Your data matches 33 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
St000921: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => 0
1 => 0
00 => 0
01 => 0
10 => 0
11 => 0
000 => 0
001 => 0
010 => 0
011 => 0
100 => 1
101 => 0
110 => 1
111 => 0
0000 => 0
0001 => 0
0010 => 0
0011 => 0
0100 => 1
0101 => 0
0110 => 1
0111 => 0
1000 => 2
1001 => 1
1010 => 2
1011 => 0
1100 => 0
1101 => 1
1110 => 2
1111 => 0
00000 => 0
00001 => 0
00010 => 0
00011 => 0
00100 => 1
00101 => 0
00110 => 1
00111 => 0
01000 => 2
01001 => 1
01010 => 2
01011 => 0
01100 => 0
01101 => 1
01110 => 2
01111 => 0
10000 => 3
10001 => 2
10010 => 3
10011 => 1
Description
The number of internal inversions of a binary word. Let $\bar w$ be the non-decreasing rearrangement of $w$, that is, $\bar w$ is sorted. An internal inversion is a pair $i < j$ such that $w_i > w_j$ and $\bar w_i = \bar w_j$. For example, the word $110$ has two inversions, but only the second is internal.
Matching statistic: St001382
Mp00178: Binary words to compositionInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00183: Skew partitions inner shapeInteger partitions
St001382: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [[2],[]]
=> []
=> 0
1 => [1,1] => [[1,1],[]]
=> []
=> 0
00 => [3] => [[3],[]]
=> []
=> 0
01 => [2,1] => [[2,2],[1]]
=> [1]
=> 0
10 => [1,2] => [[2,1],[]]
=> []
=> 0
11 => [1,1,1] => [[1,1,1],[]]
=> []
=> 0
000 => [4] => [[4],[]]
=> []
=> 0
001 => [3,1] => [[3,3],[2]]
=> [2]
=> 1
010 => [2,2] => [[3,2],[1]]
=> [1]
=> 0
011 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
100 => [1,3] => [[3,1],[]]
=> []
=> 0
101 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
110 => [1,1,2] => [[2,1,1],[]]
=> []
=> 0
111 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> 0
0000 => [5] => [[5],[]]
=> []
=> 0
0001 => [4,1] => [[4,4],[3]]
=> [3]
=> 2
0010 => [3,2] => [[4,3],[2]]
=> [2]
=> 1
0011 => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 0
0100 => [2,3] => [[4,2],[1]]
=> [1]
=> 0
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 2
0110 => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
1000 => [1,4] => [[4,1],[]]
=> []
=> 0
1001 => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
1010 => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
1100 => [1,1,3] => [[3,1,1],[]]
=> []
=> 0
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
1110 => [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> 0
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> 0
00000 => [6] => [[6],[]]
=> []
=> 0
00001 => [5,1] => [[5,5],[4]]
=> [4]
=> 3
00010 => [4,2] => [[5,4],[3]]
=> [3]
=> 2
00011 => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 2
00100 => [3,3] => [[5,3],[2]]
=> [2]
=> 1
00101 => [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 1
00110 => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 0
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 2
01000 => [2,4] => [[5,2],[1]]
=> [1]
=> 0
01001 => [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 3
01010 => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 2
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 1
01100 => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 3
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 3
10000 => [1,5] => [[5,1],[]]
=> []
=> 0
10001 => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 2
10010 => [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 1
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 0
Description
The number of boxes in the diagram of a partition that do not lie in its Durfee square.
