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Matching statistic: St000667
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St000667: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 1
[3]
=> 3
[2,1]
=> 1
[1,1,1]
=> 1
[4]
=> 4
[3,1]
=> 1
[2,2]
=> 2
[2,1,1]
=> 1
[1,1,1,1]
=> 1
[5]
=> 5
[4,1]
=> 1
[3,2]
=> 1
[3,1,1]
=> 1
[2,2,1]
=> 1
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 1
[6]
=> 6
[5,1]
=> 1
[4,2]
=> 2
[4,1,1]
=> 1
[3,3]
=> 3
[3,2,1]
=> 1
[3,1,1,1]
=> 1
[2,2,2]
=> 2
[2,2,1,1]
=> 1
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 1
[7]
=> 7
[6,1]
=> 1
[5,2]
=> 1
[5,1,1]
=> 1
[4,3]
=> 1
[4,2,1]
=> 1
[4,1,1,1]
=> 1
[3,3,1]
=> 1
[3,2,2]
=> 1
[3,2,1,1]
=> 1
[3,1,1,1,1]
=> 1
[2,2,2,1]
=> 1
[2,2,1,1,1]
=> 1
[2,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 1
[8]
=> 8
[7,1]
=> 1
[6,2]
=> 2
[6,1,1]
=> 1
[5,3]
=> 1
[5,2,1]
=> 1
Description
The greatest common divisor of the parts of the partition.
Matching statistic: St000264
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 22%●distinct values known / distinct values provided: 8%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 22%●distinct values known / distinct values provided: 8%
Values
[1]
=> 1 => [1] => ([],1)
=> ? = 1 + 2
[2]
=> 0 => [1] => ([],1)
=> ? = 2 + 2
[1,1]
=> 11 => [2] => ([],2)
=> ? = 1 + 2
[3]
=> 1 => [1] => ([],1)
=> ? = 3 + 2
[2,1]
=> 01 => [1,1] => ([(0,1)],2)
=> ? = 1 + 2
[1,1,1]
=> 111 => [3] => ([],3)
=> ? = 1 + 2
[4]
=> 0 => [1] => ([],1)
=> ? = 4 + 2
[3,1]
=> 11 => [2] => ([],2)
=> ? = 1 + 2
[2,2]
=> 00 => [2] => ([],2)
=> ? = 2 + 2
[2,1,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? = 1 + 2
[1,1,1,1]
=> 1111 => [4] => ([],4)
=> ? = 1 + 2
[5]
=> 1 => [1] => ([],1)
=> ? = 5 + 2
[4,1]
=> 01 => [1,1] => ([(0,1)],2)
=> ? = 1 + 2
[3,2]
=> 10 => [1,1] => ([(0,1)],2)
=> ? = 1 + 2
[3,1,1]
=> 111 => [3] => ([],3)
=> ? = 1 + 2
[2,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 2
[2,1,1,1]
=> 0111 => [1,3] => ([(2,3)],4)
=> ? = 1 + 2
[1,1,1,1,1]
=> 11111 => [5] => ([],5)
=> ? = 1 + 2
[6]
=> 0 => [1] => ([],1)
=> ? = 6 + 2
[5,1]
=> 11 => [2] => ([],2)
=> ? = 1 + 2
[4,2]
=> 00 => [2] => ([],2)
=> ? = 2 + 2
[4,1,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? = 1 + 2
[3,3]
=> 11 => [2] => ([],2)
=> ? = 3 + 2
[3,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[3,1,1,1]
=> 1111 => [4] => ([],4)
=> ? = 1 + 2
[2,2,2]
=> 000 => [3] => ([],3)
=> ? = 2 + 2
[2,2,1,1]
=> 0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 2
[2,1,1,1,1]
=> 01111 => [1,4] => ([(3,4)],5)
=> ? = 1 + 2
[1,1,1,1,1,1]
=> 111111 => [6] => ([],6)
=> ? = 1 + 2
[7]
=> 1 => [1] => ([],1)
=> ? = 7 + 2
[6,1]
=> 01 => [1,1] => ([(0,1)],2)
=> ? = 1 + 2
[5,2]
=> 10 => [1,1] => ([(0,1)],2)
=> ? = 1 + 2
[5,1,1]
=> 111 => [3] => ([],3)
=> ? = 1 + 2
[4,3]
=> 01 => [1,1] => ([(0,1)],2)
=> ? = 1 + 2
[4,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 2
[4,1,1,1]
=> 0111 => [1,3] => ([(2,3)],4)
=> ? = 1 + 2
[3,3,1]
=> 111 => [3] => ([],3)
=> ? = 1 + 2
[3,2,2]
=> 100 => [1,2] => ([(1,2)],3)
=> ? = 1 + 2
[3,2,1,1]
=> 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[3,1,1,1,1]
=> 11111 => [5] => ([],5)
=> ? = 1 + 2
[2,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 2
[2,2,1,1,1]
=> 00111 => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 2
[2,1,1,1,1,1]
=> 011111 => [1,5] => ([(4,5)],6)
=> ? = 1 + 2
[1,1,1,1,1,1,1]
=> 1111111 => [7] => ([],7)
=> ? = 1 + 2
[8]
=> 0 => [1] => ([],1)
=> ? = 8 + 2
[7,1]
=> 11 => [2] => ([],2)
=> ? = 1 + 2
[6,2]
=> 00 => [2] => ([],2)
=> ? = 2 + 2
[6,1,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? = 1 + 2
[5,3]
=> 11 => [2] => ([],2)
=> ? = 1 + 2
[5,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[5,1,1,1]
=> 1111 => [4] => ([],4)
=> ? = 1 + 2
[4,4]
=> 00 => [2] => ([],2)
=> ? = 4 + 2
[4,3,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? = 1 + 2
[3,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[3,2,1,1,1]
=> 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[5,2,1,1]
