searching the database
Your data matches 61 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001421
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
St001421: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001421: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 0
[1,1] => 11 => 0
[2] => 10 => 1
[1,1,1] => 111 => 0
[1,2] => 110 => 1
[2,1] => 101 => 1
[3] => 100 => 1
[1,1,1,1] => 1111 => 0
[1,1,2] => 1110 => 1
[1,2,1] => 1101 => 1
[1,3] => 1100 => 2
[2,1,1] => 1011 => 1
[2,2] => 1010 => 2
[3,1] => 1001 => 1
[4] => 1000 => 1
[1,1,1,1,1] => 11111 => 0
[1,1,1,2] => 11110 => 1
[1,1,2,1] => 11101 => 1
[1,1,3] => 11100 => 2
[1,2,1,1] => 11011 => 1
[1,2,2] => 11010 => 2
[1,3,1] => 11001 => 2
[1,4] => 11000 => 2
[2,1,1,1] => 10111 => 1
[2,1,2] => 10110 => 1
[2,2,1] => 10101 => 2
[2,3] => 10100 => 2
[3,1,1] => 10011 => 1
[3,2] => 10010 => 1
[4,1] => 10001 => 1
[5] => 10000 => 1
[1,1,1,1,1,1] => 111111 => 0
[1,1,1,1,2] => 111110 => 1
[1,1,1,2,1] => 111101 => 1
[1,1,1,3] => 111100 => 2
[1,1,2,1,1] => 111011 => 1
[1,1,2,2] => 111010 => 2
[1,1,3,1] => 111001 => 2
[1,1,4] => 111000 => 3
[1,2,1,1,1] => 110111 => 1
[1,2,1,2] => 110110 => 1
[1,2,2,1] => 110101 => 2
[1,2,3] => 110100 => 3
[1,3,1,1] => 110011 => 2
[1,3,2] => 110010 => 2
[1,4,1] => 110001 => 2
[1,5] => 110000 => 2
[2,1,1,1,1] => 101111 => 1
[2,1,1,2] => 101110 => 1
[2,1,2,1] => 101101 => 1
Description
Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word.
Matching statistic: St001933
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 80%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,1}
[2] => [1] => [[1],[]]
=> []
=> ? ∊ {0,1}
[1,1,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,1,1,1}
[1,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1}
[2,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1}
[3] => [1] => [[1],[]]
=> []
=> ? ∊ {0,1,1,1}
[1,1,1,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[1,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[1,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[2,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[2,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[3,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[4] => [1] => [[1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[1,1,1,1,1] => [5] => [[5],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2}
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2}
[1,2,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2}
[1,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2}
[2,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2}
[2,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[2,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2}
[3,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2}
[3,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2}
[4,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2}
[5] => [1] => [[1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,2,2,2,2,2}
[1,1,1,1,1,1] => [6] => [[6],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,4] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,5] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,1,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,2,1,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1
[2,2,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,3] => [2] => [[2],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[4,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[4,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[5,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[6] => [1] => [[1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,1,1,1] => [7] => [[7],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3}
[1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
=> [4]
=> 1
[1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 2
[1,1,1,1,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,2,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,1,1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,2,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,5] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,2,1,1,1,1] => [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3}
[1,2,1,1,2] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1
[2,1,1,1,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[2,1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 2
[2,1,1,3] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[2,2,1,1,1] => [2,3] => [[4,2],[1]]
=> [1]
=> 1
[2,2,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[2,2,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[2,2,3] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[3,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[3,3,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1
[1,1,1,1,1,1,2] => [6,1] => [[6,6],[5]]
=> [5]
=> 1
[1,1,1,1,1,2,1] => [5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> 2
[1,1,1,1,1,3] => [5,1] => [[5,5],[4]]
=> [4]
=> 1
[1,1,1,1,2,1,1] => [4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 2
[1,1,1,1,2,2] => [4,2] => [[5,4],[3]]
=> [3]
=> 1
[1,1,1,1,3,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 2
[1,1,1,1,4] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,2,1,2] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 3
Description
The largest multiplicity of a part in an integer partition.
