Your data matches 108 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000773
Mapping
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000773: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => 1
[2] => ([],2)
 => ([],1)
 => 1
[1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 1
[3] => ([],3)
 => ([],1)
 => 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => 1
[2,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => 1
[4] => ([],4)
 => ([],1)
 => 1
[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)
 => 3
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[2,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 1
[5] => ([],5)
 => ([],1)
 => 1
[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)
 => 4
[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)
 => 3
[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)
 => 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[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)
 => 2
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,5] => ([(4,5)],6)
 => ([(1,2)],3)
 => 1
[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)
 => 3
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[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)
 => 1
[2,4] => ([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 1
[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)
 => 2
[3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 1
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 1
[6] => ([],6)
 => ([],1)
 => 1
[1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,6] => ([(5,6)],7)
 => ([(1,2)],3)
 => 1
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
Description
The multiplicity of the largest Laplacian eigenvalue in a graph.
Matching statistic: St000326
Mapping
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 => 1 => 1
[2] => [1,1] => 11 => 11 => 1
[1,2] => [2,1] => 101 => 101 => 1
[3] => [1,1,1] => 111 => 111 => 1
[1,1,2] => [3,1] => 1001 => 0101 => 2
[1,3] => [2,1,1] => 1011 => 1011 => 1
[2,2] => [1,2,1] => 1101 => 1101 => 1
[4] => [1,1,1,1] => 1111 => 1111 => 1
[1,1,1,2] => [4,1] => 10001 => 00101 => 3
[1,1,3] => [3,1,1] => 10011 => 01011 => 2
[1,2,2] => [2,2,1] => 10101 => 11001 => 1
[1,4] => [2,1,1,1] => 10111 => 10111 => 1
[2,1,2] => [1,3,1] => 11001 => 01101 => 2
[2,3] => [1,2,1,1] => 11011 => 11011 => 1
[3,2] => [1,1,2,1] => 11101 => 11101 => 1
[5] => [1,1,1,1,1] => 11111 => 11111 => 1
[1,1,1,1,2] => [5,1] => 100001 => 000101 => 4
[1,1,1,3] => [4,1,1] => 100011 => 001011 => 3
[1,1,2,2] => [3,2,1] => 100101 => 101001 => 1
[1,1,4] => [3,1,1,1] => 100111 => 010111 => 2
[1,2,1,2] => [2,3,1] => 101001 => 011001 => 2
[1,2,3] => [2,2,1,1] => 101011 => 110011 => 1
[1,3,2] => [2,1,2,1] => 101101 => 110101 => 1
[1,5] => [2,1,1,1,1] => 101111 => 101111 => 1
[2,1,1,2] => [1,4,1] => 110001 => 001101 => 3
[2,1,3] => [1,3,1,1] => 110011 => 011011 => 2
[2,2,2] => [1,2,2,1] => 110101 => 111001 => 1
[2,4] => [1,2,1,1,1] => 110111 => 110111 => 1
[3,1,2] => [1,1,3,1] => 111001 => 011101 => 2
[3,3] => [1,1,2,1,1] => 111011 => 111011 => 1
[4,2] => [1,1,1,2,1] => 111101 => 111101 => 1
[6] => [1,1,1,1,1,1] => 111111 => 111111 => 1
[1,1,1,1,1,2] => [6,1] => 1000001 => 0000101 => 5
[1,1,1,1,3] => [5,1,1] => 1000011 => 0001011 => 4
[1,1,1,2,2] => [4,2,1] => 1000101 => 1001001 => 1
[1,1,1,4] => [4,1,1,1] => 1000111 => 0010111 => 3
[1,1,2,1,2] => [3,3,1] => 1001001 => 0101001 => 2
[1,1,2,3] => [3,2,1,1] => 1001011 => 1010011 => 1
[1,1,3,2] => [3,1,2,1] => 1001101 => 1010101 => 1
[1,1,5] => [3,1,1,1,1] => 1001111 => 0101111 => 2
[1,2,1,1,2] => [2,4,1] => 1010001 => 0011001 => 3
[1,2,1,3] => [2,3,1,1] => 1010011 => 0110011 => 2
[1,2,2,2] => [2,2,2,1] => 1010101 => 1110001 => 1
[1,2,4] => [2,2,1,1,1] => 1010111 => 1100111 => 1
[1,3,1,2] => [2,1,3,1] => 1011001 => 0110101 => 2
[1,3,3] => [2,1,2,1,1] => 1011011 => 1101011 => 1
[1,4,2] => [2,1,1,2,1] => 1011101 => 1101101 => 1
[1,6] => [2,1,1,1,1,1] => 1011111 => 1011111 => 1
[2,1,1,1,2] => [1,5,1] => 1100001 => 0001101 => 4
[2,1,1,3] => [1,4,1,1] => 1100011 => 0011011 => 3
Description
The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000382
Mapping
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => [1] => 1
[2] => ([],2)
 => ([],1)
 => [1] => 1
[1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => [1,2] => 1
[3] => ([],3)
 => ([],1)
 => [1] => 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => [2,2] => 2
[1,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => [1,2] => 1
[2,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => [1,2] => 1
[4] => ([],4)
 => ([],1)
 => [1] => 1
[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)
 => [3,2] => 3
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [2,2] => 2
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => 1
[1,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => [1,2] => 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => [2,2] => 2