=> 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[4,3,2]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[3,3,2,1]
=> 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[3,2,2,1,1]
=> 10011 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[3,2,1,1,1,1]
=> 101111 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[7,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[5,4,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[5,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[5,2,1,1,1]
=> 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[4,3,2,1]
=> 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[3,3,2,1,1]
=> 11011 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[3,2,2,2,1]
=> 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[3,2,2,1,1,1]
=> 100111 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[3,2,1,1,1,1,1]
=> 1011111 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[7,2,1,1]
=> 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[6,3,2]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[5,4,1,1]
=> 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[5,3,2,1]
=> 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[5,2,2,1,1]
=> 10011 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[5,2,1,1,1,1]
=> 101111 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[4,3,2,2]
=> 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[4,3,2,1,1]
=> 01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[3,3,2,2,1]
=> 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[3,3,2,1,1,1]
=> 110111 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[3,2,2,2,1,1]
=> 100011 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[3,2,2,1,1,1,1]
=> 1001111 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[9,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[7,4,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[7,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[7,2,1,1,1]
=> 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[6,3,2,1]
=> 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[5,4,3]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
[5,4,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[5,4,1,1,1]
=> 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[5,3,2,1,1]
=> 11011 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[5,2,2,2,1]
=> 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[5,2,2,1,1,1]
=> 100111 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[5,2,1,1,1,1,1]
=> 1011111 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[4,3,3,2]
=> 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[4,3,2,2,1]
=> 01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[4,3,2,1,1,1]
=> 010111 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[3,3,3,2,1]
=> 11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[3,3,2,2,1,1]
=> 110011 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[3,3,2,1,1,1,1]
=> 1101111 => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[3,2,2,2,2,1]
=> 100001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001051
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St001051: Set partitions ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 58%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St001051: Set partitions ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 58%
Values
[1]
=> [[1]]
=> [1] => {{1}}
=> 1
[2]
=> [[1,2]]
=> [1,2] => {{1},{2}}
=> 2
[1,1]
=> [[1],[2]]
=> [2,1] => {{1,2}}
=> 1
[3]
=> [[1,2,3]]
=> [1,2,3] => {{1},{2},{3}}
=> 3
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => {{1,2,3}}
=> 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => {{1,3},{2}}
=> 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 4
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => {{1,2,3,4}}
=> 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => {{1,3},{2,4}}
=> 2
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => {{1,2,3,4}}
=> 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 5
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => {{1,2,3,4,5}}