Mp00097: Binary words delta morphismInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
Mp00247: Graphs de-duplicateGraphs
St000771: Graphs ⟶ ℤResult quality: 59% values known / values provided: 59%distinct values known / distinct values provided: 100%
Values
0 => [1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
1 => [1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
00 => [2] => ([],2)
=> ([],1)
=> 1 = 0 + 1
01 => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
10 => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
11 => [2] => ([],2)
=> ([],1)
=> 1 = 0 + 1
000 => [3] => ([],3)
=> ([],1)
=> 1 = 0 + 1
001 => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1 = 0 + 1
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
011 => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {0,0} + 1
100 => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {0,0} + 1
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
110 => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1 = 0 + 1
111 => [3] => ([],3)
=> ([],1)
=> 1 = 0 + 1
0000 => [4] => ([],4)
=> ([],1)
=> 1 = 0 + 1
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 0 + 1
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
0011 => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,2} + 1
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,2} + 1
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
0111 => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,2} + 1
1000 => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,2} + 1
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,2} + 1
1100 => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,2} + 1
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1 = 0 + 1
1111 => [4] => ([],4)
=> ([],1)
=> 1 = 0 + 1
00000 => [5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 0 + 1
00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3} + 1
00100 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3} + 1
00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
00111 => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3} + 1
01000 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3} + 1
01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
01010 => [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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3} + 1
01100 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3} + 1
01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
01111 => [1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3} + 1
10000 => [1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3} + 1
10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
10011 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3} + 1
10100 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3} + 1
10101 => [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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3} + 1
11000 => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3} + 1
11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
11011 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3} + 1
11100 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,2,2,3,3} + 1
11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1 = 0 + 1
11111 => [5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
000000 => [6] => ([],6)
=> ([],1)
=> 1 = 0 + 1
000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1 = 0 + 1
000010 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
000011 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
000100 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
000101 => [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)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
000110 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
000111 => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
001000 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
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)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
001010 => [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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
001011 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
001100 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
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)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
001110 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
001111 => [2,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
010000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
010001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
010011 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
010100 => [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)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
010111 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
011000 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
011011 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
011100 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
011111 => [1,5] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
100000 => [1,5] => ([(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
100011 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
100100 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
100111 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
101000 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
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)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
101100 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
101111 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
110000 => [2,4] => ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
110011 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
110100 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
110111 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
111000 => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $2$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Mp00097: Binary words delta morphismInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000772: Graphs ⟶ ℤResult quality: 51% values known / values provided: 51%distinct values known / distinct values provided: 100%
Values
0 => [1] => ([],1)
=> 1 = 0 + 1
1 => [1] => ([],1)
=> 1 = 0 + 1
00 => [2] => ([],2)
=> ? ∊ {0,0} + 1
01 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
10 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
11 => [2] => ([],2)
=> ? ∊ {0,0} + 1
000 => [3] => ([],3)
=> ? ∊ {0,0,0,0} + 1
001 => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
011 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0} + 1
100 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0} + 1
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
110 => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
111 => [3] => ([],3)
=> ? ∊ {0,0,0,0} + 1
0000 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,1,1,2} + 1
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,2} + 1
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,2} + 1
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
0111 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,2} + 1
1000 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,2} + 1
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,2} + 1
1100 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,1,2} + 1
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
1111 => [4] => ([],4)
=> ? ∊ {0,0,0,0,0,1,1,2} + 1
00000 => [5] => ([],5)
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,3} + 1
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,3} + 1
00100 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,3} + 1
00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
00111 => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,3} + 1
01000 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,3} + 1
01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
01010 => [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)
=> 4 = 3 + 1
01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,3} + 1
01100 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,3} + 1
01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
01111 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,3} + 1
10000 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,3} + 1
10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
10011 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,3} + 1
10100 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,3} + 1
10101 => [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)
=> 4 = 3 + 1
10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,3} + 1
11000 => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,3} + 1
11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 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 = 0 + 1
11011 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,3} + 1
11100 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,3} + 1
11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
11111 => [5] => ([],5)
=> ? ∊ {0,0,0,0,1,1,1,1,2,2,2,2,2,2,3,3} + 1
000000 => [6] => ([],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 4 = 3 + 1
000010 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
000011 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
000100 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
000101 => [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 + 1