[2,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [1,2] => 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => [1,2] => 1
[5] => ([],5)
 => ([],1)
 => [1] => 1
[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)
 => [4,2] => 4
[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)
 => [3,2] => 3
[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)
 => [1,2,1,2] => 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [2,2] => 2
[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)
 => [2,1,1,2] => 2
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => 1
[1,5] => ([(4,5)],6)
 => ([(1,2)],3)
 => [1,2] => 1
[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)
 => [3,2] => 3
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => [2,2] => 2
[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)
 => [1,1,1,2] => 1
[2,4] => ([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [1,2] => 1
[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)
 => [2,2] => 2
[3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [1,2] => 1
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => [1,2] => 1
[6] => ([],6)
 => ([],1)
 => [1] => 1
[1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,2] => 5
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2] => 4
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,3,1,2] => 1
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => 3
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [2,2,1,2] => 2
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,1,2] => 1
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,1,2] => 1
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => [2,2] => 2
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [3,1,1,2] => 3
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,2] => 2
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,1,1,2] => 1
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => 1
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,2] => 2
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => 1
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => 1
[1,6] => ([(5,6)],7)
 => ([(1,2)],3)
 => [1,2] => 1
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2] => 4
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => 3
Description
The first part of an integer composition.
Matching statistic: St000772
(load all 6 compositions to match this statistic)
Mapping
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([],1)
 => ([],1)
 => 1
[2] => ([],2)
 => ([],1)
 => ([],1)
 => 1
[1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[3] => ([],3)
 => ([],1)
 => ([],1)
 => 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[2,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[4] => ([],4)
 => ([],1)
 => ([],1)
 => 1
[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)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[5] => ([],5)
 => ([],1)
 => ([],1)
 => 1
[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)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[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)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[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)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
[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)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,5] => ([(4,5)],6)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[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)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
[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)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[2,4] => ([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[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)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
[3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[6] => ([],6)
 => ([],1)
 => ([],1)
 => 1
[1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 5
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,6] => ([(5,6)],7)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
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: St000687
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000687: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1,0]
 => [1,0]
 => 0 = 1 - 1
[2] => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[1,2] => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => 4 = 5 - 1
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => 0 = 1 - 1
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
 => 2 = 3 - 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => 1 = 2 - 1
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => 3 = 4 - 1
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
Description
The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. In this expression, $I$ is the direct sum of all injective non-projective indecomposable modules and $P$ is the direct sum of all projective non-injective indecomposable modules. This statistic was discussed in [Theorem 5.7, 1].
