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Your data matches 21 different statistics following compositions of up to 3 maps.
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Matching statistic: St001313
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(load all 8 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
St001313: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001313: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 1
[1,1] => 11 => 1
[2] => 10 => 1
[1,1,1] => 111 => 1
[1,2] => 110 => 1
[2,1] => 101 => 2
[3] => 100 => 1
[1,1,1,1] => 1111 => 1
[1,1,2] => 1110 => 1
[1,2,1] => 1101 => 2
[1,3] => 1100 => 1
[2,1,1] => 1011 => 3
[2,2] => 1010 => 2
[3,1] => 1001 => 3
[4] => 1000 => 1
[1,1,1,1,1] => 11111 => 1
[1,1,1,2] => 11110 => 1
[1,1,2,1] => 11101 => 2
[1,1,3] => 11100 => 1
[1,2,1,1] => 11011 => 3
[1,2,2] => 11010 => 2
[1,3,1] => 11001 => 3
[1,4] => 11000 => 1
[2,1,1,1] => 10111 => 4
[2,1,2] => 10110 => 3
[2,2,1] => 10101 => 5
[2,3] => 10100 => 2
[3,1,1] => 10011 => 6
[3,2] => 10010 => 3
[4,1] => 10001 => 4
[5] => 10000 => 1
[1,1,1,1,1,1] => 111111 => 1
[1,1,1,1,2] => 111110 => 1
[1,1,1,2,1] => 111101 => 2
[1,1,1,3] => 111100 => 1
[1,1,2,1,1] => 111011 => 3
[1,1,2,2] => 111010 => 2
[1,1,3,1] => 111001 => 3
[1,1,4] => 111000 => 1
[1,2,1,1,1] => 110111 => 4
[1,2,1,2] => 110110 => 3
[1,2,2,1] => 110101 => 5
[1,2,3] => 110100 => 2
[1,3,1,1] => 110011 => 6
[1,3,2] => 110010 => 3
[1,4,1] => 110001 => 4
[1,5] => 110000 => 1
[2,1,1,1,1] => 101111 => 5
[2,1,1,2] => 101110 => 4
[2,1,2,1] => 101101 => 7
Description
The number of Dyck paths above the lattice path given by a binary word.
One may treat a binary word as a lattice path starting at the origin and treating $1$'s as steps $(1,0)$ and $0$'s as steps $(0,1)$. Given a binary word $w$, this statistic counts the number of lattice paths from the origin to the same endpoint as $w$ that stay weakly above $w$.
See [[St001312]] for this statistic on compositions treated as bounce paths.
Matching statistic: St000108
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 52%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 52%
Values
[1] => [[1],[]]
=> []
=> 1
[1,1] => [[1,1],[]]
=> []
=> 1
[2] => [[2],[]]
=> []
=> 1
[1,1,1] => [[1,1,1],[]]
=> []
=> 1
[1,2] => [[2,1],[]]
=> []
=> 1
[2,1] => [[2,2],[1]]
=> [1]
=> 2
[3] => [[3],[]]
=> []
=> 1
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> 1
[1,1,2] => [[2,1,1],[]]
=> []
=> 1
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> 2
[1,3] => [[3,1],[]]
=> []
=> 1
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 3
[2,2] => [[3,2],[1]]
=> [1]
=> 2
[3,1] => [[3,3],[2]]
=> [2]
=> 3
[4] => [[4],[]]
=> []
=> 1
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> 1
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> 1
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 2
[1,1,3] => [[3,1,1],[]]
=> []
=> 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 3
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> 2
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> 3
[1,4] => [[4,1],[]]
=> []
=> 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 4
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 3
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 5
[2,3] => [[4,2],[1]]
=> [1]
=> 2
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 6
[3,2] => [[4,3],[2]]
=> [2]
=> 3
[4,1] => [[4,4],[3]]
=> [3]
=> 4
[5] => [[5],[]]
=> []
=> 1
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> 1
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> 2
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 3
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 2
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 3
[1,1,4] => [[4,1,1],[]]
=> []
=> 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 4
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 3
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 5
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> 2
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 6
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> 3
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> 4
[1,5] => [[5,1],[]]
=> []
=> 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 5
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 4
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 7
[1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[1,1,1,2,3] => [[4,2,1,1,1],[1]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[1,2,4,1] => [[5,5,2,1],[4,1]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[1,2,5] => [[6,2,1],[1]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[1,3,4] => [[6,3,1],[2]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[1,5,1,1] => [[5,5,5,1],[4,4]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[1,5,2] => [[6,5,1],[4]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[1,6,1] => [[6,6,1],[5]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[2,1,1,1,3] => [[4,2,2,2,2],[1,1,1,1]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[2,1,1,4] => [[5,2,2,2],[1,1,1]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[2,1,4,1] => [[5,5,2,2],[4,1,1]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[2,1,5] => [[6,2,2],[1,1]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[2,2,4] => [[6,3,2],[2,1]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[2,5,1] => [[6,6,2],[5,1]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[2,6] => [[7,2],[1]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[3,1,1,1,2] => [[4,3,3,3,3],[2,2,2,2]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[3,1,4] => [[6,3,3],[2,2]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]]
=> [3,3,3,2]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[3,5] => [[7,3],[2]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]]
=> [3,3,3,3]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[4,2,1,1] => [[5,5,5,4],[4,4,3]]
=> [4,4,3]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[5,1,1,1] => [[5,5,5,5],[4,4,4]]
=> [4,4,4]
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[5,1,2] => [[6,5,5],[4,4]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[5,2,1] => [[6,6,5],[5,4]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[5,3] => [[7,5],[4]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[6,2] => [[7,6],[5]]
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,34,34,35,35}
[1,1,1,1,1,1,2,1] => [[2,2,1,1,1,1,1,1],[1]]
=> ?
=> ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
[1,1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1,1],[1,1]]
=> ?
=> ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
[1,1,1,1,1,2,2] => [[3,2,1,1,1,1,1],[1]]
=> ?
=> ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
[1,1,1,1,1,3,1] => [[3,3,1,1,1,1,1],[2]]
=> ?
=> ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
[1,1,1,1,2,1,1,1] => [[2,2,2,2,1,1,1,1],[1,1,1]]
=> ?
=> ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
[1,1,1,1,2,1,2] => [[3,2,2,1,1,1,1],[1,1]]
=> ?
=> ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
[1,1,1,1,2,2,1] => [[3,3,2,1,1,1,1],[2,1]]
=> ?
=> ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
[1,1,1,1,2,3] => [[4,2,1,1,1,1],[1]]
=> ?
=> ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
[1,1,1,1,3,1,1] => [[3,3,3,1,1,1,1],[2,2]]
=> ?
=> ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
[1,1,1,1,3,2] => [[4,3,1,1,1,1],[2]]
=> ?
=> ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
[1,1,1,1,4,1] => [[4,4,1,1,1,1],[3]]
=> ?
=> ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
[1,1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1,1],[1,1,1,1]]
=> ?
=> ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
[1,1,1,2,1,1,2] => [[3,2,2,2,1,1,1],[1,1,1]]
=> ?
=> ? ∊ {2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
Description
The number of partitions contained in the given partition.
Matching statistic: St001389
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 52%
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 52%
Values
[1] => [[1],[]]
=> [1]
=> []
=> ? = 1
[1,1] => [[1,1],[]]
=> [1,1]
=> [1]
=> 1
[2] => [[2],[]]
=> [2]
=> []
=> ? = 1
[1,1,1] => [[1,1,1],[]]
=> [1,1,1]
=> [1,1]
=> 1
[1,2] => [[2,1],[]]
=> [2,1]
=> [1]
=> 1
[2,1] => [[2,2],[1]]
=> [2,2]
=> [2]
=> 2
[3] => [[3],[]]
=> [3]
=> []
=> ? = 1
[1,1,1,1] => [[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,1,2] => [[2,1,1],[]]
=> [2,1,1]
=> [1,1]
=> 1
[1,2,1] => [[2,2,1],[1]]
=> [2,2,1]
=> [2,1]
=> 2
[1,3] => [[3,1],[]]
=> [3,1]
=> [1]
=> 1
[2,1,1] => [[2,2,2],[1,1]]
=> [2,2,2]
=> [2,2]
=> 3
[2,2] => [[3,2],[1]]
=> [3,2]
=> [2]
=> 2
[3,1] => [[3,3],[2]]
=> [3,3]
=> [3]
=> 3
[4] => [[4],[]]
=> [4]
=> []
=> ? = 1
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,1,1,2] => [[2,1,1,1],[]]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,2,1] => [[2,2,1,1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[1,1,3] => [[3,1,1],[]]
=> [3,1,1]
=> [1,1]
=> 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> [2,2,1]
=> 3
[1,2,2] => [[3,2,1],[1]]
=> [3,2,1]
=> [2,1]
=> 2
[1,3,1] => [[3,3,1],[2]]
=> [3,3,1]
=> [3,1]
=> 3
[1,4] => [[4,1],[]]
=> [4,1]
=> [1]
=> 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> [2,2,2]
=> 4
[2,1,2] => [[3,2,2],[1,1]]
=> [3,2,2]
=> [2,2]
=> 3
[2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> [3,2]
=> 5
[2,3] => [[4,2],[1]]
=> [4,2]
=> [2]
=> 2
[3,1,1] => [[3,3,3],[2,2]]
=> [3,3,3]
=> [3,3]
=> 6
[3,2] => [[4,3],[2]]
=> [4,3]
=> [3]
=> 3
[4,1] => [[4,4],[3]]
=> [4,4]
=> [4]
=> 4
[5] => [[5],[]]
=> [5]
=> []
=> ? = 1
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 2
[1,1,1,3] => [[3,1,1,1],[]]
=> [3,1,1,1]
=> [1,1,1]
=> 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> 3
[1,1,2,2] => [[3,2,1,1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> 2
[1,1,3,1] => [[3,3,1,1],[2]]
=> [3,3,1,1]
=> [3,1,1]
=> 3
[1,1,4] => [[4,1,1],[]]
=> [4,1,1]
=> [1,1]
=> 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> 4
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [3,2,2,1]
=> [2,2,1]
=> 3
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [3,3,2,1]
=> [3,2,1]
=> 5
[1,2,3] => [[4,2,1],[1]]
=> [4,2,1]
=> [2,1]
=> 2
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [3,3,3,1]
=> [3,3,1]
=> 6
[1,3,2] => [[4,3,1],[2]]
=> [4,3,1]
=> [3,1]
=> 3
