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Your data matches 35 different statistics following compositions of up to 3 maps.
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Matching statistic: St000294
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St000294: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 2
1 => 2
00 => 3
01 => 4
10 => 4
11 => 3
000 => 4
001 => 6
010 => 6
011 => 6
100 => 6
101 => 6
110 => 6
111 => 4
0000 => 5
0001 => 8
0010 => 9
0011 => 9
0100 => 9
0101 => 8
0110 => 9
0111 => 8
1000 => 8
1001 => 9
1010 => 8
1011 => 9
1100 => 9
1101 => 9
1110 => 8
1111 => 5
Description
The number of distinct factors of a binary word.
This is also known as the subword complexity of a binary word, see [1].
Matching statistic: St000228
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00262: Binary words —poset of factors⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => ([(0,1)],2)
=> [2]
=> 2
1 => ([(0,1)],2)
=> [2]
=> 2
00 => ([(0,2),(2,1)],3)
=> [3]
=> 3
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 4
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> 4
11 => ([(0,2),(2,1)],3)
=> [3]
=> 3
000 => ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 6
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> 6
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 6
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 6
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> 6
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> 6
111 => ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 8
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> [5,3,1]
=> 9
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> [5,3,1]
=> 9
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> [5,3,1]
=> 9
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> [5,3]
=> 8
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> [5,3,1]
=> 9
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 8
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 8
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> [5,3,1]
=> 9
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> [5,3]
=> 8
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> [5,3,1]
=> 9
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> [5,3,1]
=> 9
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> [5,3,1]
=> 9
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> 8
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000532
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Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000532: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000532: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [2]
=> 3 = 2 + 1
1 => [1,1] => [1,1]
=> 3 = 2 + 1
00 => [3] => [3]
=> 4 = 3 + 1
01 => [2,1] => [2,1]
=> 5 = 4 + 1
10 => [1,2] => [2,1]
=> 5 = 4 + 1
11 => [1,1,1] => [1,1,1]
=> 4 = 3 + 1
000 => [4] => [4]
=> 5 = 4 + 1
001 => [3,1] => [3,1]
=> 7 = 6 + 1
010 => [2,2] => [2,2]
=> 7 = 6 + 1
011 => [2,1,1] => [2,1,1]
=> 7 = 6 + 1
100 => [1,3] => [3,1]
=> 7 = 6 + 1
101 => [1,2,1] => [2,1,1]
=> 7 = 6 + 1
110 => [1,1,2] => [2,1,1]
=> 7 = 6 + 1
111 => [1,1,1,1] => [1,1,1,1]
=> 5 = 4 + 1
0000 => [5] => [5]
=> 6 = 5 + 1
0001 => [4,1] => [4,1]
=> 9 = 8 + 1
0010 => [3,2] => [3,2]
=> 10 = 9 + 1
0011 => [3,1,1] => [3,1,1]
=> 10 = 9 + 1
0100 => [2,3] => [3,2]
=> 10 = 9 + 1
0101 => [2,2,1] => [2,2,1]
=> 10 = 9 + 1
0110 => [2,1,2] => [2,2,1]
=> 10 = 9 + 1
0111 => [2,1,1,1] => [2,1,1,1]
=> 9 = 8 + 1
1000 => [1,4] => [4,1]
=> 9 = 8 + 1
1001 => [1,3,1] => [3,1,1]
=> 10 = 9 + 1
1010 => [1,2,2] => [2,2,1]
=> 10 = 9 + 1
1011 => [1,2,1,1] => [2,1,1,1]
=> 9 = 8 + 1
1100 => [1,1,3] => [3,1,1]
=> 10 = 9 + 1
1101 => [1,1,2,1] => [2,1,1,1]
=> 9 = 8 + 1
1110 => [1,1,1,2] => [2,1,1,1]
=> 9 = 8 + 1
1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> 6 = 5 + 1
Description
The total number of rook placements on a Ferrers board.