Matching statistic: St000993
(load all 2 compositions to match this statistic)
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
 => [1]
 => []
 => ? = 1
[2] => [[2],[]]
 => [2]
 => []
 => ? = 1
[1,2] => [[2,1],[]]
 => [2,1]
 => [1]
 => ? = 1
[3] => [[3],[]]
 => [3]
 => []
 => ? = 1
[1,1,2] => [[2,1,1],[]]
 => [2,1,1]
 => [1,1]
 => 2
[1,3] => [[3,1],[]]
 => [3,1]
 => [1]
 => ? = 1
[2,2] => [[3,2],[1]]
 => [3,2]
 => [2]
 => 1
[4] => [[4],[]]
 => [4]
 => []
 => ? = 1
[1,1,1,2] => [[2,1,1,1],[]]
 => [2,1,1,1]
 => [1,1,1]
 => 3
[1,1,3] => [[3,1,1],[]]
 => [3,1,1]
 => [1,1]
 => 2
[1,2,2] => [[3,2,1],[1]]
 => [3,2,1]
 => [2,1]
 => 1
[1,4] => [[4,1],[]]
 => [4,1]
 => [1]
 => ? = 1
[2,1,2] => [[3,2,2],[1,1]]
 => [3,2,2]
 => [2,2]
 => 2
[2,3] => [[4,2],[1]]
 => [4,2]
 => [2]
 => 1
[3,2] => [[4,3],[2]]
 => [4,3]
 => [3]
 => 1
[5] => [[5],[]]
 => [5]
 => []
 => ? = 1
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 4
[1,1,1,3] => [[3,1,1,1],[]]
 => [3,1,1,1]
 => [1,1,1]
 => 3
[1,1,2,2] => [[3,2,1,1],[1]]
 => [3,2,1,1]
 => [2,1,1]
 => 1
[1,1,4] => [[4,1,1],[]]
 => [4,1,1]
 => [1,1]
 => 2
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [3,2,2,1]
 => [2,2,1]
 => 2
[1,2,3] => [[4,2,1],[1]]
 => [4,2,1]
 => [2,1]
 => 1
[1,3,2] => [[4,3,1],[2]]
 => [4,3,1]
 => [3,1]
 => 1
[1,5] => [[5,1],[]]
 => [5,1]
 => [1]
 => ? = 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [3,2,2,2]
 => [2,2,2]
 => 3
[2,1,3] => [[4,2,2],[1,1]]
 => [4,2,2]
 => [2,2]
 => 2
[2,2,2] => [[4,3,2],[2,1]]
 => [4,3,2]
 => [3,2]
 => 1
[2,4] => [[5,2],[1]]
 => [5,2]
 => [2]
 => 1
[3,1,2] => [[4,3,3],[2,2]]
 => [4,3,3]
 => [3,3]
 => 2
[3,3] => [[5,3],[2]]
 => [5,3]
 => [3]
 => 1
[4,2] => [[5,4],[3]]
 => [5,4]
 => [4]
 => 1
[6] => [[6],[]]
 => [6]
 => []
 => ? = 1
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 5
[1,1,1,1,3] => [[3,1,1,1,1],[]]
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 4
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [3,2,1,1,1]
 => [2,1,1,1]
 => 1
[1,1,1,4] => [[4,1,1,1],[]]
 => [4,1,1,1]
 => [1,1,1]
 => 3
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => [3,2,2,1,1]
 => [2,2,1,1]
 => 2
[1,1,2,3] => [[4,2,1,1],[1]]
 => [4,2,1,1]
 => [2,1,1]
 => 1
[1,1,3,2] => [[4,3,1,1],[2]]
 => [4,3,1,1]
 => [3,1,1]
 => 1
[1,1,5] => [[5,1,1],[]]
 => [5,1,1]
 => [1,1]
 => 2
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
 => [3,2,2,2,1]
 => [2,2,2,1]
 => 3
[1,2,1,3] => [[4,2,2,1],[1,1]]
 => [4,2,2,1]
 => [2,2,1]
 => 2
[1,2,2,2] => [[4,3,2,1],[2,1]]
 => [4,3,2,1]
 => [3,2,1]
 => 1
[1,2,4] => [[5,2,1],[1]]
 => [5,2,1]
 => [2,1]
 => 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
 => [4,3,3,1]
 => [3,3,1]
 => 2
[1,3,3] => [[5,3,1],[2]]
 => [5,3,1]
 => [3,1]
 => 1
[1,4,2] => [[5,4,1],[3]]
 => [5,4,1]
 => [4,1]
 => 1
[1,6] => [[6,1],[]]
 => [6,1]
 => [1]
 => ? = 1
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]]
 => [3,2,2,2,2]
 => [2,2,2,2]
 => 4
[2,1,1,3] => [[4,2,2,2],[1,1,1]]
 => [4,2,2,2]
 => [2,2,2]
 => 3
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
 => [4,3,2,2]
 => [3,2,2]
 => 1
[2,1,4] => [[5,2,2],[1,1]]
 => [5,2,2]
 => [2,2]
 => 2
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