[1,4,1] => [[4,4,1],[3]]
=> [4,4,1]
=> [4,1]
=> 4
[1,5] => [[5,1],[]]
=> [5,1]
=> [1]
=> 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2]
=> [2,2,2,2]
=> 5
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [3,2,2,2]
=> [2,2,2]
=> 4
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [3,3,2,2]
=> [3,2,2]
=> 7
[2,1,3] => [[4,2,2],[1,1]]
=> [4,2,2]
=> [2,2]
=> 3
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [3,3,3,2]
=> [3,3,2]
=> 9
[2,2,2] => [[4,3,2],[2,1]]
=> [4,3,2]
=> [3,2]
=> 5
[2,3,1] => [[4,4,2],[3,1]]
=> [4,4,2]
=> [4,2]
=> 7
[2,4] => [[5,2],[1]]
=> [5,2]
=> [2]
=> 2
[6] => [[6],[]]
=> [6]
=> []
=> ? = 1
[7] => [[7],[]]
=> [7]
=> []
=> ? = 1
[1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[1,1,1,2,3] => [[4,2,1,1,1],[1]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[1,2,4,1] => [[5,5,2,1],[4,1]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[1,2,5] => [[6,2,1],[1]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[1,3,4] => [[6,3,1],[2]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[1,5,1,1] => [[5,5,5,1],[4,4]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[1,5,2] => [[6,5,1],[4]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[1,6,1] => [[6,6,1],[5]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[2,1,1,1,3] => [[4,2,2,2,2],[1,1,1,1]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[2,1,1,4] => [[5,2,2,2],[1,1,1]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[2,1,4,1] => [[5,5,2,2],[4,1,1]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[2,1,5] => [[6,2,2],[1,1]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[2,2,4] => [[6,3,2],[2,1]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[2,5,1] => [[6,6,2],[5,1]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[2,6] => [[7,2],[1]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[3,1,1,1,2] => [[4,3,3,3,3],[2,2,2,2]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[3,1,2,1,1] => [[4,4,4,3,3],[3,3,2,2]]
=> [4,4,4,3,3]
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[3,1,4] => [[6,3,3],[2,2]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]]
=> [4,4,4,4,3]
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[3,3,1,1] => [[5,5,5,3],[4,4,2]]
=> [5,5,5,3]
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[3,5] => [[7,3],[2]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[4,1,2,1] => [[5,5,4,4],[4,3,3]]
=> [5,5,4,4]
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[4,2,1,1] => [[5,5,5,4],[4,4,3]]
=> [5,5,5,4]
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[5,1,2] => [[6,5,5],[4,4]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[5,2,1] => [[6,6,5],[5,4]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[5,3] => [[7,5],[4]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[6,2] => [[7,6],[5]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[8] => [[8],[]]
=> [8]
=> []
=> ? ∊ {1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20,30,31,31,34,34}
[1,1,1,1,1,1,2,1] => [[2,2,1,1,1,1,1,1],[1]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
[1,1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1,1],[1,1]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
[1,1,1,1,1,2,2] => [[3,2,1,1,1,1,1],[1]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
[1,1,1,1,1,3,1] => [[3,3,1,1,1,1,1],[2]]
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,12,12,12,12,12,12,13,13,13,13,13,13,14,14,14,14,14,14,14,14,14,15,15,15,15,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,20,21,21,21,21,22,22,22,22,22,22,23,23,23,23,24,24,25,25,25,25,25,25,26,27,27,28,28,28,28,28,28,28,30,30,30,30,30,30,31,31,31,31,32,32,34,34,34,34,34,34,35,35,35,35,35,35,36,36,37,37,40,40,42,43,43,46,46,46,47,47,48,48,50,50,52,52,53,53,53,55,55,55,55,56,56,62,65,65,69,70}
Description
The number of partitions of the same length below the given integer partition.