000110 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
000111 => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
001000 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
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)
=> 1 = 0 + 1
001010 => [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)
=> 1 = 0 + 1
001011 => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
001100 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
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 = 0 + 1
001110 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
001111 => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
010000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
010001 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
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 = 0 + 1
010011 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
010100 => [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)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
010101 => [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)
=> 5 = 4 + 1
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)
=> 1 = 0 + 1
010111 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
011000 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
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 = 0 + 1
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 = 0 + 1
011011 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
011100 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
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)
=> 2 = 1 + 1
011110 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
011111 => [1,5] => ([(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
100000 => [1,5] => ([(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
100001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 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)
=> 2 = 1 + 1
100011 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
100100 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
100111 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4} + 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $1$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$. The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St000681
Mp00097: Binary words delta morphismInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00183: Skew partitions inner shapeInteger partitions
St000681: Integer partitions ⟶ ℤResult quality: 51% values known / values provided: 51%distinct values known / distinct values provided: 100%
Values
0 => [1] => [[1],[]]
=> []
=> ? ∊ {0,0}
1 => [1] => [[1],[]]
=> []
=> ? ∊ {0,0}
00 => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0,0}
01 => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0}
10 => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,0,0}
11 => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,0,0}
000 => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
001 => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1}
010 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
011 => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
100 => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
101 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
110 => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1}
111 => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1}
0000 => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2}
0001 => [3,1] => [[3,3],[2]]
=> [2]
=> 1
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
0011 => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2}
0100 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2}
0110 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2}
0111 => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2}
1000 => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2}
1001 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2}
1010 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2}
1011 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2}
1100 => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2}
1101 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
1110 => [3,1] => [[3,3],[2]]
=> [2]
=> 1
1111 => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,2,2,2}
00000 => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3}
00001 => [4,1] => [[4,4],[3]]
=> [3]
=> 2
00010 => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
00011 => [3,2] => [[4,3],[2]]
=> [2]
=> 1
00100 => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
00110 => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
00111 => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3}
01000 => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3}
01001 => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3}
01010 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3}
01011 => [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3}
01100 => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3}
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
01110 => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
01111 => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3}
10000 => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3}
10001 => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
10011 => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3}
10100 => [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3}
10101 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3}
10110 => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3}
10111 => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3}
11000 => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3}
11001 => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
11010 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
11011 => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
11100 => [3,2] => [[4,3],[2]]
=> [2]
=> 1
11101 => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
11110 => [4,1] => [[4,4],[3]]
=> [3]
=> 2
11111 => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,2,2,3,3,3,3}
000000 => [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,2,3,3,3,3,4,4,4,4}
000001 => [5,1] => [[5,5],[4]]
=> [4]
=> 3
000010 => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 4
000011 => [4,2] => [[5,4],[3]]
=> [3]
=> 2
000100 => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 2
000101 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 4
000110 => [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 0
000111 => [3,3] => [[5,3],[2]]
=> [2]
=> 1
001000 => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1
001001 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 3
001010 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 3
001011 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 2
001100 => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 0
001101 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 0
001110 => [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 3
001111 => [2,4] => [[5,2],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,2,3,3,3,3,4,4,4,4}
010000 => [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,2,3,3,3,3,4,4,4,4}
010001 => [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 1
010010 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1
010011 => [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,2,3,3,3,3,4,4,4,4}
010100 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,2,3,3,3,3,4,4,4,4}
010101 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,2,3,3,3,3,4,4,4,4}
010110 => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,2,3,3,3,3,4,4,4,4}
010111 => [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,2,2,2,2,3,3,3,3,4,4,4,4}
011001 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 0
011010 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 2
011011 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
011100 => [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 1
011101 => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 2
011110 => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 2
100001 => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 2
100010 => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 2
100011 => [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 1
100100 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
100101 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 2
100110 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 0
101101 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1
101110 => [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 1
Description
The Grundy value of Chomp on Ferrers diagrams. Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1]. This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
Matching statistic: St001629
Mp00224: Binary words runsortBinary words
Mp00097: Binary words delta morphismInteger compositions
Mp00133: Integer compositions delta morphismInteger compositions
St001629: Integer compositions ⟶ ℤResult quality: 23% values known / values provided: 23%distinct values known / distinct values provided: 60%
Values
0 => 0 => [1] => [1] => ? ∊ {0,0}
1 => 1 => [1] => [1] => ? ∊ {0,0}
00 => 00 => [2] => [1] => ? ∊ {0,0,0,0}
01 => 01 => [1,1] => [2] => ? ∊ {0,0,0,0}
10 => 01 => [1,1] => [2] => ? ∊ {0,0,0,0}
11 => 11 => [2] => [1] => ? ∊ {0,0,0,0}
000 => 000 => [3] => [1] => ? ∊ {0,0,0,0,0,0,1,1}
001 => 001 => [2,1] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1}