=> 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => {{1,3,5},{2,4}}
=> 1
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => {{1,2,3,4,5}}
=> 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => {{1,2,4,5},{3}}
=> 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => {{1},{2},{3},{4},{5},{6}}
=> 6
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => {{1,2,3,4,5,6}}
=> 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => {{1,3,5},{2,4,6}}
=> 2
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => {{1,2,3,4,5,6}}
=> 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => {{1,4},{2,5},{3,6}}
=> 3
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => {{1,2,3,4,5,6}}
=> 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => {{1,3,4,6},{2,5}}
=> 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => {{1,5},{2,6},{3},{4}}
=> 2
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => {{1,2,5,6},{3},{4}}
=> 1
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => {{1,2,5,6},{3,4}}
=> 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => {{1,6},{2,5},{3,4}}
=> 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => {{1},{2},{3},{4},{5},{6},{7}}
=> 7
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => {{1,2,3,4,5,6,7}}
=> 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => {{1,2,3,4,5,6,7}}
=> 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => {{1,3,5,7},{2,4,6}}
=> 1
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => {{1,2,3,4,5,6,7}}
=> 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => {{1,4,7},{2,5},{3,6}}
=> 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => {{1,4,7},{2,3,5,6}}
=> 1
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => {{1,3,5,7},{2,4,6}}
=> 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => {{1,2,3,4,5,6,7}}
=> 1
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => {{1,3,4,5,7},{2,6}}
=> 1
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => {{1,3,5,7},{2,6},{4}}
=> 1
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => {{1,2,3,4,5,6,7}}
=> 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => {{1,2,6,7},{3,4,5}}
=> 1
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => {{1,2,6,7},{3,5},{4}}
=> 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => {{1,7},{2,6},{3,5},{4}}
=> 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 8
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => {{1,2,3,4,5,6,7,8}}
=> 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => {{1,3,5,7},{2,4,6,8}}
=> ? = 2
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => {{1,2,3,4,5,6,7,8}}
=> 1
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => {{1,2,3,4,5,6,7,8}}
=> 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => {{1,2,3,4,5,6,7,8}}
=> 1
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [8,7,6,1,2,3,4,5] => {{1,2,4,5,7,8},{3,6}}
=> 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
=> 4
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => {{1,3,5,7},{2,4,6,8}}
=> ? = 2
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => {{1,4,5,8},{2,3,6,7}}
=> ? = 1
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => {{1,2,3,6,7,8},{4},{5}}
=> ? = 1
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => {{1,3,6,8},{2,7},{4},{5}}
=> ? = 1
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => {{1,3,6,8},{2,7},{4,5}}
=> ? = 1
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => {{1,2,7,8},{3,5},{4,6}}
=> ? = 1
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => {{1,2,7,8},{3,4,5,6}}
=> ? = 1
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => {{1,2,7,8},{3,6},{4,5}}
=> ? = 1
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 9
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [9,1,2,3,4,5,6,7,8] => {{1,2,3,4,5,6,7,8,9}}
=> ? = 1
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [8,9,1,2,3,4,5,6,7] => {{1,2,3,4,5,6,7,8,9}}
=> ? = 1
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [9,8,1,2,3,4,5,6,7] => {{1,3,5,7,9},{2,4,6,8}}
=> ? = 1