For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is
$$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St000420
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 54%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 54%
Values
[1] => [[1],[]]
=> []
=> []
=> ? = 1
[1,1] => [[1,1],[]]
=> []
=> []
=> ? ∊ {1,1}
[2] => [[2],[]]
=> []
=> []
=> ? ∊ {1,1}
[1,1,1] => [[1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1}
[1,2] => [[2,1],[]]
=> []
=> []
=> ? ∊ {1,1,1}
[2,1] => [[2,2],[1]]
=> [1]
=> [1,0,1,0]
=> 2
[3] => [[3],[]]
=> []
=> []
=> ? ∊ {1,1,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1}
[1,1,2] => [[2,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 2
[1,3] => [[3,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[2,2] => [[3,2],[1]]
=> [1]
=> [1,0,1,0]
=> 2
[3,1] => [[3,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[4] => [[4],[]]
=> []
=> []
=> ? ∊ {1,1,1,1}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 2
[1,1,3] => [[3,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 2
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[1,4] => [[4,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 5
[2,3] => [[4,2],[1]]
=> [1]
=> [1,0,1,0]
=> 2
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 6
[3,2] => [[4,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[4,1] => [[4,4],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4
[5] => [[5],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 2
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 2
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[1,1,4] => [[4,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 5
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 2
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 6
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4
[1,5] => [[5,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 7
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 9
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 5
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 7
[2,4] => [[5,2],[1]]
=> [1]
=> [1,0,1,0]
=> 2
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 10
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 6
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 9
[3,3] => [[5,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 10
[4,2] => [[5,4],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 4
[5,1] => [[5,5],[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
[6] => [[6],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 2
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 2
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 3
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 4
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 3
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 5
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 2
[1,1,5] => [[5,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1}
[1,6] => [[6,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1}
[7] => [[7],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1}
[1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20}
[1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20}
[1,1,1,1,1,3] => [[3,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20}
[1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20}
[1,1,1,1,4] => [[4,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20}
[1,1,1,2,3] => [[4,2,1,1,1],[1]]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20}
[1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20}
[1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20}
[1,1,1,5] => [[5,1,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20}
[1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20}
[1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20}
[1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20}
[1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20}
[1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20}
[1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20}
[1,1,6] => [[6,1,1],[]]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20}
[1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20}
[1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20}
[1,2,4,1] => [[5,5,2,1],[4,1]]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20}
[1,2,5] => [[6,2,1],[1]]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20}
[1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20}
[1,3,4] => [[6,3,1],[2]]
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,9,10,10,11,13,15,15,15,15,20}
Description
The number of Dyck paths that are weakly above a Dyck path.
Matching statistic: St000456
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 12%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 12%
Values
[1] => [1,0]
=> [1] => ([],1)
=> ? = 1
[1,1] => [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1
[2] => [1,1,0,0]
=> [1,2] => ([],2)
=> ? = 1
[1,1,1] => [1,0,1,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,2}
[1,2] => [1,0,1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[2,1] => [1,1,0,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[3] => [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ? ∊ {1,2}
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {2,3,3}
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {2,3,3}
[1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? ∊ {2,3,3}
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? ∊ {1,2,3,3,4,4,5,6}
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,3,3,4,4,5,6}
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {1,2,3,3,4,4,5,6}
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {1,2,3,3,4,4,5,6}
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {1,2,3,3,4,4,5,6}
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,2,3,3,4,4,5,6}
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,2,3,3,4,4,5,6}
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([],5)
=> ? ∊ {1,2,3,3,4,4,5,6}
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
=> ? ∊ {2,2,4,4,5,5,5,6,6,7,7,9,9,10,10}
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? ∊ {2,2,4,4,5,5,5,6,6,7,7,9,9,10,10}
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? ∊ {2,2,4,4,5,5,5,6,6,7,7,9,9,10,10}
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> ? ∊ {2,2,4,4,5,5,5,6,6,7,7,9,9,10,10}
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {2,2,4,4,5,5,5,6,6,7,7,9,9,10,10}
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,5,6,3] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? ∊ {2,2,4,4,5,5,5,6,6,7,7,9,9,10,10}
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? ∊ {2,2,4,4,5,5,5,6,6,7,7,9,9,10,10}
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {2,2,4,4,5,5,5,6,6,7,7,9,9,10,10}
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,6,3,4,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? ∊ {2,2,4,4,5,5,5,6,6,7,7,9,9,10,10}
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,5,6] => ([(4,5)],6)
=> ? ∊ {2,2,4,4,5,5,5,6,6,7,7,9,9,10,10}
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,4,2,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? ∊ {2,2,4,4,5,5,5,6,6,7,7,9,9,10,10}
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,4,1,2,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {2,2,4,4,5,5,5,6,6,7,7,9,9,10,10}
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [3,1,4,2,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {2,2,4,4,5,5,5,6,6,7,7,9,9,10,10}
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [3,4,1,2,5,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {2,2,4,4,5,5,5,6,6,7,7,9,9,10,10}
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 2
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,5,1,2,6,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [4,1,5,6,2,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 3
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 5
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,1,6,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([],6)
=> ? ∊ {2,2,4,4,5,5,5,6,6,7,7,9,9,10,10}
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7] => ([(1,6),(2,5),(3,4)],7)
=> ? ∊ {1,2,3,3,3,3,3,3,5,6,6,6,6,6,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,7] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? ∊ {1,2,3,3,3,3,3,3,5,6,6,6,6,6,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,7] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? ∊ {1,2,3,3,3,3,3,3,5,6,6,6,6,6,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,7] => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? ∊ {1,2,3,3,3,3,3,3,5,6,6,6,6,6,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => ([(0,3),(1,2),(4,6),(5,6)],7)
=> ? ∊ {1,2,3,3,3,3,3,3,5,6,6,6,6,6,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,6,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ? ∊ {1,2,3,3,3,3,3,3,5,6,6,6,6,6,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,6,7,3,5] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {1,2,3,3,3,3,3,3,5,6,6,6,6,6,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,6,7,1,3,5] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 4