010 => 001 => [2,1] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1}
011 => 011 => [1,2] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1}
100 => 001 => [2,1] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1}
101 => 011 => [1,2] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1}
110 => 011 => [1,2] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1}
111 => 111 => [3] => [1] => ? ∊ {0,0,0,0,0,0,1,1}
0000 => 0000 => [4] => [1] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
0001 => 0001 => [3,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
0010 => 0001 => [3,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
0011 => 0011 => [2,2] => [2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
0100 => 0001 => [3,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
0101 => 0101 => [1,1,1,1] => [4] => 1
0110 => 0011 => [2,2] => [2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
0111 => 0111 => [1,3] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
1000 => 0001 => [3,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
1001 => 0011 => [2,2] => [2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
1010 => 0011 => [2,2] => [2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
1011 => 0111 => [1,3] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
1100 => 0011 => [2,2] => [2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
1101 => 0111 => [1,3] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
1110 => 0111 => [1,3] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
1111 => 1111 => [4] => [1] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2}
00000 => 00000 => [5] => [1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00001 => 00001 => [4,1] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00010 => 00001 => [4,1] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00011 => 00011 => [3,2] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00100 => 00001 => [4,1] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00101 => 00101 => [2,1,1,1] => [1,3] => 0
00110 => 00011 => [3,2] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00111 => 00111 => [2,3] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01000 => 00001 => [4,1] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01001 => 00101 => [2,1,1,1] => [1,3] => 0
01010 => 00101 => [2,1,1,1] => [1,3] => 0
01011 => 01011 => [1,1,1,2] => [3,1] => 0
01100 => 00011 => [3,2] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01101 => 01011 => [1,1,1,2] => [3,1] => 0
01110 => 00111 => [2,3] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01111 => 01111 => [1,4] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10000 => 00001 => [4,1] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10001 => 00011 => [3,2] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10010 => 00011 => [3,2] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10011 => 00111 => [2,3] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10100 => 00011 => [3,2] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10101 => 01011 => [1,1,1,2] => [3,1] => 0
10110 => 00111 => [2,3] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10111 => 01111 => [1,4] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
11000 => 00011 => [3,2] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
11001 => 00111 => [2,3] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
11010 => 00111 => [2,3] => [1,1] => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
000101 => 000101 => [3,1,1,1] => [1,3] => 0
001001 => 001001 => [2,1,2,1] => [1,1,1,1] => 0
001010 => 000101 => [3,1,1,1] => [1,3] => 0
001011 => 001011 => [2,1,1,2] => [1,2,1] => 2
001101 => 001101 => [2,2,1,1] => [2,2] => 1
010001 => 000101 => [3,1,1,1] => [1,3] => 0
010010 => 000101 => [3,1,1,1] => [1,3] => 0
010011 => 001101 => [2,2,1,1] => [2,2] => 1
010100 => 000101 => [3,1,1,1] => [1,3] => 0
010101 => 010101 => [1,1,1,1,1,1] => [6] => 1
010110 => 001011 => [2,1,1,2] => [1,2,1] => 2
010111 => 010111 => [1,1,1,3] => [3,1] => 0
011001 => 001011 => [2,1,1,2] => [1,2,1] => 2
011010 => 001011 => [2,1,1,2] => [1,2,1] => 2
011011 => 011011 => [1,2,1,2] => [1,1,1,1] => 0
011101 => 010111 => [1,1,1,3] => [3,1] => 0
100101 => 001011 => [2,1,1,2] => [1,2,1] => 2
101001 => 001011 => [2,1,1,2] => [1,2,1] => 2
101010 => 001011 => [2,1,1,2] => [1,2,1] => 2
101011 => 010111 => [1,1,1,3] => [3,1] => 0
101101 => 010111 => [1,1,1,3] => [3,1] => 0
110101 => 010111 => [1,1,1,3] => [3,1] => 0
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Matching statistic: St001630
Mp00097: Binary words delta morphismInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00192: Skew partitions dominating sublatticeLattices
St001630: Lattices ⟶ ℤResult quality: 22% values known / values provided: 22%distinct values known / distinct values provided: 40%
Values
0 => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,0}
1 => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,0}
00 => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
01 => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
10 => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
11 => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
000 => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1}
001 => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1}
010 => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1}
011 => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1}
100 => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1}
101 => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1}
110 => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1}
111 => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1}
0000 => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
0001 => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
0011 => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
0100 => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
0110 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
0111 => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1000 => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1001 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1010 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1011 => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1100 => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1101 => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1110 => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1111 => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
00000 => [5] => [[5],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00001 => [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00010 => [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00011 => [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00100 => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00110 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
00111 => [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01000 => [1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01001 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01010 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01011 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01100 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01110 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01111 => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10000 => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10001 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10011 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
10100 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10101 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10110 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
11001 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
000110 => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
000111 => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
001001 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
001100 => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
001101 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
001110 => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
010010 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
010011 => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
011000 => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
011001 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
011011 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
011100 => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
100011 => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
100100 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
100110 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
100111 => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
101100 => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
101101 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
110001 => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
110010 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
110011 => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
110110 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
111000 => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
111001 => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001876
Mp00097: Binary words delta morphismInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00192: Skew partitions dominating sublatticeLattices