 => [4,3,3,2]
 => [3,3,2]
 => 2
[2,2,3] => [[5,3,2],[2,1]]
 => [5,3,2]
 => [3,2]
 => 1
[2,3,2] => [[5,4,2],[3,1]]
 => [5,4,2]
 => [4,2]
 => 1
[2,5] => [[6,2],[1]]
 => [6,2]
 => [2]
 => 1
[3,1,1,2] => [[4,3,3,3],[2,2,2]]
 => [4,3,3,3]
 => [3,3,3]
 => 3
[3,1,3] => [[5,3,3],[2,2]]
 => [5,3,3]
 => [3,3]
 => 2
[3,2,2] => [[5,4,3],[3,2]]
 => [5,4,3]
 => [4,3]
 => 1
[3,4] => [[6,3],[2]]
 => [6,3]
 => [3]
 => 1
[4,1,2] => [[5,4,4],[3,3]]
 => [5,4,4]
 => [4,4]
 => 2
[7] => [[7],[]]
 => [7]
 => []
 => ? = 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000731
(load all 2 compositions to match this statistic)
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000731: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1] => [1] => 0 = 1 - 1
[2] => [1,1,0,0]
 => [1,2] => [1,2] => 0 = 1 - 1
[1,2] => [1,0,1,1,0,0]
 => [2,1,3] => [2,1,3] => 0 = 1 - 1
[3] => [1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [3,1,2,4] => 0 = 1 - 1
[4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,3,4,1,5] => 2 = 3 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,3,1,4,5] => 1 = 2 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [4,2,1,3,5] => 0 = 1 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => 0 = 1 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [3,1,4,2,5] => 1 = 2 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [3,1,2,4,5] => 0 = 1 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => [4,1,2,3,5] => 0 = 1 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => 0 = 1 - 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,5,1,6] => [2,3,4,5,1,6] => 3 = 4 - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,3,4,1,5,6] => [2,3,4,1,5,6] => 2 = 3 - 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,3,1,5,4,6] => [5,2,3,1,4,6] => 0 = 1 - 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,3,1,4,5,6] => [2,3,1,4,5,6] => 1 = 2 - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,1,4,5,3,6] => [4,2,1,5,3,6] => 1 = 2 - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [2,1,4,3,5,6] => [4,2,1,3,5,6] => 0 = 1 - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [2,1,3,5,4,6] => [5,2,1,3,4,6] => 0 = 1 - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 0 = 1 - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,3,4,5,2,6] => [3,1,4,5,2,6] => 2 = 3 - 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,3,4,2,5,6] => [3,1,4,2,5,6] => 1 = 2 - 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4,6] => [5,3,1,2,4,6] => 0 = 1 - 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,3,2,4,5,6] => [3,1,2,4,5,6] => 0 = 1 - 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,2,4,5,3,6] => [4,1,2,5,3,6] => 1 = 2 - 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,2,4,3,5,6] => [4,1,2,3,5,6] => 0 = 1 - 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,2,3,5,4,6] => [5,1,2,3,4,6] => 0 = 1 - 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,5,6,1,7] => [2,3,4,5,6,1,7] => 4 = 5 - 1
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,3,4,5,1,6,7] => [2,3,4,5,1,6,7] => 3 = 4 - 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,3,4,1,6,5,7] => [6,2,3,4,1,5,7] => ? = 1 - 1
