,2,2,3] => ([(2,8),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> [1,1,2,2,3] => ([(2,8),(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> [1,1,1,1,2,3] => ([(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,3] => ([(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [1,2,2,2,2] => ([(1,8),(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> [1,1,1,2,2,2] => ([(1,8),(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> [1,1,1,1,1,2,2] => ([(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
Description
The number of acyclic orientations of a graph.
Matching statistic: St001109
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001109: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 22%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001109: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 22%
Values
[2]
=> [[1,2]]
=> [2] => ([],2)
=> 1
[1,1]
=> [[1],[2]]
=> [1,1] => ([(0,1)],2)
=> 2
[3]
=> [[1,2,3]]
=> [3] => ([],3)
=> 1
[2,1]
=> [[1,2],[3]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 6
[4]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> 1
[3,1]
=> [[1,2,3],[4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 4
[2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 24
[5]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> 1
[4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 12
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 24
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 120
[6]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 8
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 12
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 24
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 36
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 48
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => ([(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)
=> 120
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => ([(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)
=> 720
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => ([],7)
=> 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 4
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 8
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 12
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 24
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 24
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 36
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 48
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 120
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 144
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> 240
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1] => ([(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)
=> 720
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => ([(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)
=> 5040
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [8] => ([],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [5,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [5,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [4,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [4,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [4,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [3,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [3,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [3,2,2,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [3,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [3,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [2,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [2,2,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [9] => ([],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [7,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [7,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [6,3] => ([(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [6,2,1] => ([(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [6,1,1,1] => ([(0,6),(0,7),(0,8),(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [5,4] => ([(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [5,3,1] => ([(0,8),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [5,2,2] => ([(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [5,2,1,1] => ([(0,7),(0,8),(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [5,1,1,1,1] => ([(0,5),(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [4,4,1] => ([(0,8),(1,8),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [4,3,2] => ([(1,8),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [4,3,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [4,2,2,1] => ([(0,8),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [4,2,1,1,1] => ([(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> [4,1,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [3,3,3] => ([(2,8),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [3,3,2,1] => ([(0,8),(1,7),(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [3,3,1,1,1] => ([(0,6),(0,7),(0,8),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [3,2,2,2] => ([(1,8),(2,7),(2,8),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [3,2,2,1,1] => ([(0,7),(0,8),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [3,2,1,1,1,1] => ([(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> [3,1,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [2,2,2,2,1] => ([(0,8),(1,7),(1,8),(2,6),(2,7),(2,8),(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [2,2,2,1,1,1] => ([(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> [2,2,1,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(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)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
Description
The number of proper colourings of a graph with as few colours as possible.
By definition, this is the evaluation of the chromatic polynomial at the first nonnegative integer which is not a zero of the polynomial.
Matching statistic: St001813
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001813: Posets ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 17%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001813: Posets ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 17%
Values
[2]
=> [[1,2]]
=> [1,2] => ([(0,1)],2)
=> 2
[1,1]
=> [[1],[2]]
=> [2,1] => ([],2)
=> 1
[3]
=> [[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 6
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> 2
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => ([],3)
=> 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 24
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 6
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 4
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => ([(2,3)],4)
=> 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => ([],4)
=> 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 120
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 24
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> 12
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 6
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 4
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => ([(3,4)],5)
=> 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ([],5)
=> 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 720
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> 120
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> 48
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6)
=> 24
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