Matching statistic: St000993
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 80%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 80%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,1}
[2] => [1] => [[1],[]]
=> []
=> ? ∊ {0,1}
[1,1,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,1,1,1}
[1,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1}
[2,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1}
[3] => [1] => [[1],[]]
=> []
=> ? ∊ {0,1,1,1}
[1,1,1,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2}
[1,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,1,2,2}
[1,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2}
[1,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2}
[2,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2}
[2,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2}
[3,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2}
[4] => [1] => [[1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2}
[1,1,1,1,1] => [5] => [[5],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,2,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5] => [1] => [[1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,1,1,1,1,1] => [6] => [[6],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,4] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,5] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,1,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,2,1,1] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,2,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
=> [4]
=> 1
[1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 2
[1,1,1,1,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,2,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,1,1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,2,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[2,1,1,1,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[2,1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 2
[2,2,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[2,2,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,1,1,1,1,2] => [6,1] => [[6,6],[5]]
=> [5]
=> 1
[1,1,1,1,1,2,1] => [5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> 2
[1,1,1,1,1,3] => [5,1] => [[5,5],[4]]
=> [4]
=> 1
[1,1,1,1,2,1,1] => [4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 2
[1,1,1,1,2,2] => [4,2] => [[5,4],[3]]
=> [3]
=> 1
[1,1,1,1,3,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 2
[1,1,1,1,4] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,2,1,2] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 3
[1,1,1,2,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 1
[1,1,1,2,3] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,3,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,3,2] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,4,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
[1,1,1,5] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1,1,1] => [2,1,4] => [[5,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,2,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 1
[1,1,2,1,2,1] => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 4
[1,1,2,1,3] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[1,1,2,2,1,1] => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,2,3,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[1,1,2,4] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[1,1,3,2,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St001568
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 60%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,1}
[2] => [1] => [[1],[]]
=> []
=> ? ∊ {0,1}
[1,1,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,1,1,1}
[1,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1}
[2,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1}
[3] => [1] => [[1],[]]
=> []
=> ? ∊ {0,1,1,1}
[1,1,1,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2}
[1,1,2] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,1,2,2}
[1,2,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2}
[1,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2}
[2,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2}
[2,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2}
[3,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2}
[4] => [1] => [[1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,2,2}
[1,1,1,1,1] => [5] => [[5],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,2,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,3] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,1] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,2] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,1] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5] => [1] => [[1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,1,1,1,1,1] => [6] => [[6],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,4] => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,5] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,1,3] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,2,1,1] => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,2,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,3,1] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[2,4] => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,1,1,1] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[3,1,2] => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,1,2] => [5,1] => [[5,5],[4]]
=> [4]
=> 1
[1,1,1,1,2,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 1
[1,1,1,1,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,1,2,2] => [3,2] => [[4,3],[2]]
=> [2]
=> 1
[1,1,1,3,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,2,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,2,3] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,4,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[2,1,1,1,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
[2,1,1,2,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 2
[2,2,1,2] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[2,2,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,1,1,1,1,2] => [6,1] => [[6,6],[5]]
=> [5]
=> 1
[1,1,1,1,1,2,1] => [5,1,1] => [[5,5,5],[4,4]]
=> [4,4]
=> 1
[1,1,1,1,1,3] => [5,1] => [[5,5],[4]]
=> [4]
=> 1
[1,1,1,1,2,1,1] => [4,1,2] => [[5,4,4],[3,3]]
=> [3,3]
=> 1
[1,1,1,1,2,2] => [4,2] => [[5,4],[3]]
=> [3]
=> 1
[1,1,1,1,3,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 1
[1,1,1,1,4] => [4,1] => [[4,4],[3]]
=> [3]
=> 1
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,1,2,1,2] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 1
[1,1,1,2,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 1
[1,1,1,2,3] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,1,3,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,1,3,2] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,1,4,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
[1,1,1,5] => [3,1] => [[3,3],[2]]
=> [2]
=> 1
[1,1,2,1,1,1,1] => [2,1,4] => [[5,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,2,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 2
[1,1,2,1,2,1] => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 2
[1,1,2,1,3] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
[1,1,2,2,1,1] => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,2,3,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
[1,1,2,4] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3,1,1,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,3,1,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
[1,1,3,2,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St000259
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 60%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,1] => [2] => [1,1] => ([(0,1)],2)
=> 1
[2] => [1] => [1] => ([],1)
=> 0
[1,1,1] => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1}
[2,1] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1}
[3] => [1] => [1] => ([],1)
=> 0
[1,1,1,1] => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,2] => [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,2}
[1,2,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,2}
[1,3] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,2}
[2,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,2] => [2] => [1,1] => ([(0,1)],2)
=> 1
[3,1] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,2}
[4] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1] => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,2] => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2}
[1,1,2,1] => [2,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2}
[1,1,3] => [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2}
[1,2,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,2,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,3,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2}
[1,4] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2}
[2,1,1,1] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,2] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2}