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,1,5,6,7] => [2,3,4,1,5,6,7] => 2 = 3 - 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,3,1,5,6,4,7] => [5,2,3,1,6,4,7] => ? = 2 - 1
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [2,3,1,5,4,6,7] => [5,2,3,1,4,6,7] => ? = 1 - 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [2,3,1,4,6,5,7] => [6,2,3,1,4,5,7] => ? = 1 - 1
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,1,4,5,6,7] => [2,3,1,4,5,6,7] => 1 = 2 - 1
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,4,5,6,3,7] => [4,2,1,5,6,3,7] => ? = 3 - 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,1,4,5,3,6,7] => [4,2,1,5,3,6,7] => ? = 2 - 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,7] => [6,4,2,1,3,5,7] => ? = 1 - 1
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,4,3,5,6,7] => [4,2,1,3,5,6,7] => ? = 1 - 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,1,3,5,6,4,7] => [5,2,1,3,6,4,7] => ? = 2 - 1
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => [5,2,1,3,4,6,7] => ? = 1 - 1
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,1,3,4,6,5,7] => [6,2,1,3,4,5,7] => ? = 1 - 1
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => 0 = 1 - 1
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,3,4,5,6,2,7] => [3,1,4,5,6,2,7] => ? = 4 - 1
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,3,4,5,2,6,7] => [3,1,4,5,2,6,7] => ? = 3 - 1
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,3,4,2,6,5,7] => [6,3,1,4,2,5,7] => ? = 1 - 1
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,3,4,2,5,6,7] => [3,1,4,2,5,6,7] => ? = 2 - 1
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,3,2,5,6,4,7] => [5,3,1,2,6,4,7] => ? = 2 - 1
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,5,4,6,7] => [5,3,1,2,4,6,7] => ? = 1 - 1
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,3,2,4,6,5,7] => [6,3,1,2,4,5,7] => ? = 1 - 1
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,3,2,4,5,6,7] => [3,1,2,4,5,6,7] => 0 = 1 - 1
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,2,4,5,6,3,7] => [4,1,2,5,6,3,7] => ? = 3 - 1
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,2,4,5,3,6,7] => [4,1,2,5,3,6,7] => ? = 2 - 1
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5,7] => [6,4,1,2,3,5,7] => ? = 1 - 1
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => 0 = 1 - 1
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [1,2,3,5,6,4,7] => [5,1,2,3,6,4,7] => 1 = 2 - 1
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => 0 = 1 - 1
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => 0 = 1 - 1
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0 = 1 - 1
Description
The number of double exceedences of a permutation. A double exceedence is an index $\sigma(i)$ such that $i < \sigma(i) < \sigma(\sigma(i))$.
Matching statistic: St000297
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 80%
Values
[1] => [[1],[]]
 => []
 => => ? = 1
[2] => [[2],[]]
 => []
 => => ? = 1
[1,2] => [[2,1],[]]
 => []
 => => ? = 1
[3] => [[3],[]]
 => []
 => => ? = 1
[1,1,2] => [[2,1,1],[]]
 => []
 => => ? = 2
[1,3] => [[3,1],[]]
 => []
 => => ? = 1
[2,2] => [[3,2],[1]]