St001876: Lattices ⟶ ℤResult quality: 22% values known / values provided: 22%distinct values known / distinct values provided: 40%
Values
0 => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,0}
1 => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,0}
00 => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
01 => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
10 => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
11 => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
000 => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1}
001 => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1}
010 => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1}
011 => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1}
100 => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1}
101 => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1}
110 => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1}
111 => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1}
0000 => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
0001 => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
0011 => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
0100 => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
0110 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
0111 => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1000 => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1001 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1010 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1011 => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1100 => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1101 => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1110 => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1111 => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
00000 => [5] => [[5],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00001 => [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00010 => [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00011 => [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00100 => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00110 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
00111 => [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01000 => [1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01001 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01010 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01011 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01100 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01110 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01111 => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10000 => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10001 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10011 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
10100 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10101 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10110 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
11001 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
000110 => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
000111 => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
001001 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
001100 => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
001101 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 0
001110 => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
010010 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 0
010011 => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
011000 => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
011001 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
011011 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
011100 => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
100011 => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
100100 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
100110 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
100111 => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
101100 => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
101101 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 0
110001 => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
110010 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 0
110011 => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
110110 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
111000 => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
111001 => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St001877
Mp00097: Binary words delta morphismInteger compositions
Mp00180: Integer compositions to ribbonSkew partitions
Mp00192: Skew partitions dominating sublatticeLattices
St001877: Lattices ⟶ ℤResult quality: 22% values known / values provided: 22%distinct values known / distinct values provided: 40%
Values
0 => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,0}
1 => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {0,0}
00 => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
01 => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
10 => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
11 => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {0,0,0,0}
000 => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1}
001 => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1}
010 => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1}
011 => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1}
100 => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1}
101 => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1}
110 => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1}
111 => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,1,1}
0000 => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
0001 => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
0011 => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
0100 => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
0110 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
0111 => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1000 => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1001 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1010 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1011 => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1100 => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1101 => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1110 => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1111 => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2}
00000 => [5] => [[5],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00001 => [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00010 => [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00011 => [3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00100 => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00110 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
00111 => [2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01000 => [1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01001 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01010 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01011 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01100 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01110 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01111 => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10000 => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10001 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10011 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
10100 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10101 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10110 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
11001 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
000110 => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
000111 => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
001001 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
001100 => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
001101 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 0
001110 => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
010010 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 0
010011 => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
011000 => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
011001 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
011011 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
011100 => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
100011 => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
100100 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
100110 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
100111 => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
101100 => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
101101 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 0
110001 => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
110010 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 0
110011 => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
110110 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
111000 => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
111001 => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
Description
Number of indecomposable injective modules with projective dimension 2.