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [7,8,9,1,2,3,4,5,6] => {{1,4,7},{2,5,8},{3,6,9}}
=> ? = 3
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [9,7,8,1,2,3,4,5,6] => {{1,2,3,4,5,6,7,8,9}}
=> ? = 1
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [9,8,7,1,2,3,4,5,6] => {{1,3,4,6,7,9},{2,5,8}}
=> ? = 1
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [6,7,8,9,1,2,3,4,5] => {{1,2,3,4,5,6,7,8,9}}
=> ? = 1
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [9,6,7,8,1,2,3,4,5] => {{1,5,9},{2,6},{3,7},{4,8}}
=> ? = 1
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [8,9,6,7,1,2,3,4,5] => {{1,2,3,4,5,6,7,8,9}}
=> ? = 1
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [9,8,6,7,1,2,3,4,5] => {{1,5,9},{2,3,4,6,7,8}}
=> ? = 1
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [9,8,7,6,1,2,3,4,5] => {{1,5,9},{2,4,6,8},{3,7}}
=> ? = 1
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [9,5,6,7,8,1,2,3,4] => {{1,2,3,4,5,6,7,8,9}}
=> ? = 1
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [8,9,5,6,7,1,2,3,4] => {{1,2,3,4,5,6,7,8,9}}
=> ? = 1
[4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [9,8,5,6,7,1,2,3,4] => {{1,4,6,9},{2,3,5,7,8}}
=> ? = 1
[4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => {{1,4,5,6,9},{2,7},{3,8}}
=> ? = 1
[4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,1,2,3,4] => {{1,4,5,6,9},{2,3,7,8}}
=> ? = 1
[4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,1,2,3,4] => {{1,4,6,9},{2,3,7,8},{5}}
=> ? = 1
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => {{1,7},{2,8},{3,9},{4},{5},{6}}
=> ? = 3
[3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [9,7,8,4,5,6,1,2,3] => {{1,2,3,7,8,9},{4},{5},{6}}
=> ? = 1
[3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [9,8,7,4,5,6,1,2,3] => {{1,3,7,9},{2,8},{4},{5},{6}}
=> ? = 1
[3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [8,9,6,7,4,5,1,2,3] => {{1,2,3,4,5,6,7,8,9}}
=> ? = 1
[3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [9,8,6,7,4,5,1,2,3] => {{1,3,4,5,6,7,9},{2,8}}
=> ? = 1
[3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [9,8,7,6,4,5,1,2,3] => {{1,3,7,9},{2,8},{4,5,6}}
=> ? = 1
[3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,1,2,3] => {{1,3,7,9},{2,8},{4,6},{5}}
=> ? = 1
[2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,3,4,1,2] => {{1,2,3,4,5,6,7,8,9}}
=> ? = 1
[2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,3,4,1,2] => {{1,2,8,9},{3,4,5,6,7}}
=> ? = 1
[2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,3,4,1,2] => {{1,2,8,9},{3,4,6,7},{5}}
=> ? = 1
[2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,1,2] => {{1,2,8,9},{3,7},{4,6},{5}}
=> ? = 1
[1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,2,1] => {{1,9},{2,8},{3,7},{4,6},{5}}
=> ? = 1
[10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [1,2,3,4,5,6,7,8,9,10] => {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> ? = 10
[9,1]
=> [[1,2,3,4,5,6,7,8,9],[10]]
=> [10,1,2,3,4,5,6,7,8,9] => {{1,2,3,4,5,6,7,8,9,10}}
=> ? = 1
[8,2]
=> [[1,2,3,4,5,6,7,8],[9,10]]
=> [9,10,1,2,3,4,5,6,7,8] => {{1,3,5,7,9},{2,4,6,8,10}}
=> ? = 2
[8,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10]]
=> [10,9,1,2,3,4,5,6,7,8] => {{1,2,3,4,5,6,7,8,9,10}}
=> ? = 1
[7,3]
=> [[1,2,3,4,5,6,7],[8,9,10]]
=> [8,9,10,1,2,3,4,5,6,7] => {{1,2,3,4,5,6,7,8,9,10}}
=> ? = 1
[7,2,1]
=> [[1,2,3,4,5,6,7],[8,9],[10]]
=> [10,8,9,1,2,3,4,5,6,7] => {{1,4,7,10},{2,5,8},{3,6,9}}
=> ? = 1
[7,1,1,1]
=> [[1,2,3,4,5,6,7],[8],[9],[10]]
=> [10,9,8,1,2,3,4,5,6,7] => {{1,4,7,10},{2,3,5,6,8,9}}
=> ? = 1
[6,4]
=> [[1,2,3,4,5,6],[7,8,9,10]]
=> [7,8,9,10,1,2,3,4,5,6] => {{1,3,5,7,9},{2,4,6,8,10}}
=> ? = 2
[6,3,1]
=> [[1,2,3,4,5,6],[7,8,9],[10]]
=> [10,7,8,9,1,2,3,4,5,6] => {{1,2,3,4,5,6,7,8,9,10}}
=> ? = 1
[6,2,2]
=> [[1,2,3,4,5,6],[7,8],[9,10]]
=> [9,10,7,8,1,2,3,4,5,6] => {{1,5,9},{2,6,10},{3,7},{4,8}}
=> ? = 2
Description
The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition.
The bijection between set partitions of $\{1,\dots,n\}$ into $k$ blocks and trees with $n+1-k$ leaves is described in Theorem 1 of [1].