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,7,5,6] => ([(0,3),(1,2),(4,6),(5,6)],7)
=> ? ∊ {1,2,3,3,3,3,3,3,5,6,6,6,6,6,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,3,7,5,6] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
=> ? ∊ {1,2,3,3,3,3,3,3,5,6,6,6,6,6,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,7,3,5,6] => ([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {1,2,3,3,3,3,3,3,5,6,6,6,6,6,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,7,1,3,5,6] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 2
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ([(3,6),(4,5)],7)
=> ? ∊ {1,2,3,3,3,3,3,3,5,6,6,6,6,6,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,4,1,3,5,6,7] => ([(3,6),(4,5),(5,6)],7)
=> ? ∊ {1,2,3,3,3,3,3,3,5,6,6,6,6,6,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,4,5,6,7,3] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,2,3,3,3,3,3,3,5,6,6,6,6,6,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,5,6,7,1,3] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4,7] => ([(1,2),(3,6),(4,5),(5,6)],7)
=> ? ∊ {1,2,3,3,3,3,3,3,5,6,6,6,6,6,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4,7] => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {1,2,3,3,3,3,3,3,5,6,6,6,6,6,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4,7] => ([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {1,2,3,3,3,3,3,3,5,6,6,6,6,6,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,6,1,3,4,7] => ([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ? ∊ {1,2,3,3,3,3,3,3,5,6,6,6,6,6,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,6,7,4] => ([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {1,2,3,3,3,3,3,3,5,6,6,6,6,6,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,6,7,4] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,6,7,3,4] => ([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {1,2,3,3,3,3,3,3,5,6,6,6,6,6,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,5,6,7,1,3,4] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> 5
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,1,6,3,7,4,5] => ([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {1,2,3,3,3,3,3,3,5,6,6,6,6,6,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,6,1,3,7,4,5] => ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> 2
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,6,7,1,3,4,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,7,1,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 1
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,7,2,5] => ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> 2
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,6,7,1,2,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 5
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,7,2,5,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> 1
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [3,4,7,1,2,5,6] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> 3
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 1
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 5
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [4,1,5,2,6,7,3] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 2
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [4,5,1,2,6,7,3] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> 3
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St001581
Values
[1] => ([],1)
=> ([],0)
=> ([],0)
=> ? = 1
[1,1] => ([(0,1)],2)
=> ([],1)
=> ([],1)
=> 1
[2] => ([],2)
=> ([],0)
=> ([],0)
=> ? = 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2] => ([(1,2)],3)
=> ([],1)
=> ([],1)
=> 1
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 1
[3] => ([],3)
=> ([],0)
=> ([],0)
=> ? = 2
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 3
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3] => ([(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 2
[2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[4] => ([],4)
=> ([],0)
=> ([],0)
=> ? = 3
[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)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ? ∊ {3,4,5,6}
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 3
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 4
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {3,4,5,6}
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,4] => ([(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,5),(1,6),(2,3),(2,4),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? ∊ {3,4,5,6}
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 2
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 3
[2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> 3
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1
[5] => ([],5)
=> ([],0)
=> ([],0)
=> ? ∊ {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)
=> ?
=> ?
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[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)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 3
[1,1,2,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 4
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,8),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,8),(4,6),(5,6),(6,7),(6,8),(7,8)],9)
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,9),(1,4),(1,7),(1,8),(2,3),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ([(2,5),(3,5),(4,5)],6)
=> 2
[1,5] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 1
[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)
=> ?
=> ?
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,5),(1,6),(2,3),(2,4),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(1,5),(1,6),(2,4),(2,6),(2,8),(3,4),(3,5),(3,7),(4,9),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 2
[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)
=> ?
=> ?
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 3
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 1
[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)
=> ?
=> ?
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> 3
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,5),(4,6),(4,7)],8)
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ([(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7)],9)
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 1
[6] => ([],6)
=> ([],0)
=> ([],0)
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[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)
=> ?
=> ?
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[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)
=> ?
=> ?
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,1,2,1] => ([(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)
=> ?
=> ?
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[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)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,2,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[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)
=> ?
=> ?
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 3
[1,1,2,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,2,2,1] => ([(0,6),(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)
=> ?
=> ?
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 4
[1,1,3,1,1] => ([(0,5),(0,6),(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)
=> ?
=> ?
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,6),(1,8),(2,6),(2,7),(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)
=> ([(0,8),(1,7),(2,6),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2,1,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,1,2,1] => ([(0,6),(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)
=> ?
=> ?
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,2,1,1] => ([(0,5),(0,6),(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)
=> ?
=> ?
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,8),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,8),(4,6),(5,6),(6,7),(6,8),(7,8)],9)
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,7),(0,8),(1,4),(1,5),(1,6),(1,9),(2,3),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,9),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(8,9)],10)
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> ?
=> ?
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,9),(1,4),(1,7),(1,8),(2,3),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(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)
=> ([(2,5),(3,5),(4,5)],6)
=> 2
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(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)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,6] => ([(5,6)],7)
=> ([],1)
=> ([],1)
=> 1
[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)
=> ?
=> ?
=> ? ∊ {3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 3
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(0,7),(1,5),(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)
=> ([(1,7),(2,6),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> 3
Description
The achromatic number of a graph.
This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.