Matching statistic: St000548
Mp00262: Binary words —poset of factors⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000548: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000548: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => ([(0,1)],2)
=> [2]
=> [1,1]
=> 2
1 => ([(0,1)],2)
=> [2]
=> [1,1]
=> 2
00 => ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 4
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 4
11 => ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3
000 => ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> [2,2,1,1]
=> 6
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> [2,2,1,1]
=> 6
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> [2,2,1,1]
=> 6
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> [2,2,1,1]
=> 6
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> [2,2,1,1]
=> 6
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [4,2]
=> [2,2,1,1]
=> 6
111 => ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> [2,2,2,1,1]
=> 8
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> [5,3,1]
=> [3,2,2,1,1]
=> 9
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> [5,3,1]
=> [3,2,2,1,1]
=> 9
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> [5,3,1]
=> [3,2,2,1,1]
=> 9
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> [5,3]
=> [2,2,2,1,1]
=> 8
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> [5,3,1]
=> [3,2,2,1,1]
=> 9
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> [2,2,2,1,1]
=> 8
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> [2,2,2,1,1]
=> 8
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> [5,3,1]
=> [3,2,2,1,1]
=> 9
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> [5,3]
=> [2,2,2,1,1]
=> 8
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> [5,3,1]
=> [3,2,2,1,1]
=> 9
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> [5,3,1]
=> [3,2,2,1,1]
=> 9
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> [5,3,1]
=> [3,2,2,1,1]
=> 9
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> [5,3]
=> [2,2,2,1,1]
=> 8
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
Description
The number of different non-empty partial sums of an integer partition.
Matching statistic: St001658
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001658: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001658: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [2]
=> [[2],[]]
=> 3 = 2 + 1
1 => [1,1] => [1,1]
=> [[1,1],[]]
=> 3 = 2 + 1
00 => [3] => [3]
=> [[3],[]]
=> 4 = 3 + 1
01 => [2,1] => [2,1]
=> [[2,1],[]]
=> 5 = 4 + 1
10 => [1,2] => [2,1]
=> [[2,1],[]]
=> 5 = 4 + 1
11 => [1,1,1] => [1,1,1]
=> [[1,1,1],[]]
=> 4 = 3 + 1
000 => [4] => [4]
=> [[4],[]]
=> 5 = 4 + 1
001 => [3,1] => [3,1]
=> [[3,1],[]]
=> 7 = 6 + 1
010 => [2,2] => [2,2]
=> [[2,2],[]]
=> 7 = 6 + 1
011 => [2,1,1] => [2,1,1]
=> [[2,1,1],[]]
=> 7 = 6 + 1
100 => [1,3] => [3,1]
=> [[3,1],[]]
=> 7 = 6 + 1
101 => [1,2,1] => [2,1,1]
=> [[2,1,1],[]]
=> 7 = 6 + 1
110 => [1,1,2] => [2,1,1]
=> [[2,1,1],[]]
=> 7 = 6 + 1
111 => [1,1,1,1] => [1,1,1,1]
=> [[1,1,1,1],[]]
=> 5 = 4 + 1
0000 => [5] => [5]
=> [[5],[]]
=> 6 = 5 + 1
0001 => [4,1] => [4,1]
=> [[4,1],[]]
=> 9 = 8 + 1
0010 => [3,2] => [3,2]
=> [[3,2],[]]
=> 10 = 9 + 1
0011 => [3,1,1] => [3,1,1]
=> [[3,1,1],[]]
=> 10 = 9 + 1
0100 => [2,3] => [3,2]
=> [[3,2],[]]
=> 10 = 9 + 1
0101 => [2,2,1] => [2,2,1]
=> [[2,2,1],[]]
=> 10 = 9 + 1
0110 => [2,1,2] => [2,2,1]
=> [[2,2,1],[]]
=> 10 = 9 + 1
0111 => [2,1,1,1] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 9 = 8 + 1
1000 => [1,4] => [4,1]
=> [[4,1],[]]
=> 9 = 8 + 1
1001 => [1,3,1] => [3,1,1]
=> [[3,1,1],[]]
=> 10 = 9 + 1
1010 => [1,2,2] => [2,2,1]
=> [[2,2,1],[]]
=> 10 = 9 + 1
1011 => [1,2,1,1] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 9 = 8 + 1
1100 => [1,1,3] => [3,1,1]
=> [[3,1,1],[]]
=> 10 = 9 + 1
1101 => [1,1,2,1] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 9 = 8 + 1
1110 => [1,1,1,2] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 9 = 8 + 1
1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 6 = 5 + 1
Description
The total number of rook placements on a Ferrers board.
Matching statistic: St000479
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(load all 2 compositions to match this statistic)
Values
0 => ([(0,1)],2)
=> ([],2)
=> 2
1 => ([(0,1)],2)
=> ([],2)
=> 2
00 => ([(0,2),(2,1)],3)
=> ([],3)
=> 3
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 4
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 4
11 => ([(0,2),(2,1)],3)
=> ([],3)
=> 3
000 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 6
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 6
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 6
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 6
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 6
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 6
111 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {8,8,8,8,9,9,9,9,9,9,9,9}
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {8,8,8,8,9,9,9,9,9,9,9,9}
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? ∊ {8,8,8,8,9,9,9,9,9,9,9,9}
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {8,8,8,8,9,9,9,9,9,9,9,9}
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> 8
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {8,8,8,8,9,9,9,9,9,9,9,9}
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {8,8,8,8,9,9,9,9,9,9,9,9}
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {8,8,8,8,9,9,9,9,9,9,9,9}
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {8,8,8,8,9,9,9,9,9,9,9,9}
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> 8
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {8,8,8,8,9,9,9,9,9,9,9,9}
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? ∊ {8,8,8,8,9,9,9,9,9,9,9,9}
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? ∊ {8,8,8,8,9,9,9,9,9,9,9,9}
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? ∊ {8,8,8,8,9,9,9,9,9,9,9,9}
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
Description
The Ramsey number of a graph.