=> 36
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ([(1,3),(2,4),(4,5)],6)
=> 12
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ([(3,4),(4,5)],6)
=> 6
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6)
=> 8
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ([(2,5),(3,4)],6)
=> 4
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ([(4,5)],6)
=> 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ([],6)
=> 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {8,12,24,36,48,120,144,240,720,5040}
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? ∊ {8,12,24,36,48,120,144,240,720,5040}
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? ∊ {8,12,24,36,48,120,144,240,720,5040}
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? ∊ {8,12,24,36,48,120,144,240,720,5040}
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? ∊ {8,12,24,36,48,120,144,240,720,5040}
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? ∊ {8,12,24,36,48,120,144,240,720,5040}
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ([(3,4),(4,6),(6,5)],7)
=> 24
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? ∊ {8,12,24,36,48,120,144,240,720,5040}
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? ∊ {8,12,24,36,48,120,144,240,720,5040}
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ([(2,4),(3,5),(5,6)],7)
=> ? ∊ {8,12,24,36,48,120,144,240,720,5040}
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ([(4,5),(5,6)],7)
=> 6
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
=> ? ∊ {8,12,24,36,48,120,144,240,720,5040}
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ([(3,6),(4,5)],7)
=> 4
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ([(5,6)],7)
=> 2
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ([],7)
=> 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ([(1,7),(3,4),(4,6),(5,3),(6,2),(7,5)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => ([(2,7),(4,6),(5,4),(6,3),(7,5)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ([(0,7),(1,6),(4,5),(5,3),(6,4),(7,2)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => ([(1,7),(2,4),(5,6),(6,3),(7,5)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [8,7,6,1,2,3,4,5] => ([(3,4),(4,7),(6,5),(7,6)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => ([(1,6),(2,7),(5,4),(6,5),(7,3)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => ([(0,5),(1,4),(2,7),(6,3),(7,6)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => ([(2,4),(3,5),(5,6),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => ([(4,5),(5,7),(7,6)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => ([(0,5),(1,7),(2,6),(6,3),(7,4)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => ([(2,5),(3,4),(4,6),(5,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ([(1,5),(2,4),(3,6),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => ([(3,5),(4,6),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => ([(5,6),(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ([(2,7),(3,6),(4,5)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ([(4,7),(5,6)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => ([(6,7)],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => ([],8)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [9,1,2,3,4,5,6,7,8] => ([(1,8),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [8,9,1,2,3,4,5,6,7] => ([(0,8),(1,3),(4,5),(5,7),(6,4),(7,2),(8,6)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [9,8,1,2,3,4,5,6,7] => ([(2,8),(4,5),(5,7),(6,4),(7,3),(8,6)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [7,8,9,1,2,3,4,5,6] => ([(0,8),(1,7),(4,6),(5,4),(6,3),(7,5),(8,2)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [9,7,8,1,2,3,4,5,6] => ([(1,8),(2,4),(5,7),(6,5),(7,3),(8,6)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [9,8,7,1,2,3,4,5,6] => ([(3,8),(5,7),(6,5),(7,4),(8,6)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [6,7,8,9,1,2,3,4,5] => ([(0,7),(1,8),(4,5),(5,2),(6,3),(7,6),(8,4)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [9,6,7,8,1,2,3,4,5] => ([(1,8),(2,7),(5,6),(6,4),(7,5),(8,3)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [8,9,6,7,1,2,3,4,5] => ([(0,5),(1,4),(2,8),(6,7),(7,3),(8,6)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [9,8,6,7,1,2,3,4,5] => ([(2,8),(3,5),(6,7),(7,4),(8,6)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [9,8,7,6,1,2,3,4,5] => ([(4,5),(5,8),(7,6),(8,7)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [9,5,6,7,8,1,2,3,4] => ([(1,8),(2,7),(5,3),(6,4),(7,5),(8,6)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [8,9,5,6,7,1,2,3,4] => ([(0,5),(1,7),(2,8),(6,3),(7,4),(8,6)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [9,8,5,6,7,1,2,3,4] => ([(2,7),(3,8),(6,5),(7,6),(8,4)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => ([(1,6),(2,5),(3,8),(7,4),(8,7)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,1,2,3,4] => ([(3,5),(4,6),(6,7),(7,8)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,1,2,3,4] => ([(5,6),(6,8),(8,7)],9)
=> ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
Description
The product of the sizes of the principal order filters in a poset.
Matching statistic: St001346
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001346: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 17%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001346: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 17%
Values
[2]
=> [[1,2]]
=> [1,2] => 2
[1,1]
=> [[1],[2]]
=> [2,1] => 1
[3]
=> [[1,2,3]]
=> [1,2,3] => 6
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 24
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 6
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 4
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 120
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 24
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 12
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 6
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 4
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 720
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 120
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => 48
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 24
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 36
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 12
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 6
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 8
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 4
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [8,7,6,1,2,3,4,5] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [9,1,2,3,4,5,6,7,8] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [8,9,1,2,3,4,5,6,7] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [9,8,1,2,3,4,5,6,7] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [7,8,9,1,2,3,4,5,6] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [9,7,8,1,2,3,4,5,6] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [9,8,7,1,2,3,4,5,6] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [6,7,8,9,1,2,3,4,5] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [9,6,7,8,1,2,3,4,5] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [8,9,6,7,1,2,3,4,5] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [9,8,6,7,1,2,3,4,5] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [9,8,7,6,1,2,3,4,5] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [9,5,6,7,8,1,2,3,4] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
Description
The number of parking functions that give the same permutation.