 => [1]
 => 10 => 1
[4] => [[4],[]]
 => []
 => => ? = 1
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => => ? = 3
[1,1,3] => [[3,1,1],[]]
 => []
 => => ? = 2
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => 10 => 1
[1,4] => [[4,1],[]]
 => []
 => => ? = 1
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 110 => 2
[2,3] => [[4,2],[1]]
 => [1]
 => 10 => 1
[3,2] => [[4,3],[2]]
 => [2]
 => 100 => 1
[5] => [[5],[]]
 => []
 => => ? = 1
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => => ? = 4
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => => ? = 3
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => 10 => 1
[1,1,4] => [[4,1,1],[]]
 => []
 => => ? = 2
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 110 => 2
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => 10 => 1
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => 100 => 1
[1,5] => [[5,1],[]]
 => []
 => => ? = 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1110 => 3
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 110 => 2
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 1010 => 1
[2,4] => [[5,2],[1]]
 => [1]
 => 10 => 1
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 1100 => 2
[3,3] => [[5,3],[2]]
 => [2]
 => 100 => 1
[4,2] => [[5,4],[3]]
 => [3]
 => 1000 => 1
[6] => [[6],[]]
 => []
 => => ? = 1
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
 => []
 => => ? = 5
[1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => => ? = 4
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => 10 => 1
[1,1,1,4] => [[4,1,1,1],[]]
 => []
 => => ? = 3
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => [1,1]
 => 110 => 2
[1,1,2,3] => [[4,2,1,1],[1]]
 => [1]
 => 10 => 1
[1,1,3,2] => [[4,3,1,1],[2]]
 => [2]
 => 100 => 1
[1,1,5] => [[5,1,1],[]]
 => []
 => => ? = 2
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 1110 => 3
[1,2,1,3] => [[4,2,2,1],[1,1]]
 => [1,1]
 => 110 => 2
[1,2,2,2] => [[4,3,2,1],[2,1]]
 => [2,1]
 => 1010 => 1
[1,2,4] => [[5,2,1],[1]]
 => [1]
 => 10 => 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
 => [2,2]
 => 1100 => 2
[1,3,3] => [[5,3,1],[2]]
 => [2]
 => 100 => 1
[1,4,2] => [[5,4,1],[3]]
 => [3]
 => 1000 => 1
[1,6] => [[6,1],[]]
 => []
 => => ? = 1
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 11110 => 4
[2,1,1,3] => [[4,2,2,2],[1,1,1]]
 => [1,1,1]
 => 1110 => 3
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
 => [2,1,1]
 => 10110 => 1
[2,1,4] => [[5,2,2],[1,1]]
 => [1,1]
 => 110 => 2
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
 => [2,2,1]
 => 11010 => 2
[2,2,3] => [[5,3,2],[2,1]]
 => [2,1]
 => 1010 => 1
[2,3,2] => [[5,4,2],[3,1]]
 => [3,1]
 => 10010 => 1
[2,5] => [[6,2],[1]]
 => [1]
 => 10 => 1
[3,1,1,2] => [[4,3,3,3],[2,2,2]]
 => [2,2,2]
 => 11100 => 3
[3,1,3] => [[5,3,3],[2,2]]
 => [2,2]
 => 1100 => 2
[3,2,2] => [[5,4,3],[3,2]]
 => [3,2]
 => 10100 => 1
[3,4] => [[6,3],[2]]
 => [2]
 => 100 => 1
[4,1,2] => [[5,4,4],[3,3]]
 => [3,3]
 => 11000 => 2
[4,3] => [[6,4],[3]]
 => [3]
 => 1000 => 1
[5,2] => [[6,5],[4]]
 => [4]
 => 10000 => 1
[7] => [[7],[]]
 => []
 => => ? = 1
Description
The number of leading ones in a binary word.
