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Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000519
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Mp00094: Integer compositions —to binary word⟶ Binary words
St000519: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000519: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 0
[1,1] => 11 => 1
[2] => 10 => 1
[1,1,1] => 111 => 2
[1,2] => 110 => 2
[2,1] => 101 => 2
[3] => 100 => 2
[1,1,1,1] => 1111 => 3
[1,1,2] => 1110 => 3
[1,2,1] => 1101 => 2
[1,3] => 1100 => 2
[2,1,1] => 1011 => 2
[2,2] => 1010 => 3
[3,1] => 1001 => 2
[4] => 1000 => 3
[1,1,1,1,1] => 11111 => 4
[1,1,1,2] => 11110 => 4
[1,1,2,1] => 11101 => 3
[1,1,3] => 11100 => 3
[1,2,1,1] => 11011 => 3
[1,2,2] => 11010 => 3
[1,3,1] => 11001 => 2
[1,4] => 11000 => 3
[2,1,1,1] => 10111 => 3
[2,1,2] => 10110 => 3
[2,2,1] => 10101 => 4
[2,3] => 10100 => 3
[3,1,1] => 10011 => 2
[3,2] => 10010 => 3
[4,1] => 10001 => 3
[5] => 10000 => 4
[1,1,1,1,1,1] => 111111 => 5
[1,1,1,1,2] => 111110 => 5
[1,1,1,2,1] => 111101 => 4
[1,1,1,3] => 111100 => 4
[1,1,2,1,1] => 111011 => 3
[1,1,2,2] => 111010 => 3
[1,1,3,1] => 111001 => 3
[1,1,4] => 111000 => 3
[1,2,1,1,1] => 110111 => 3
[1,2,1,2] => 110110 => 4
[1,2,2,1] => 110101 => 4
[1,2,3] => 110100 => 3
[1,3,1,1] => 110011 => 3
[1,3,2] => 110010 => 3
[1,4,1] => 110001 => 3
[1,5] => 110000 => 4
[2,1,1,1,1] => 101111 => 4
[2,1,1,2] => 101110 => 3
[2,1,2,1] => 101101 => 4
Description
The largest length of a factor maximising the subword complexity.
Let $p_w(n)$ be the number of distinct factors of length $n$. Then the statistic is the largest $n$ such that $p_w(n)$ is maximal:
$$
H_w = \max\{n: p_w(n)\text{ is maximal}\}
$$
A related statistic is the number of distinct factors of arbitrary length, also known as subword complexity, [[St000294]].
Matching statistic: St001875
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(load all 2 compositions to match this statistic)
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 44%
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 44%
Values
[1] => [[1],[]]
=> ([],1)
=> ? = 0
[1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {1,1}
[2] => [[2],[]]
=> ([],1)
=> ? ∊ {1,1}
[1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {2,2,2,2}
[1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {2,2,2,2}
[2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {2,2,2,2}
[3] => [[3],[]]
=> ([],1)
=> ? ∊ {2,2,2,2}
[1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {2,2,2,2,3,3,3,3}
[1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {2,2,2,2,3,3,3,3}
[1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {2,2,2,2,3,3,3,3}
[1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {2,2,2,2,3,3,3,3}
[2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {2,2,2,2,3,3,3,3}
[2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {2,2,2,2,3,3,3,3}
[3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {2,2,2,2,3,3,3,3}
[4] => [[4],[]]
=> ([],1)
=> ? ∊ {2,2,2,2,3,3,3,3}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3
[2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[5] => [[5],[]]
=> ([],1)
=> ? ∊ {2,2,3,3,3,3,3,3,3,3,4,4,4,4}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ([],1)
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> ([],1)
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,1,3] => [[3,1,1,1],[]]
=> ([],1)
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,3,1] => [[3,3,1,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,4] => [[4,1,1],[]]
=> ([],1)
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> ([(0,1)],2)
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5}
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4
[1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> ([(0,1)],2)
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5}
[1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,4,1] => [[4,4,1],[3]]
=> ([(0,1)],2)
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5}
[1,5] => [[5,1],[]]
=> ([],1)
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> ([],1)
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5}
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5}
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[2,1,3] => [[4,2,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5}
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 3
[2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
[2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[2,4] => [[5,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5}
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ([],1)
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5}
[3,1,2] => [[4,3,3],[2,2]]
=> ([(0,1)],2)
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5}
[3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 3
[3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 3
[4,1,1] => [[4,4,4],[3,3]]
=> ([],1)
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5}
[4,2] => [[5,4],[3]]
=> ([(0,1)],2)
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5}
[5,1] => [[5,5],[4]]
=> ([],1)
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5}
[6] => [[6],[]]
=> ([],1)
=> ? ∊ {3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> ([],1)
=> ? ∊ {3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,6,6}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4
[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)
=> 5
[1,1,2,3] => [[4,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,3,2] => [[4,3,1,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[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)
=> 5
[1,2,1,3] => [[4,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[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)
=> 5
[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)
=> 6
[1,2,3,1] => [[4,4,2,1],[3,1]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
[1,2,4] => [[5,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 5
[1,3,3] => [[5,3,1],[2]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
[1,4,2] => [[5,4,1],[3]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[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)
=> 4
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> ([(0,2),(2,1)],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)
=> 5
[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)
=> 6
[2,2,3] => [[5,3,2],[2,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5
[2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[2,3,2] => [[5,4,2],[3,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5
[2,4,1] => [[5,5,2],[4,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3
[3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> 3
[3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> ([(0,2),(2,1)],3)
=> 3
[3,2,2] => [[5,4,3],[3,2]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5
[3,3,1] => [[5,5,3],[4,2]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4
[3,4] => [[6,3],[2]]
=> ([(0,2),(2,1)],3)
=> 3
Description
The number of simple modules with projective dimension at most 1.
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