A '''parking function''' (a_1,\dots,a_n) is a list of preferred parking spots of n cars entering a one-way street. Once the cars have parked, the order of the cars gives a permutation of \{1,\dots,n\}. This statistic records the number of parking functions that yield the same permutation of cars.
Matching statistic: St000109
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000109: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 17%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000109: Permutations ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 17%
Values
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 1
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 2
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 1
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => [1,3,2] => 2
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 6
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 1
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => 2
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [1,4,2,3] => 4
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => 6
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 24
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,5,4] => 2
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,2,5,3,4] => 4
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => 6
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,5,2,4,3] => 12
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => 24
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 120
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => 2
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [1,2,3,6,4,5] => 4
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [1,2,3,6,5,4] => 6
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [1,2,6,3,4,5] => 8
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [1,2,6,3,5,4] => 12
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [1,2,6,5,4,3] => 24
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [1,6,2,5,3,4] => 36
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [1,6,2,5,4,3] => 48
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [1,6,5,4,3,2] => 120
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => 720
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [1,2,3,4,7,5,6] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => [1,2,3,4,7,6,5] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [1,2,3,7,4,5,6] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => [1,2,3,7,4,6,5] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => [1,2,7,3,4,6,5] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [1,2,7,3,6,4,5] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => [1,2,7,3,6,5,4] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => [1,2,7,6,5,4,3] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => [1,7,2,6,3,5,4] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => [1,7,2,6,5,4,3] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => [1,7,6,5,4,3,2] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? ∊ {1,2,4,6,8,12,24,24,36,48,120,144,240,720,5040}
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,8,7] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [1,2,3,4,5,8,6,7] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => [1,2,3,4,5,8,7,6] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => [1,2,3,4,8,5,6,7] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => [1,2,3,4,8,5,7,6] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [8,7,6,1,2,3,4,5] => [1,2,3,4,8,7,6,5] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [1,2,3,8,4,5,6,7] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => [1,2,3,8,4,5,7,6] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => [1,2,3,8,4,7,5,6] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => [1,2,3,8,4,7,6,5] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => [1,2,3,8,7,6,5,4] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => [1,2,8,3,4,7,5,6] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => [1,2,8,3,4,7,6,5] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => [1,2,8,3,7,4,6,5] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => [1,2,8,3,7,6,5,4] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => [1,2,8,7,6,5,4,3] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [1,8,2,7,3,6,4,5] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => [1,8,2,7,3,6,5,4] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => [1,8,2,7,6,5,4,3] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => [1,8,7,6,5,4,3,2] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,72,96,120,144,240,576,720,720,1440,5040,40320}
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,9,8] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [8,9,1,2,3,4,5,6,7] => [1,2,3,4,5,6,9,7,8] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [9,8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,9,8,7] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [7,8,9,1,2,3,4,5,6] => [1,2,3,4,5,9,6,7,8] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [9,7,8,1,2,3,4,5,6] => [1,2,3,4,5,9,6,8,7] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [9,8,7,1,2,3,4,5,6] => [1,2,3,4,5,9,8,7,6] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [6,7,8,9,1,2,3,4,5] => [1,2,3,4,9,5,6,7,8] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [9,6,7,8,1,2,3,4,5] => [1,2,3,4,9,5,6,8,7] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [8,9,6,7,1,2,3,4,5] => [1,2,3,4,9,5,8,6,7] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [9,8,6,7,1,2,3,4,5] => [1,2,3,4,9,5,8,7,6] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [9,8,7,6,1,2,3,4,5] => [1,2,3,4,9,8,7,6,5] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [9,5,6,7,8,1,2,3,4] => [1,2,3,9,4,5,6,8,7] => ? ∊ {1,2,4,6,8,12,16,24,24,36,48,48,72,96,120,144,216,240,288,480,576,720,720,1440,2880,4320,5040,10080,40320,362880}
Description
The number of elements less than or equal to the given element in Bruhat order.
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