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Matching statistic: St001621
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
1010 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
00100 => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
00101 => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
01001 => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
01010 => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
10010 => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
10100 => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
10101 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
10110 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
11010 => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
000100 => [4,3] => [[6,4],[3]]
=> ([(0,2),(2,1)],3)
=> 1
000101 => [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> 1
001000 => [3,4] => [[6,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
001001 => [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
001010 => [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
001011 => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> ([(0,2),(2,1)],3)
=> 1
001100 => [3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
001101 => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
010001 => [2,4,1] => [[5,5,2],[4,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
010010 => [2,3,2] => [[5,4,2],[3,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
010011 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
010100 => [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
010101 => [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)
=> 1
010110 => [2,2,1,2] => [[4,3,3,2],[2,2,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 2
010111 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
011001 => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
011010 => [2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 2
011011 => [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)
=> 1
011101 => [2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
100010 => [1,4,2] => [[5,4,1],[3]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
100100 => [1,3,3] => [[5,3,1],[2]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
100101 => [1,3,2,1] => [[4,4,3,1],[3,2]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 1
100110 => [1,3,1,2] => [[4,3,3,1],[2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
101000 => [1,2,4] => [[5,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
101001 => [1,2,3,1] => [[4,4,2,1],[3,1]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 1
101010 => [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)
=> 1
101011 => [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)
=> 1
101100 => [1,2,1,3] => [[4,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
101101 => [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)
=> 1
101110 => [1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
110010 => [1,1,3,2] => [[4,3,1,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
110011 => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
110100 => [1,1,2,3] => [[4,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
110101 => [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)
=> 1
110110 => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
110111 => [1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> 1
Description
The number of atoms of a lattice.
An element of a lattice is an '''atom''' if it covers the least element.
Matching statistic: St001878
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
1010 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
00100 => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
00101 => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
01001 => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
01010 => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
10010 => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
10100 => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
10101 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
10110 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
11010 => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
000100 => [4,3] => [[6,4],[3]]
=> ([(0,2),(2,1)],3)
=> 1
000101 => [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> 1
001000 => [3,4] => [[6,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
001001 => [3,3,1] => [[5,5,3],[4,2]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
001010 => [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
001011 => [3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> ([(0,2),(2,1)],3)
=> 1
001100 => [3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
001101 => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
010001 => [2,4,1] => [[5,5,2],[4,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
010010 => [2,3,2] => [[5,4,2],[3,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
010011 => [2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
010100 => [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
010101 => [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)
=> 1
010110 => [2,2,1,2] => [[4,3,3,2],[2,2,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 2
010111 => [2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
011001 => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
011010 => [2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 2
011011 => [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)
=> 1
011101 => [2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
100010 => [1,4,2] => [[5,4,1],[3]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
100100 => [1,3,3] => [[5,3,1],[2]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
100101 => [1,3,2,1] => [[4,4,3,1],[3,2]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 1
100110 => [1,3,1,2] => [[4,3,3,1],[2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
101000 => [1,2,4] => [[5,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
101001 => [1,2,3,1] => [[4,4,2,1],[3,1]]
=> ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 1
101010 => [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)
=> 1
101011 => [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)
=> 1
101100 => [1,2,1,3] => [[4,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
101101 => [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)
=> 1
101110 => [1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
110010 => [1,1,3,2] => [[4,3,1,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
110011 => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
110100 => [1,1,2,3] => [[4,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
110101 => [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)
=> 1
110110 => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
110111 => [1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> ([(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: St000455
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
0101 => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 - 1
1010 => 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 - 1
00100 => 01000 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
00101 => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
01001 => 01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
01010 => 01100 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
01011 => 10011 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
01101 => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
10010 => 00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
10100 => 10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
10101 => 01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
10110 => 01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
11010 => 10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
11011 => 11011 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 1 - 1
000100 => 010000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
000101 => 100001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
001000 => 001000 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
001001 => 010001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
001010 => 011000 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
001011 => 100011 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
001100 => 101000 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
001101 => 110001 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
010001 => 001001 => [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)
=> ? = 2 - 1
010010 => 001100 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
010011 => 010011 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
010100 => 100100 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
010101 => 011001 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
010110 => 011100 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
010111 => 100111 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
011001 => 101001 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
011010 => 101100 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
011011 => 110011 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
011101 => 111001 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
100010 => 000110 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
100100 => 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)
=> ? = 2 - 1
100101 => 001101 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
100110 => 001110 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
101000 => 010010 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
101001 => 100101 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
101010 => 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 - 1
101011 => 011011 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
101100 => 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)
=> ? = 2 - 1
101101 => 011101 => [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 - 1
101110 => 011110 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
110010 => 010110 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 1
110011 => 101011 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
110100 => 011010 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
110101 => 101101 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
110110 => 101110 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 - 1
110111 => 110111 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
111010 => 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 - 1
111011 => 111011 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 1 - 1
0001000 => 0010000 => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
0001001 => 0100001 => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
0001010 => 0110000 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
0001011 => 1000011 => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1 - 1
0001100 => 1010000 => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
0001101 => 1100001 => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
0010001 => 0010001 => [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)
=> ? = 2 - 1
0010011 => 0100011 => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 1
0011100 => 1101000 => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
1101111 => 1101111 => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
1110111 => 1110111 => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
1111011 => 1111011 => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 1 - 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
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