[2,2,1] => [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2}
[2,3] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2}
[3,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,2] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2}
[4,1] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2}
[5] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1,1] => [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
[1,1,1,1,2] => [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[1,1,1,2,1] => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[1,1,1,3] => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[1,1,2,1,1] => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,2,2] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,3,1] => [2,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[1,1,4] => [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[1,2,1,1,1] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,1,2] => [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}
[1,2,2,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[1,2,3] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[1,3,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,2] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[1,4,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[1,5] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[2,1,1,1,1] => [1,4] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,1,2] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[2,1,2,1] => [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}
[2,1,3] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[2,2,1,1] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,2,2] => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,3,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[2,4] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[3,1,1,1] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,2] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[3,2,1] => [1,1,1] => [3] => ([],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[3,3] => [2] => [1,1] => ([(0,1)],2)
=> 1
[4,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[4,2] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[5,1] => [1,1] => [2] => ([],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[6] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[1,1,1,1,1,2] => [5,1] => [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)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,1,1,2,1] => [4,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,1,1,3] => [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,1,2,1,1] => [3,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,1,1,2,2] => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,1,3,1] => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,1,4] => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,2,1,1,1] => [2,1,3] => [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)
=> 2
[1,1,2,1,2] => [2,1,1,1] => [1,4] => ([(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,2,2,1] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,2,3] => [2,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,3,1,1] => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,1,3,2] => [2,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,4,1] => [2,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,5] => [2,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,1,1,1,1] => [1,1,4] => [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
[1,2,1,1,2] => [1,1,2,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,1,2,1] => [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,1,3] => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,2,1,1] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,2,2,2] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,1,1,1] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,3,3] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,4,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,1,1,1,1,1] => [1,5] => [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
[2,1,2,1,1] => [1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[2,1,2,2] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,2,1,1,1] => [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,3,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,1,1,1,1] => [1,4] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,2,1,1] => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,2,2] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[4,1,1,1] => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[5,1,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[7] => [1] => [1] => ([],1)
=> 0
[1,1,1,1,2,1,1] => [4,1,2] => [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
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001630
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 40%
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 40%
Values
[1] => [[1],[]]
=> ([],1)
=> ([],1)
=> ? = 0
[1,1] => [[1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1}
[2] => [[2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1}
[1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1}
[1,2] => [[2,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1}
[2,1] => [[2,2],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1}
[3] => [[3],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2}
[1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2}
[1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,2,2}
[1,3] => [[3,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2}
[2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2}
[2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,2,2}
[3,1] => [[3,3],[2]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2}
[4] => [[4],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,4] => [[4,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,1] => [[4,4],[3]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[5] => [[5],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[1,1,1,3] => [[3,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[1,1,4] => [[4,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,4,1] => [[4,4,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[1,5] => [[5,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3] => [[4,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4] => [[5,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[3,1,2] => [[4,3,3],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[4,1,1] => [[4,4,4],[3,3]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[4,2] => [[5,4],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[5,1] => [[5,5],[4]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[6] => [[6],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2
[1,1,2,3] => [[4,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,3,2] => [[4,3,1,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2
[1,2,1,3] => [[4,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2
[1,2,2,2] => [[4,3,2,1],[2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[1,2,3,1] => [[4,4,2,1],[3,1]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,2,4] => [[5,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,3,3] => [[5,3,1],[2]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,4,2] => [[5,4,1],[3]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[2,2,3] => [[5,3,2],[2,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,2] => [[5,4,2],[3,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[2,4,1] => [[5,5,2],[4,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[3,2,2] => [[5,4,3],[3,2]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[3,3,1] => [[5,5,3],[4,2]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[3,4] => [[6,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001878
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 40%
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 40%
Values
[1] => [[1],[]]
=> ([],1)
=> ([],1)
=> ? = 0
[1,1] => [[1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1}
[2] => [[2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1}
[1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1}
[1,2] => [[2,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1}
[2,1] => [[2,2],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1}
[3] => [[3],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2}
[1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2}
[1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,2,2}
[1,3] => [[3,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2}
[2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2}
[2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,2,2}
[3,1] => [[3,3],[2]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2}