Matching statistic: St000260
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 11%●distinct values known / distinct values provided: 8%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 11%●distinct values known / distinct values provided: 8%
Values
[1]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[2]
=> 100 => [1,2] => ([(1,2)],3)
=> ? = 2
[1,1]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1
[3]
=> 1000 => [1,3] => ([(2,3)],4)
=> ? = 3
[2,1]
=> 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1]
=> 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4]
=> 10000 => [1,4] => ([(3,4)],5)
=> ? = 4
[3,1]
=> 10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,2]
=> 1100 => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[2,1,1]
=> 10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,1,1]
=> 11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[5]
=> 100000 => [1,5] => ([(4,5)],6)
=> ? = 5
[4,1]
=> 100010 => [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)
=> 1
[3,2]
=> 10100 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[3,1,1]
=> 100110 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,2,1]
=> 11010 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,1,1,1]
=> 101110 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,1,1,1,1]
=> 111110 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[6]
=> 1000000 => [1,6] => ([(5,6)],7)
=> ? = 6
[5,1]
=> 1000010 => [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[4,2]
=> 100100 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[4,1,1]
=> 1000110 => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[3,3]
=> 11000 => [2,3] => ([(2,4),(3,4)],5)
=> ? = 3
[3,2,1]
=> 101010 => [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
[3,1,1,1]
=> 1001110 => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[2,2,2]
=> 11100 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[2,2,1,1]
=> 110110 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,1,1,1,1]
=> 1011110 => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[1,1,1,1,1,1]
=> 1111110 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[7]
=> 10000000 => [1,7] => ([(6,7)],8)
=> ? = 7
[6,1]
=> 10000010 => [1,5,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[5,2]
=> 1000100 => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[5,1,1]
=> 10000110 => [1,4,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[4,3]
=> 101000 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[4,2,1]
=> 1001010 => [1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[4,1,1,1]
=> 10001110 => [1,3,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[3,3,1]
=> 110010 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[3,2,2]
=> 101100 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[3,2,1,1]
=> 1010110 => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[3,1,1,1,1]
=> 10011110 => [1,2,4,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[2,2,2,1]
=> 111010 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[2,2,1,1,1]
=> 1101110 => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[2,1,1,1,1,1]
=> 10111110 => [1,1,5,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[1,1,1,1,1,1,1]
=> 11111110 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 1
[8]
=> 100000000 => [1,8] => ([(7,8)],9)
=> ? = 8
[7,1]
=> 100000010 => [1,6,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[6,2]
=> 10000100 => [1,4,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
[6,1,1]
=> 100000110 => [1,5,2,1] => ([(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[5,3]
=> 1001000 => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[5,2,1]
=> 10001010 => [1,3,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[5,1,1,1]
=> 100001110 => [1,4,3,1] => ([(0,8),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[4,4]
=> 110000 => [2,4] => ([(3,5),(4,5)],6)
=> ? = 4
[4,3,1]
=> 1010010 => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[4,2,2]
=> 1001100 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[4,2,1,1]
=> 10010110 => [1,2,1,1,2,1] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[4,1,1,1,1]
=> 100011110 => [1,3,4,1] => ([(0,8),(1,8),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[3,3,2]
=> 110100 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[3,3,1,1]
=> 1100110 => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[3,2,2,1]
=> 1011010 => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[3,2,1,1,1]
=> 10101110 => [1,1,1,1,3,1] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[3,1,1,1,1,1]
=> 100111110 => [1,2,5,1] => ([(0,8),(1,8),(2,8),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[2,2,2,2]
=> 111100 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2
[2,2,2,1,1]
=> 1110110 => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[2,2,1,1,1,1]
=> 11011110 => [2,1,4,1] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[2,1,1,1,1,1,1]
=> 101111110 => [1,1,6,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[1,1,1,1,1,1,1,1]
=> 111111110 => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 1
[9]
=> 1000000000 => [1,9] => ([(8,9)],10)
=> ? = 9
[8,1]
=> 1000000010 => [1,7,1,1] => ([(0,8),(0,9),(1,8),(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[7,2]
=> 100000100 => [1,5,1,2] => ([(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[7,1,1]
=> 1000000110 => [1,6,2,1] => ([(0,9),(1,8),(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[6,3]
=> 10001000 => [1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
[6,2,1]
=> 100001010 => [1,4,1,1,1,1] => ([(0,5),(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[6,1,1,1]
=> 1000001110 => [1,5,3,1] => ([(0,9),(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[5,4]
=> 1010000 => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[5,3,1]
=> 10010010 => [1,2,1,2,1,1] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[5,2,2]
=> 10001100 => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 1
[5,2,1,1]
=> 100010110 => [1,3,1,1,2,1] => ([(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 1
[5,1,1,1,1]
=> 1000011110 => [1,4,4,1] => ([(0,9),(1,9),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 1
[4,4,1]
=> 1100010 => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[3,3,2,1]
=> 1101010 => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[2,2,2,2,1]
=> 1111010 => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[3,3,3,1]
=> 1110010 => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St001232
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 50%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 50%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ? = 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? = 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ? = 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ? = 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 7
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? = 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ? = 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ? = 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? = 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ? = 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? = 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? = 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> ? = 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? = 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 5
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2
[4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 4
[3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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