Matching statistic: St000733
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 80%
Values
[1] => [[1],[]]
 => []
 => []
 => ? = 1
[2] => [[2],[]]
 => []
 => []
 => ? = 1
[1,2] => [[2,1],[]]
 => []
 => []
 => ? = 1
[3] => [[3],[]]
 => []
 => []
 => ? = 1
[1,1,2] => [[2,1,1],[]]
 => []
 => []
 => ? = 2
[1,3] => [[3,1],[]]
 => []
 => []
 => ? = 1
[2,2] => [[3,2],[1]]
 => [1]
 => [[1]]
 => 1
[4] => [[4],[]]
 => []
 => []
 => ? = 1
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => []
 => ? = 3
[1,1,3] => [[3,1,1],[]]
 => []
 => []
 => ? = 2
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => [[1]]
 => 1
[1,4] => [[4,1],[]]
 => []
 => []
 => ? = 1
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [[1],[2]]
 => 2
[2,3] => [[4,2],[1]]
 => [1]
 => [[1]]
 => 1
[3,2] => [[4,3],[2]]
 => [2]
 => [[1,2]]
 => 1
[5] => [[5],[]]
 => []
 => []
 => ? = 1
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => []
 => ? = 4
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => []
 => ? = 3
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => [[1]]
 => 1
[1,1,4] => [[4,1,1],[]]
 => []
 => []
 => ? = 2
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [[1],[2]]
 => 2
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => [[1]]
 => 1
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => [[1,2]]
 => 1
[1,5] => [[5,1],[]]
 => []
 => []
 => ? = 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [[1],[2],[3]]
 => 3
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [[1],[2]]
 => 2
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [[1,3],[2]]
 => 1
[2,4] => [[5,2],[1]]
 => [1]
 => [[1]]
 => 1
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
[3,3] => [[5,3],[2]]
 => [2]
 => [[1,2]]
 => 1
[4,2] => [[5,4],[3]]
 => [3]
 => [[1,2,3]]
 => 1
[6] => [[6],[]]
 => []
 => []
 => ? = 1
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
 => []
 => []
 => ? = 5
[1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => []
 => ? = 4
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => [[1]]
 => 1
[1,1,1,4] => [[4,1,1,1],[]]
 => []
 => []
 => ? = 3
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => [1,1]
 => [[1],[2]]
 => 2
[1,1,2,3] => [[4,2,1,1],[1]]
 => [1]
 => [[1]]
 => 1
[1,1,3,2] => [[4,3,1,1],[2]]
 => [2]
 => [[1,2]]
 => 1
[1,1,5] => [[5,1,1],[]]
 => []
 => []
 => ? = 2
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [[1],[2],[3]]
 => 3
[1,2,1,3] => [[4,2,2,1],[1,1]]
 => [1,1]
 => [[1],[2]]
 => 2
[1,2,2,2] => [[4,3,2,1],[2,1]]
 => [2,1]
 => [[1,3],[2]]
 => 1
[1,2,4] => [[5,2,1],[1]]
 => [1]
 => [[1]]
 => 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
[1,3,3] => [[5,3,1],[2]]
 => [2]
 => [[1,2]]
 => 1
[1,4,2] => [[5,4,1],[3]]
 => [3]
 => [[1,2,3]]
 => 1
[1,6] => [[6,1],[]]
 => []
 => []
 => ? = 1
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 4
[2,1,1,3] => [[4,2,2,2],[1,1,1]]
 => [1,1,1]
 => [[1],[2],[3]]
 => 3
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 1
[2,1,4] => [[5,2,2],[1,1]]
 => [1,1]
 => [[1],[2]]
 => 2
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2
[2,2,3] => [[5,3,2],[2,1]]
 => [2,1]
 => [[1,3],[2]]
 => 1
[2,3,2] => [[5,4,2],[3,1]]
 => [3,1]
 => [[1,3,4],[2]]
 => 1
[2,5] => [[6,2],[1]]
 => [1]
 => [[1]]
 => 1
[3,1,1,2] => [[4,3,3,3],[2,2,2]]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => 3
[3,1,3] => [[5,3,3],[2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
[3,2,2] => [[5,4,3],[3,2]]
 => [3,2]
 => [[1,2,5],[3,4]]
 => 1
[3,4] => [[6,3],[2]]
 => [2]
 => [[1,2]]
 => 1
[4,1,2] => [[5,4,4],[3,3]]
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => 2
[4,3] => [[6,4],[3]]
 => [3]
 => [[1,2,3]]
 => 1
[5,2] => [[6,5],[4]]
 => [4]
 => [[1,2,3,4]]
 => 1
[7] => [[7],[]]
 => []
 => []
 => ? = 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St001264
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001264: Dyck paths ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 80%
Values
[1] => [[1],[]]
 => []
 => []
 => ? = 1 - 1
[2] => [[2],[]]
 => []
 => []
 => ? = 1 - 1
[1,2] => [[2,1],[]]
 => []
 => []
 => ? = 1 - 1
[3] => [[3],[]]
 => []
 => []
 => ? = 1 - 1
[1,1,2] => [[2,1,1],[]]
 => []
 => []
 => ? = 2 - 1
[1,3] => [[3,1],[]]
 => []
 => []
 => ? = 1 - 1
[2,2] => [[3,2],[1]]
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
[4] => [[4],[]]
 => []
 => []
 => ? = 1 - 1
[1,1,1,2] => [[2,1,1,1],[]]
 => []
 => []
 => ? = 3 - 1
[1,1,3] => [[3,1,1],[]]
 => []
 => []
 => ? = 2 - 1
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
[1,4] => [[4,1],[]]
 => []
 => []
 => ? = 1 - 1
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[2,3] => [[4,2],[1]]
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
[3,2] => [[4,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[5] => [[5],[]]
 => []
 => []
 => ? = 1 - 1
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => []
 => ? = 4 - 1
[1,1,1,3] => [[3,1,1,1],[]]
 => []
 => []
 => ? = 3 - 1
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
[1,1,4] => [[4,1,1],[]]