[4] => [[4],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,4] => [[4,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[4,1] => [[4,4],[3]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[5] => [[5],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,2,2,2,2,2,2}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,3] => [[3,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[1,1,4] => [[4,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,4,1] => [[4,4,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[1,5] => [[5,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3] => [[4,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4] => [[5,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[3,1,2] => [[4,3,3],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[4,1,1] => [[4,4,4],[3,3]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[4,2] => [[5,4],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[5,1] => [[5,5],[4]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[6] => [[6],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,1,3,2] => [[4,3,1,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
[1,2,1,3] => [[4,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
[1,2,2,2] => [[4,3,2,1],[2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1
[1,2,3,1] => [[4,4,2,1],[3,1]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,2,4] => [[5,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,3,3] => [[5,3,1],[2]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,4,2] => [[5,4,1],[3]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1
[2,2,3] => [[5,3,2],[2,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,2] => [[5,4,2],[3,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[2,4,1] => [[5,5,2],[4,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[3,2,2] => [[5,4,3],[3,2]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[3,3,1] => [[5,5,3],[4,2]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[3,4] => [[6,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St000996
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [1] => 0
[1,1] => [1,0,1,0]
=> [2,1] => [2,1] => 1
[2] => [1,1,0,0]
=> [1,2] => [1,2] => 0
[1,1,1] => [1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => 1
[1,2] => [1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => 1
[2,1] => [1,1,0,0,1,0]
=> [3,1,2] => [3,2,1] => 1
[3] => [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [4,3,1,2] => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,2,1,3] => 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => 2
[3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,3,2,1] => 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [4,3,1,2,5] => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [4,1,2,5,3] => 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,2,1,3,5] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,2,1,3,4] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,2,4,1,3] => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [4,2,1,5,3] => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [5,3,1,4,2] => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [5,4,3,2,1] => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 3
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5] => [4,3,1,2,6,5] => 2
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5] => [2,1,6,5,3,4] => 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5] => [4,1,2,6,5,3] => 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => [2,1,4,3,5,6] => 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,5,6] => [4,3,1,2,5,6] => 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,5,6,3] => [2,1,6,3,4,5] => 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => [6,3,5,1,2,4] => 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,5,3,6,4] => [2,1,6,4,3,5] => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6,4,3,1,2,5] => 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,5,6,3,4] => [2,1,5,3,6,4] => 3
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,5,6,1,3,4] => [6,4,1,2,5,3] => 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,6,3,4,5] => [2,1,6,5,4,3] => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,6,1,3,4,5] => [6,5,4,3,1,2] => 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,1,2,3,4,5] => 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,1,4,2,6,5] => [4,2,1,3,6,5] => 2
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [3,4,1,2,6,5] => [3,1,4,2,6,5] => 3
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [3,1,4,6,2,5] => [6,5,2,1,3,4] => 1
[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] => [2,1,4,3,6,5,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [4,3,1,2,6,5,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [2,1,6,5,3,4,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [4,1,2,6,5,3,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [2,1,4,3,7,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [4,3,1,2,7,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [2,1,6,3,4,7,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [6,3,7,5,1,2,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [2,1,4,3,7,6,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [4,3,1,2,7,6,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [2,1,7,6,5,3,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [4,1,2,7,6,5,3] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [2,1,4,3,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [4,3,1,2,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,5,6,7,1,3] => [6,1,2,4,7,3,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [2,1,6,4,3,5,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [6,4,3,1,2,5,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [2,1,5,3,6,4,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [6,4,1,2,5,3,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [2,1,7,4,3,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [7,4,3,1,2,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [2,1,7,4,6,3,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,5,6,7,1,3,4] => [5,1,2,6,3,7,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [2,1,6,4,3,7,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [6,4,3,1,2,7,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,6,7,3,4,5] => [2,1,7,5,3,6,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,6,7,1,3,4,5] => [6,4,1,2,7,5,3] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,4,2,6,5,7] => [4,2,1,3,6,5,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [3,4,1,2,6,5,7] => [3,1,4,2,6,5,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [3,1,4,6,2,5,7] => [6,5,2,1,3,4,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [3,4,6,1,2,5,7] => [6,5,2,4,1,3,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [3,1,4,2,6,7,5] => [4,2,1,3,7,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [3,4,1,2,6,7,5] => [3,1,4,2,7,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [6,2,1,3,4,7,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,6,7,1,2,5] => [7,5,1,3,6,2,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,7,5,6] => [4,2,1,3,7,6,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,7,5,6] => [3,1,4,2,7,6,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [7,6,5,2,1,3,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [3,4,7,1,2,5,6] => [7,6,5,2,4,1,3] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,2,5,6,7] => [4,2,1,3,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [3,4,1,2,5,6,7] => [3,1,4,2,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => [7,2,4,6,1,3,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [4,1,5,2,6,3,7] => [4,2,1,6,3,5,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [4,5,1,2,6,3,7] => [6,3,1,4,2,5,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [4,1,5,6,2,3,7] => [6,3,5,2,1,4,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3,7] => [4,1,5,2,6,3,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[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] => [4,2,1,7,3,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [4,5,1,2,6,7,3] => [7,3,1,4,2,5,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [4,1,5,6,7,2,3] => [6,2,1,4,7,3,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,1,2,3] => [7,3,6,2,5,1,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3}
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St001637
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001637: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 60%
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001637: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 60%
Values
[1] => [[1],[]]
=> ([],1)
=> ([],1)
=> ? = 0
[1,1] => [[1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1}
[2] => [[2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1}
[1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1}
[1,2] => [[2,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1}
[2,1] => [[2,2],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1}
[3] => [[3],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2}
[1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2}
[1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3] => [[3,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2}