Matching statistic: St001670
Values
[1] => ([],1)
=> ([],0)
=> ([],0)
=> ? = 1
[1,1] => ([(0,1)],2)
=> ([],1)
=> ([],1)
=> 1
[2] => ([],2)
=> ([],0)
=> ([],0)
=> ? = 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2] => ([(1,2)],3)
=> ([],1)
=> ([],1)
=> 1
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 1
[3] => ([],3)
=> ([],0)
=> ([],0)
=> ? = 2
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 3
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3] => ([(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 2
[2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[4] => ([],4)
=> ([],0)
=> ([],0)
=> ? = 3
[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)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ? ∊ {3,4,5,6}
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 3
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 4
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {3,4,5,6}
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,4] => ([(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,5),(1,6),(2,3),(2,4),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? ∊ {3,4,5,6}
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 2
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 3
[2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> 3
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1
[5] => ([],5)
=> ([],0)
=> ([],0)
=> ? ∊ {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)
=> ?
=> ?
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[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)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 3
[1,1,2,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 4
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,8),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,8),(4,6),(5,6),(6,7),(6,8),(7,8)],9)
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,9),(1,4),(1,7),(1,8),(2,3),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ([(2,5),(3,5),(4,5)],6)
=> 2
[1,5] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 1
[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)
=> ?
=> ?
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,5),(1,6),(2,3),(2,4),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(1,5),(1,6),(2,4),(2,6),(2,8),(3,4),(3,5),(3,7),(4,9),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 2
[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)
=> ?
=> ?
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 3
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 1
[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)
=> ?
=> ?
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> 3
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,5),(4,6),(4,7)],8)
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ([(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7)],9)
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 1
[6] => ([],6)
=> ([],0)
=> ([],0)
=> ? ∊ {3,3,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[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)
=> ?
=> ?
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[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)
=> ?
=> ?
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,1,2,1] => ([(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)
=> ?
=> ?
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[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)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,2,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[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)
=> ?
=> ?
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 3
[1,1,2,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,2,2,1] => ([(0,6),(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)
=> ?
=> ?
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 4
[1,1,3,1,1] => ([(0,5),(0,6),(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)
=> ?
=> ?
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,6),(1,8),(2,6),(2,7),(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)
=> ([(0,8),(1,7),(2,6),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2,1,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,1,2,1] => ([(0,6),(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)
=> ?
=> ?
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,2,1,1] => ([(0,5),(0,6),(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)
=> ?
=> ?
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,8),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,8),(4,6),(5,6),(6,7),(6,8),(7,8)],9)
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,7),(0,8),(1,4),(1,5),(1,6),(1,9),(2,3),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,9),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(8,9)],10)
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> ?
=> ?
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,9),(1,4),(1,7),(1,8),(2,3),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(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)
=> ([(2,5),(3,5),(4,5)],6)
=> 2
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(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)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,6] => ([(5,6)],7)
=> ([],1)
=> ([],1)
=> 1
[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)
=> ?
=> ?
=> ? ∊ {3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 3
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> 3
[2,5] => ([(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([],2)
=> 1
Description
The connected partition number of a graph.
This is the maximal number of blocks of a set partition $P$ of the set of vertices of a graph such that contracting each block of $P$ to a single vertex yields a clique.
Also called the pseudoachromatic number of a graph. This is the largest $n$ such that there exists a (not necessarily proper) $n$-coloring of the graph so that every two distinct colors are adjacent.
Matching statistic: St001286
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],0)
=> ([],0)
=> ? = 1
[1,1] => ([(0,1)],2)
=> ([],1)
=> ([],1)
=> 1
[2] => ([],2)
=> ([],0)
=> ([],0)
=> ? = 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => ([(1,2)],3)
=> ([],1)
=> ([],1)
=> 1
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3] => ([],3)
=> ([],0)
=> ([],0)
=> ? = 2
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3] => ([(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[4] => ([],4)
=> ([],0)
=> ([],0)
=> ? = 3
[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)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ? ∊ {1,3,4,5,6}
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 4
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,3,4,5,6}
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4] => ([(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,5),(1,6),(2,3),(2,4),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? ∊ {1,3,4,5,6}
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,3,4,5,6}
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[5] => ([],5)
=> ([],0)
=> ([],0)
=> ? ∊ {1,3,4,5,6}
[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)
=> ?
=> ?
=> ? ∊ {1,1,3,3,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[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)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ? ∊ {1,1,3,3,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? ∊ {1,1,3,3,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[1,1,2,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,3,3,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 4
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? ∊ {1,1,3,3,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,3,3,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,1,3,3,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,8),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,8),(4,6),(5,6),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,3,3,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,9),(1,4),(1,7),(1,8),(2,3),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)
=> ? ∊ {1,1,3,3,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ([(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)
=> 3
[1,5] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 1
[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)
=> ?
=> ?
=> ? ∊ {1,1,3,3,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,5),(1,6),(2,3),(2,4),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? ∊ {1,1,3,3,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(1,5),(1,6),(2,4),(2,6),(2,8),(3,4),(3,5),(3,7),(4,9),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)
=> ? ∊ {1,1,3,3,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[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)
=> ?
=> ?
=> ? ∊ {1,1,3,3,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[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)
=> ?
=> ?