This is the smallest integer $n$ such that every two-colouring of the edges of the complete graph $K_n$ contains a (not necessarily induced) monochromatic copy of the given graph. [1]
Thus, the Ramsey number of the complete graph $K_n$ is the ordinary Ramsey number $R(n,n)$. Very few of these numbers are known, in particular, it is only known that $43\leq R(5,5)\leq 48$. [2,3,4,5]
Matching statistic: St000189
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00262: Binary words —poset of factors⟶ Posets
St000189: Posets ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 71%
St000189: Posets ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 71%
Values
0 => ([(0,1)],2)
=> 2
1 => ([(0,1)],2)
=> 2
00 => ([(0,2),(2,1)],3)
=> 3
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
11 => ([(0,2),(2,1)],3)
=> 3
000 => ([(0,3),(2,1),(3,2)],4)
=> 4
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 6
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 6
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
111 => ([(0,3),(2,1),(3,2)],4)
=> 4
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
Description
The number of elements in the poset.
Matching statistic: St001717
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00262: Binary words —poset of factors⟶ Posets
St001717: Posets ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 71%
St001717: Posets ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 71%
Values
0 => ([(0,1)],2)
=> 2
1 => ([(0,1)],2)
=> 2
00 => ([(0,2),(2,1)],3)
=> 3
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
11 => ([(0,2),(2,1)],3)
=> 3
000 => ([(0,3),(2,1),(3,2)],4)
=> 4
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 6
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 6
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
111 => ([(0,3),(2,1),(3,2)],4)
=> 4
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
Description
The largest size of an interval in a poset.
Matching statistic: St001300
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00262: Binary words —poset of factors⟶ Posets
St001300: Posets ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 71%
St001300: Posets ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 71%
Values
0 => ([(0,1)],2)
=> 1 = 2 - 1
1 => ([(0,1)],2)
=> 1 = 2 - 1
00 => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 4 - 1
11 => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5 = 6 - 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 5 = 6 - 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5 = 6 - 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5 = 6 - 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 5 = 6 - 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 5 = 6 - 1
111 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9} - 1
0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9} - 1
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9} - 1
0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9} - 1
0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9} - 1
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9} - 1
0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9} - 1
1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9} - 1
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9} - 1
1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9} - 1
1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9} - 1
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9} - 1
1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9} - 1
1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9} - 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
Description
The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset.
Matching statistic: St000656
Mp00224: Binary words —runsort⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000656: Posets ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 71%
Mp00262: Binary words —poset of factors⟶ Posets
St000656: Posets ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 71%
Values
0 => 0 => ([(0,1)],2)
=> 2
1 => 1 => ([(0,1)],2)
=> 2
00 => 00 => ([(0,2),(2,1)],3)
=> 3
01 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
10 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
11 => 11 => ([(0,2),(2,1)],3)
=> 3
000 => 000 => ([(0,3),(2,1),(3,2)],4)
=> 4
001 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
010 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
011 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
100 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
101 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
110 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 6
111 => 111 => ([(0,3),(2,1),(3,2)],4)
=> 4
0000 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
0001 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
0010 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
0011 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
0100 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
0101 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
0110 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
0111 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
1000 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
1001 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
1010 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
1011 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
1100 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
1101 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
1110 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {8,8,8,8,8,8,9,9,9,9,9,9,9,9}
1111 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
Description
The number of cuts of a poset.
A cut is a subset $A$ of the poset such that the set of lower bounds of the set of upper bounds of $A$ is exactly $A$.
The following 25 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001342The number of vertices in the center of a graph. St001622The number of join-irreducible elements of a lattice. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001120The length of a longest path in a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St000171The degree of the graph. St000362The size of a minimal vertex cover of a graph. St001706The number of closed sets in a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001645The pebbling number of a connected graph. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001875The number of simple modules with projective dimension at most 1. St000422The energy of a graph, if it is integral. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St000455The second largest eigenvalue of a graph if it is integral.
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