Mp00262: Binary words poset of factorsPosets
Mp00198: Posets incomparability graphGraphs
St001327: Graphs ⟶ ℤResult quality: 16% values known / values provided: 16%distinct values known / distinct values provided: 40%
Values
0 => ([(0,1)],2)
=> ([],2)
=> 0
1 => ([(0,1)],2)
=> ([],2)
=> 0
00 => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 0
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 0
11 => ([(0,2),(2,1)],3)
=> ([],3)
=> 0
000 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 0
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 0
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 0
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 0
111 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
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)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2}
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)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2}
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)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2}
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)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2}
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)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2}
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)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2}
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)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2}
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)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2}
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)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2}
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)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2}
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)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2}
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)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2}
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)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2}
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)
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2}
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 0
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00010 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00100 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(2,5),(3,4),(3,10),(4,9),(5,11),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,10),(8,11),(9,10)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00101 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01000 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(2,9),(3,8),(4,7),(5,6)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01101 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01110 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(2,6),(3,6),(3,9),(3,10),(4,5),(4,8),(4,10),(5,7),(5,9),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
01111 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ([(2,6),(3,6),(3,9),(3,10),(4,5),(4,8),(4,10),(5,7),(5,9),(6,11),(7,8),(7,10),(7,11),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(2,9),(3,8),(4,7),(5,6)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ([(2,5),(3,4),(3,11),(4,9),(5,10),(6,8),(6,9),(6,11),(7,8),(7,10),(7,11),(8,10),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ([(2,5),(3,6),(3,11),(4,7),(4,9),(5,11),(6,10),(6,11),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ([(2,5),(3,4),(3,10),(4,9),(5,11),(6,9),(6,10),(6,11),(7,8),(7,9),(7,11),(8,10),(8,11),(9,10)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(2,11),(3,6),(3,10),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,11),(7,9),(7,10),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ([(2,7),(3,6),(3,11),(4,8),(4,10),(4,11),(5,9),(5,10),(5,11),(6,8),(6,10),(7,9),(7,11),(8,9),(8,11),(9,10),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3}
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 0
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 0
000001 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ([(2,11),(3,10),(4,9),(4,10),(5,8),(5,11),(6,7),(6,9),(6,10),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
000010 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ([(2,10),(3,9),(3,14),(4,7),(4,9),(4,12),(4,14),(5,6),(5,10),(5,11),(5,13),(6,12),(6,13),(6,14),(7,11),(7,13),(7,14),(8,11),(8,12),(8,13),(8,14),(9,11),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
000011 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ([(2,10),(2,14),(3,9),(3,13),(4,6),(4,9),(4,12),(4,13),(5,7),(5,10),(5,11),(5,14),(6,11),(6,13),(6,14),(7,12),(7,13),(7,14),(8,11),(8,12),(8,13),(8,14),(9,11),(9,14),(10,12),(10,13),(11,12),(11,13),(12,14),(13,14)],15)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
000100 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ([(2,5),(3,4),(3,12),(3,14),(4,13),(4,15),(5,7),(5,15),(6,12),(6,13),(6,14),(6,15),(7,8),(7,10),(7,11),(8,11),(8,14),(8,15),(9,10),(9,11),(9,12),(9,14),(9,15),(10,13),(10,14),(10,15),(11,13),(11,15),(12,13),(12,15),(13,14),(14,15)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
000101 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ([(2,6),(3,4),(3,13),(3,15),(4,12),(4,14),(5,8),(5,11),(5,15),(6,11),(6,15),(7,12),(7,13),(7,14),(7,15),(8,10),(8,11),(8,13),(8,15),(9,10),(9,11),(9,13),(9,14),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(12,13),(12,15),(13,14),(14,15)],16)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
000110 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ([(2,4),(2,13),(3,5),(3,12),(3,16),(4,11),(4,16),(5,10),(5,14),(5,15),(6,10),(6,12),(6,14),(6,15),(6,16),(7,11),(7,13),(7,14),(7,15),(7,16),(8,9),(8,10),(8,13),(8,14),(8,15),(8,16),(9,11),(9,12),(9,14),(9,15),(9,16),(10,11),(10,12),(10,16),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,15),(13,16),(14,16),(15,16)],17)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 0
Description
The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. A graph is a split graph if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ is an edge and $(b,c)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
The following 23 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001656The monophonic position number of a graph. St000370The genus of a graph. St001095The number of non-isomorphic posets with precisely one further covering relation. St001871The number of triconnected components of a graph. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000322The skewness of a graph. St001305The number of induced cycles on four vertices in a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001271The competition number of a graph. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000741The Colin de Verdière graph invariant. St001964The interval resolution global dimension of a poset. St001625The Möbius invariant of a lattice. St001621The number of atoms of a lattice. St001812The biclique partition number of a graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.