=> ? ∊ {1,1,3,3,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {1,1,3,3,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {1,1,3,3,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ? ∊ {1,1,3,3,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[6] => ([],6)
=> ([],0)
=> ([],0)
=> ? ∊ {1,1,3,3,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[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)
=> ?
=> ?
=> ? ∊ {1,1,1,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[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)
=> ?
=> ?
=> ? ∊ {1,1,1,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,1,2,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[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)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ? ∊ {1,1,1,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,2,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {1,1,1,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {1,1,1,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[1,1,2,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {1,1,1,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {1,1,1,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 4
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {1,1,1,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? ∊ {1,1,1,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,6),(1,8),(2,6),(2,7),(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)
=> ([(0,8),(1,7),(2,6),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ? ∊ {1,1,1,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,1,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {1,1,1,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {1,1,1,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {1,1,1,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {1,1,1,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,8),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,8),(4,6),(5,6),(6,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,7),(0,8),(1,4),(1,5),(1,6),(1,9),(2,3),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,9),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(8,9)],10)
=> ? ∊ {1,1,1,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {1,1,1,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,9),(1,4),(1,7),(1,8),(2,3),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {1,1,1,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(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)
=> ([(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)
=> 3
[1,6] => ([(5,6)],7)
=> ([],1)
=> ([],1)
=> 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4
[2,5] => ([(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,4] => ([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
Description
The annihilation number of a graph.
For a graph on $m$ edges with degree sequence $d_1\leq\dots\leq d_n$, this is the largest number $k\leq n$ such that $\sum_{i=1}^k d_i \leq m$.
Matching statistic: St001725
Values
[1] => ([],1)
=> ([],0)
=> ([],0)
=> ? = 1
[1,1] => ([(0,1)],2)
=> ([],1)
=> ([],1)
=> 1
[2] => ([],2)
=> ([],0)
=> ([],0)
=> ? = 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2] => ([(1,2)],3)
=> ([],1)
=> ([],1)
=> 1
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 1
[3] => ([],3)
=> ([],0)
=> ([],0)
=> ? = 2
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 3
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3] => ([(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 3
[2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[4] => ([],4)
=> ([],0)
=> ([],0)
=> ? = 2
[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)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ? ∊ {2,2,3,4,5,6}
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 3
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? ∊ {2,2,3,4,5,6}
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {2,2,3,4,5,6}
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 3
[1,4] => ([(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,5),(1,6),(2,3),(2,4),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? ∊ {2,2,3,4,5,6}
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 3
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 4
[2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ? ∊ {2,2,3,4,5,6}
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1
[5] => ([],5)
=> ([],0)
=> ([],0)
=> ? ∊ {2,2,3,4,5,6}
[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)
=> ?
=> ?
=> ? ∊ {2,2,2,3,3,3,4,4,5,5,5,6,6,7,7,9,9,10,10}
[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)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ? ∊ {2,2,2,3,3,3,4,4,5,5,5,6,6,7,7,9,9,10,10}
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? ∊ {2,2,2,3,3,3,4,4,5,5,5,6,6,7,7,9,9,10,10}
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 3
[1,1,2,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {2,2,2,3,3,3,4,4,5,5,5,6,6,7,7,9,9,10,10}
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,3,3,3,4,4,5,5,5,6,6,7,7,9,9,10,10}
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? ∊ {2,2,2,3,3,3,4,4,5,5,5,6,6,7,7,9,9,10,10}
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {2,2,2,3,3,3,4,4,5,5,5,6,6,7,7,9,9,10,10}
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {2,2,2,3,3,3,4,4,5,5,5,6,6,7,7,9,9,10,10}
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,8),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,8),(4,6),(5,6),(6,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,3,3,3,4,4,5,5,5,6,6,7,7,9,9,10,10}
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,9),(1,4),(1,7),(1,8),(2,3),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)
=> ? ∊ {2,2,2,3,3,3,4,4,5,5,5,6,6,7,7,9,9,10,10}
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 3
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> ([(2,5),(3,5),(4,5)],6)
=> 4
[1,5] => ([(4,5)],6)
=> ([],1)
=> ([],1)
=> 1
[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)
=> ?
=> ?
=> ? ∊ {2,2,2,3,3,3,4,4,5,5,5,6,6,7,7,9,9,10,10}
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,5),(1,6),(2,3),(2,4),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? ∊ {2,2,2,3,3,3,4,4,5,5,5,6,6,7,7,9,9,10,10}
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(1,5),(1,6),(2,4),(2,6),(2,8),(3,4),(3,5),(3,7),(4,9),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)
=> ? ∊ {2,2,2,3,3,3,4,4,5,5,5,6,6,7,7,9,9,10,10}
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 3
[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)
=> ?
=> ?
=> ? ∊ {2,2,2,3,3,3,4,4,5,5,5,6,6,7,7,9,9,10,10}
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 4
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> 5
[2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 1
[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)
=> ?
=> ?
=> ? ∊ {2,2,2,3,3,3,4,4,5,5,5,6,6,7,7,9,9,10,10}
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ? ∊ {2,2,2,3,3,3,4,4,5,5,5,6,6,7,7,9,9,10,10}
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,5),(4,6),(4,7)],8)
=> ? ∊ {2,2,2,3,3,3,4,4,5,5,5,6,6,7,7,9,9,10,10}
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ([(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7)],9)
=> ? ∊ {2,2,2,3,3,3,4,4,5,5,5,6,6,7,7,9,9,10,10}
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 1
[6] => ([],6)
=> ([],0)
=> ([],0)
=> ? ∊ {2,2,2,3,3,3,4,4,5,5,5,6,6,7,7,9,9,10,10}
[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)
=> ?
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[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)
=> ?
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,1,2,1] => ([(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)
=> ?
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[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)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,2,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[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)
=> ?
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 3
[1,1,2,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,2,2,1] => ([(0,6),(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)
=> ?
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,3,1,1] => ([(0,5),(0,6),(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)
=> ?
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,6),(1,8),(2,6),(2,7),(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)
=> ([(0,8),(1,7),(2,6),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[1,2,1,1,1,1] => ([(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)
=> ?
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,1,2,1] => ([(0,6),(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)
=> ?
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,2,1,1] => ([(0,5),(0,6),(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)
=> ?
=> ?
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,8),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,8),(4,6),(5,6),(6,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,7),(0,8),(1,4),(1,5),(1,6),(1,9),(2,3),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,9),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(8,9)],10)
=> ? ∊ {2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 3
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(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)
=> ([(2,5),(3,5),(4,5)],6)
=> 4
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(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)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> 5
[1,6] => ([(5,6)],7)
=> ([],1)
=> ([],1)
=> 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 3
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 4
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> 5
[2,5] => ([(4,6),(5,6)],7)
=> ([(0,1)],2)
=> ([],2)
=> 1
[3,4] => ([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1
[5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 1
[6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([],6)
=> 1
Description
The harmonious chromatic number of a graph.
A harmonious colouring is a proper vertex colouring such that any pair of colours appears at most once on adjacent vertices.
Matching statistic: St001734
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],0)
=> ? = 1
[1,1] => ([(0,1)],2)
=> ([],1)
=> 1
[2] => ([],2)
=> ([],0)
=> ? = 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => ([(1,2)],3)
=> ([],1)
=> 1
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[3] => ([],3)
=> ([],0)
=> ? = 2
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3] => ([(2,3)],4)
=> ([],1)
=> 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
[2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[4] => ([],4)
=> ([],0)
=> ? = 2
[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)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {2,3,4,4,5,6}
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,3,4,4,5,6}
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {2,3,4,4,5,6}
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4] => ([(3,4)],5)
=> ([],1)
=> 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {2,3,4,4,5,6}
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {2,3,4,4,5,6}
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[5] => ([],5)
=> ([],0)
=> ? ∊ {2,3,4,4,5,6}
[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)
=> ?
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[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)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[1,1,2,1,1] => ([(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)
=> ?
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,1,1,1] => ([(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)
=> ?
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(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)
=> 2
[1,5] => ([(4,5)],6)
=> ([],1)
=> 1
[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)
=> ?
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
[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)
=> ?
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[2,4] => ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
[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)
=> ?
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[6] => ([],6)
=> ([],0)
=> ? ∊ {2,3,3,3,4,4,4,4,5,5,5,5,6,6,7,7,9,9,10,10}
[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)
=> ?
=> ? ∊ {2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[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)
=> ?
=> ? ∊ {2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,1,2,1] => ([(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)
=> ?
=> ? ∊ {2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[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)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,2,1,1] => ([(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)
=> ?
=> ? ∊ {2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[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)
=> ?
=> ? ∊ {2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
[1,1,2,1,1,1] => ([(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)
=> ?
=> ? ∊ {2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,2,2,1] => ([(0,6),(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)
=> ?
=> ? ∊ {2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,3,1,1] => ([(0,5),(0,6),(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)
=> ?
=> ? ∊ {2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,6),(1,8),(2,6),(2,7),(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)
=> ? ∊ {2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,1,1,1,1] => ([(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)
=> ?
=> ? ∊ {2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? ∊ {2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,1,2,1] => ([(0,6),(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)
=> ?
=> ? ∊ {2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,2,1,1] => ([(0,5),(0,6),(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)
=> ?
=> ? ∊ {2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,7,7,7,7,9,9,9,9,9,9,10,10,10,10,10,12,12,14,14,14,15,15,16,16,19,20}
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(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)
=> 2
[1,6] => ([(5,6)],7)
=> ([],1)
=> 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[2,5] => ([(4,6),(5,6)],7)
=> ([(0,1)],2)
=> 1
[3,4] => ([(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(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)
=> 1
Description
The lettericity of a graph.
Let $D$ be a digraph on $k$ vertices, possibly with loops and let $w$ be a word of length $n$ whose letters are vertices of $D$.
The letter graph corresponding to $D$ and $w$ is the graph with vertex set $\{1,\dots,n\}$ whose edges are the pairs $(i,j)$ with $i < j$ sucht that $(w_i, w_j)$ is a (directed) edge of $D$.
The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001596The number of two-by-two squares inside a skew partition. St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001118The acyclic chromatic index of a graph. St001881The number of factors of a lattice as a Cartesian product of lattices. St001845The number of join irreducibles minus the rank of a lattice. St001868The number of alignments of type NE of a signed permutation.
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