[2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2}
[2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1] => [[3,3],[2]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2}
[4] => [[4],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,2,2}
[1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,2,2}
[1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,2,2}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,4] => [[4,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,2,2}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,2,2}
[2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,2,2}
[3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[4,1] => [[4,4],[3]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,2,2}
[5] => [[5],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,2,2}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2,3,3}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2,3,3}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,1,3] => [[3,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2,3,3}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,3,1] => [[3,3,1,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,4] => [[4,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2,3,3}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3
[1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,4,1] => [[4,4,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,5] => [[5,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2,3,3}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2,3,3}
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3] => [[4,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
[2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4] => [[5,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2,3,3}
[3,1,2] => [[4,3,3],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[4,1,1] => [[4,4,4],[3,3]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2,3,3}
[4,2] => [[5,4],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[5,1] => [[5,5],[4]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2,3,3}
[6] => [[6],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2,3,3}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,1,4] => [[4,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,1,2,3] => [[4,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,3,2] => [[4,3,1,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,4,1] => [[4,4,1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,5] => [[5,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,1,3] => [[4,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,2,2] => [[4,3,2,1],[2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,3,1] => [[4,4,2,1],[3,1]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,2,4] => [[5,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[1,3,3] => [[5,3,1],[2]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
[1,4,1,1] => [[4,4,4,1],[3,3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,6] => [[6,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[2,2,3] => [[5,3,2],[2,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[2,3,2] => [[5,4,2],[3,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3}
Description
The number of (upper) dissectors of a poset.
Matching statistic: St001668
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001668: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 60%
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001668: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 60%
Values
[1] => [[1],[]]
=> ([],1)
=> ([],1)
=> ? = 0
[1,1] => [[1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1}
[2] => [[2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1}
[1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1}
[1,2] => [[2,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1}
[2,1] => [[2,2],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1}
[3] => [[3],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2}
[1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2}
[1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3] => [[3,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2}
[2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2}
[2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1] => [[3,3],[2]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2}
[4] => [[4],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,2,2}
[1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,2,2}
[1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,2,2}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,4] => [[4,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,2,2}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,2,2}
[2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,2,2}
[3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[4,1] => [[4,4],[3]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,2,2}
[5] => [[5],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,2,2,2,2}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2,3,3}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2,3,3}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,1,3] => [[3,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2,3,3}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,3,1] => [[3,3,1,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,4] => [[4,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2,3,3}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3
[1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,4,1] => [[4,4,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,5] => [[5,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2,3,3}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2,3,3}
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,3] => [[4,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
[2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4] => [[5,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2,3,3}
[3,1,2] => [[4,3,3],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[4,1,1] => [[4,4,4],[3,3]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2,3,3}
[4,2] => [[5,4],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[5,1] => [[5,5],[4]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2,3,3}
[6] => [[6],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,2,2,3,3}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,1,4] => [[4,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
[1,1,2,3] => [[4,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,1,3,2] => [[4,3,1,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,4,1] => [[4,4,1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,5] => [[5,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
[1,2,1,3] => [[4,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
[1,2,2,2] => [[4,3,2,1],[2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
[1,2,3,1] => [[4,4,2,1],[3,1]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
[1,2,4] => [[5,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 2
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 3
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
[1,3,3] => [[5,3,1],[2]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3
[1,4,1,1] => [[4,4,4,1],[3,3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,6] => [[6,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
[2,2,3] => [[5,3,2],[2,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
[2,3,2] => [[5,4,2],[3,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
Description
The number of points of the poset minus the width of the poset.
The following 51 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000884The number of isolated descents of a permutation. St000782The indicator function of whether a given perfect matching is an L & P matching. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St000454The largest eigenvalue of a graph if it is integral. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000619The number of cyclic descents of a permutation. St000260The radius of a connected graph. St001057The Grundy value of the game of creating an independent set in a graph. St001877Number of indecomposable injective modules with projective dimension 2. St001271The competition number of a graph. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000741The Colin de Verdière graph invariant. St000456The monochromatic index of a connected graph. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000099The number of valleys of a permutation, including the boundary. St000325The width of the tree associated to a permutation. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001530The depth of a Dyck path. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001720The minimal length of a chain of small intervals in a lattice. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St000455The second largest eigenvalue of a graph if it is integral.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!