 => []
 => []
 => ? = 2 - 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[1,5] => [[5,1],[]]
 => []
 => []
 => ? = 1 - 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[2,4] => [[5,2],[1]]
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[3,3] => [[5,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[4,2] => [[5,4],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[6] => [[6],[]]
 => []
 => []
 => ? = 1 - 1
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
 => []
 => []
 => ? = 5 - 1
[1,1,1,1,3] => [[3,1,1,1,1],[]]
 => []
 => []
 => ? = 4 - 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
[1,1,1,4] => [[4,1,1,1],[]]
 => []
 => []
 => ? = 3 - 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[1,1,2,3] => [[4,2,1,1],[1]]
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
[1,1,3,2] => [[4,3,1,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1,5] => [[5,1,1],[]]
 => []
 => []
 => ? = 2 - 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[1,2,1,3] => [[4,2,2,1],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[1,2,2,2] => [[4,3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[1,2,4] => [[5,2,1],[1]]
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[1,3,3] => [[5,3,1],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[1,4,2] => [[5,4,1],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[1,6] => [[6,1],[]]
 => []
 => []
 => ? = 1 - 1
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
[2,1,1,3] => [[4,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0 = 1 - 1
[2,1,4] => [[5,2,2],[1,1]]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[2,2,3] => [[5,3,2],[2,1]]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[2,3,2] => [[5,4,2],[3,1]]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[2,5] => [[6,2],[1]]
 => [1]
 => [1,0,1,0]
 => 0 = 1 - 1
[3,1,1,2] => [[4,3,3,3],[2,2,2]]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[3,1,3] => [[5,3,3],[2,2]]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[3,2,2] => [[5,4,3],[3,2]]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[3,4] => [[6,3],[2]]
 => [2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[4,1,2] => [[5,4,4],[3,3]]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 2 - 1
[4,3] => [[6,4],[3]]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[5,2] => [[6,5],[4]]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0 = 1 - 1
[7] => [[7],[]]
 => []
 => []
 => ? = 1 - 1
Description
The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra.
The following 98 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000732The number of double deficiencies of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St001130The number of two successive successions in a permutation. St000366The number of double descents of a permutation. St001118The acyclic chromatic index of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001549The number of restricted non-inversions between exceedances. St001845The number of join irreducibles minus the rank of a lattice. St000359The number of occurrences of the pattern 23-1. St001271The competition number of a graph. St000648The number of 2-excedences of a permutation. St001330The hat guessing number of a graph. St001651The Frankl number of a lattice. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St000365The number of double ascents of a permutation. St000485The length of the longest cycle of a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001846The number of elements which do not have a complement in the lattice. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001811The Castelnuovo-Mumford regularity of a permutation. St001948The number of augmented double ascents of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001625The Möbius invariant of a lattice. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001866The nesting alignments of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001875The number of simple modules with projective dimension at most 1. St000068The number of minimal elements in a poset. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000905The number of different multiplicities of parts of an integer composition. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St000488The number of cycles of a permutation of length at most 2. St000664The number of right ropes of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001822The number of alignments of a signed permutation. St001856The number of edges in the reduced word graph of a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000456The monochromatic index of a connected graph. St000633The size of the automorphism group of a poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001399The distinguishing number of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001488The number of corners of a skew partition. St001472The permanent of the Coxeter matrix of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset.