- FindStat

Your data matches 168 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000392
(load all 7 compositions to match this statistic)

Mapping

Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 1 => 1
[2] => [2] => 10 => 1
[1,2] => [2,1] => 101 => 1
[3] => [3] => 100 => 1
[1,1,2] => [1,2,1] => 1101 => 2
[1,3] => [3,1] => 1001 => 1
[2,2] => [2,2] => 1010 => 1
[4] => [4] => 1000 => 1
[1,1,1,2] => [1,1,2,1] => 11101 => 3
[1,1,3] => [1,3,1] => 11001 => 2
[1,2,2] => [2,2,1] => 10101 => 1
[1,4] => [4,1] => 10001 => 1
[2,1,2] => [1,2,2] => 11010 => 2
[2,3] => [3,2] => 10010 => 1
[3,2] => [2,3] => 10100 => 1
[5] => [5] => 10000 => 1
[1,1,1,1,2] => [1,1,1,2,1] => 111101 => 4
[1,1,1,3] => [1,1,3,1] => 111001 => 3
[1,1,2,2] => [1,2,2,1] => 110101 => 2
[1,1,4] => [1,4,1] => 110001 => 2
[1,2,1,2] => [2,1,2,1] => 101101 => 2
[1,2,3] => [2,3,1] => 101001 => 1
[1,3,2] => [3,2,1] => 100101 => 1
[1,5] => [5,1] => 100001 => 1
[2,1,1,2] => [1,1,2,2] => 111010 => 3
[2,1,3] => [1,3,2] => 110010 => 2
[2,2,2] => [2,2,2] => 101010 => 1
[2,4] => [4,2] => 100010 => 1
[3,1,2] => [1,2,3] => 110100 => 2
[3,3] => [3,3] => 100100 => 1
[4,2] => [2,4] => 101000 => 1
[6] => [6] => 100000 => 1
[1,1,1,1,1,2] => [1,1,1,1,2,1] => 1111101 => 5
[1,1,1,1,3] => [1,1,1,3,1] => 1111001 => 4
[1,1,1,2,2] => [1,1,2,2,1] => 1110101 => 3
[1,1,1,4] => [1,1,4,1] => 1110001 => 3
[1,1,2,1,2] => [1,2,1,2,1] => 1101101 => 2
[1,1,2,3] => [1,2,3,1] => 1101001 => 2
[1,1,3,2] => [1,3,2,1] => 1100101 => 2
[1,1,5] => [1,5,1] => 1100001 => 2
[1,2,1,1,2] => [2,1,1,2,1] => 1011101 => 3
[1,2,1,3] => [2,1,3,1] => 1011001 => 2
[1,2,2,2] => [2,2,2,1] => 1010101 => 1
[1,2,4] => [2,4,1] => 1010001 => 1
[1,3,1,2] => [3,1,2,1] => 1001101 => 2
[1,3,3] => [3,3,1] => 1001001 => 1
[1,4,2] => [4,2,1] => 1000101 => 1
[1,6] => [6,1] => 1000001 => 1
[2,1,1,1,2] => [1,1,1,2,2] => 1111010 => 4
[2,1,1,3] => [1,1,3,2] => 1110010 => 3

Description

The length of the longest run of ones in a binary word.

Matching statistic: St001933
(load all 3 compositions to match this statistic)

Mapping

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [[1],[]]
 => [1]
 => 1
[2] => [[2],[]]
 => [2]
 => 1
[1,2] => [[2,1],[]]
 => [2,1]
 => 1
[3] => [[3],[]]
 => [3]
 => 1
[1,1,2] => [[2,1,1],[]]
 => [2,1,1]
 => 2
[1,3] => [[3,1],[]]
 => [3,1]
 => 1
[2,2] => [[3,2],[1]]
 => [3,2]
 => 1
[4] => [[4],[]]
 => [4]
 => 1
[1,1,1,2] => [[2,1,1,1],[]]
 => [2,1,1,1]
 => 3
[1,1,3] => [[3,1,1],[]]
 => [3,1,1]
 => 2
[1,2,2] => [[3,2,1],[1]]
 => [3,2,1]
 => 1
[1,4] => [[4,1],[]]
 => [4,1]
 => 1
[2,1,2] => [[3,2,2],[1,1]]
 => [3,2,2]
 => 2
[2,3] => [[4,2],[1]]
 => [4,2]
 => 1
[3,2] => [[4,3],[2]]
 => [4,3]
 => 1
[5] => [[5],[]]
 => [5]
 => 1
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => [2,1,1,1,1]
 => 4
[1,1,1,3] => [[3,1,1,1],[]]
 => [3,1,1,1]
 => 3
[1,1,2,2] => [[3,2,1,1],[1]]
 => [3,2,1,1]
 => 2
[1,1,4] => [[4,1,1],[]]
 => [4,1,1]
 => 2
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [3,2,2,1]
 => 2
[1,2,3] => [[4,2,1],[1]]
 => [4,2,1]
 => 1
[1,3,2] => [[4,3,1],[2]]
 => [4,3,1]
 => 1
[1,5] => [[5,1],[]]
 => [5,1]
 => 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [3,2,2,2]
 => 3
[2,1,3] => [[4,2,2],[1,1]]
 => [4,2,2]
 => 2
[2,2,2] => [[4,3,2],[2,1]]
 => [4,3,2]
 => 1
[2,4] => [[5,2],[1]]
 => [5,2]
 => 1
[3,1,2] => [[4,3,3],[2,2]]
 => [4,3,3]
 => 2
[3,3] => [[5,3],[2]]
 => [5,3]
 => 1
[4,2] => [[5,4],[3]]
 => [5,4]
 => 1
[6] => [[6],[]]
 => [6]
 => 1
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
 => [2,1,1,1,1,1]
 => 5
[1,1,1,1,3] => [[3,1,1,1,1],[]]
 => [3,1,1,1,1]
 => 4
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [3,2,1,1,1]
 => 3
[1,1,1,4] => [[4,1,1,1],[]]
 => [4,1,1,1]
 => 3
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => [3,2,2,1,1]
 => 2
[1,1,2,3] => [[4,2,1,1],[1]]
 => [4,2,1,1]
 => 2
[1,1,3,2] => [[4,3,1,1],[2]]
 => [4,3,1,1]
 => 2
[1,1,5] => [[5,1,1],[]]
 => [5,1,1]
 => 2
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
 => [3,2,2,2,1]
 => 3
[1,2,1,3] => [[4,2,2,1],[1,1]]
 => [4,2,2,1]
 => 2
[1,2,2,2] => [[4,3,2,1],[2,1]]
 => [4,3,2,1]
 => 1
[1,2,4] => [[5,2,1],[1]]
 => [5,2,1]
 => 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
 => [4,3,3,1]
 => 2
[1,3,3] => [[5,3,1],[2]]
 => [5,3,1]
 => 1
[1,4,2] => [[5,4,1],[3]]
 => [5,4,1]
 => 1
[1,6] => [[6,1],[]]
 => [6,1]
 => 1
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]]
 => [3,2,2,2,2]
 => 4
[2,1,1,3] => [[4,2,2,2],[1,1,1]]
 => [4,2,2,2]
 => 3

Description

The largest multiplicity of a part in an integer partition.

Matching statistic: St000969
(load all 3 compositions to match this statistic)

Mapping

Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000969: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1,0]
 => 2 = 1 + 1
[2] => [2] => [1,1,0,0]
 => 2 = 1 + 1
[1,2] => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[3] => [3] => [1,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[4] => [4] => [1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,2] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,1,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,2,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,4] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,2,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,3,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[2,1,1,2] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[2,1,3] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[2,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[3,1,2] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,1,2] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[1,1,1,1,3] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,1,1,2,2] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,1,4] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,1,2,1,2] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,2,3] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,3,2] => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,1,5] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,2,1,1,2] => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,2,1,3] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,2,2,2] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,2,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,3,1,2] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,3,3] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,4,2] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,6] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[2,1,1,1,2] => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[2,1,1,3] => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => 4 = 3 + 1

Description

We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n-1}]$ by adding

$c_0$ to

$c_{n-1}$ . Then we calculate the global dimension of that CNakayama algebra.

Matching statistic: St000723
(load all 4 compositions to match this statistic)

Mapping

Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000723: 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] => ([(0,1)],2)
 => 1
[1,2] => [2,1] => [1,2] => ([(1,2)],3)
 => 1
[3] => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,1,2] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[1,3] => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[2,2] => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4] => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,1,2] => [1,1,2,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,1,3] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,2,2] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,4] => [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[2,1,2] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[2,3] => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[3,2] => [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[5] => [5] => [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)
 => 1
[1,1,1,1,2] => [1,1,1,2,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[1,1,1,3] => [1,1,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,1,2,2] => [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,1,4] => [1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,2,1,2] => [2,1,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[1,2,3] => [2,3,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,3,2] => [3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,5] => [5,1] => [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
[2,1,1,2] => [1,1,2,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[2,1,3] => [1,3,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[2,2,2] => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[2,4] => [4,2] => [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
[3,1,2] => [1,2,3] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[3,3] => [3,3] => [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
[4,2] => [2,4] => [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
[6] => [6] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,1,1,1,2] => [1,1,1,1,2,1] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 5
[1,1,1,1,3] => [1,1,1,3,1] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 4
[1,1,1,2,2] => [1,1,2,2,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
[1,1,1,4] => [1,1,4,1] => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
[1,1,2,1,2] => [1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[1,1,2,3] => [1,2,3,1] => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[1,1,3,2] => [1,3,2,1] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[1,1,5] => [1,5,1] => [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)
 => 2
[1,2,1,1,2] => [2,1,1,2,1] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => 3
[1,2,1,3] => [2,1,3,1] => [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)
 => 2
[1,2,2,2] => [2,2,2,1] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,2,4] => [2,4,1] => [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
[1,3,1,2] => [3,1,2,1] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 2
[1,3,3] => [3,3,1] => [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
[1,4,2] => [4,2,1] => [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
[1,6] => [6,1] => [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
[2,1,1,1,2] => [1,1,1,2,2] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 4
[2,1,1,3] => [1,1,3,2] => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3

Description

The maximal cardinality of a set of vertices with the same neighbourhood in a graph. The set of so called mating graphs, for which this statistic equals

$1$ , is enumerated by [1].

Matching statistic: St000771
(load all 8 compositions to match this statistic)

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000771: 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)
 => 2
[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)
 => 3
[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)
 => 2
[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)
 => 2
[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 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

$1$ .

Matching statistic: St000774

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000774: 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)
 => 2
[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)
 => 3
[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)
 => 2
[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)
 => 2
[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 maximal multiplicity of a Laplacian eigenvalue in a graph.

Matching statistic: St001372

Mapping

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [[1],[]]
 => [1]
 => 10 => 1
[2] => [[2],[]]
 => [2]
 => 100 => 1
[1,2] => [[2,1],[]]
 => [2,1]
 => 1010 => 1
[3] => [[3],[]]
 => [3]
 => 1000 => 1
[1,1,2] => [[2,1,1],[]]
 => [2,1,1]
 => 10110 => 2
[1,3] => [[3,1],[]]
 => [3,1]
 => 10010 => 1
[2,2] => [[3,2],[1]]
 => [3,2]
 => 10100 => 1
[4] => [[4],[]]
 => [4]
 => 10000 => 1
[1,1,1,2] => [[2,1,1,1],[]]
 => [2,1,1,1]
 => 101110 => 3
[1,1,3] => [[3,1,1],[]]
 => [3,1,1]
 => 100110 => 2
[1,2,2] => [[3,2,1],[1]]
 => [3,2,1]
 => 101010 => 1
[1,4] => [[4,1],[]]
 => [4,1]
 => 100010 => 1
[2,1,2] => [[3,2,2],[1,1]]
 => [3,2,2]
 => 101100 => 2
[2,3] => [[4,2],[1]]
 => [4,2]
 => 100100 => 1
[3,2] => [[4,3],[2]]
 => [4,3]
 => 101000 => 1
[5] => [[5],[]]
 => [5]
 => 100000 => 1
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => [2,1,1,1,1]
 => 1011110 => 4
[1,1,1,3] => [[3,1,1,1],[]]
 => [3,1,1,1]
 => 1001110 => 3
[1,1,2,2] => [[3,2,1,1],[1]]
 => [3,2,1,1]
 => 1010110 => 2
[1,1,4] => [[4,1,1],[]]
 => [4,1,1]
 => 1000110 => 2
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [3,2,2,1]
 => 1011010 => 2
[1,2,3] => [[4,2,1],[1]]
 => [4,2,1]
 => 1001010 => 1
[1,3,2] => [[4,3,1],[2]]
 => [4,3,1]
 => 1010010 => 1
[1,5] => [[5,1],[]]
 => [5,1]
 => 1000010 => 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [3,2,2,2]
 => 1011100 => 3
[2,1,3] => [[4,2,2],[1,1]]
 => [4,2,2]
 => 1001100 => 2
[2,2,2] => [[4,3,2],[2,1]]
 => [4,3,2]
 => 1010100 => 1
[2,4] => [[5,2],[1]]
 => [5,2]
 => 1000100 => 1
[3,1,2] => [[4,3,3],[2,2]]
 => [4,3,3]
 => 1011000 => 2
[3,3] => [[5,3],[2]]
 => [5,3]
 => 1001000 => 1
[4,2] => [[5,4],[3]]
 => [5,4]
 => 1010000 => 1
[6] => [[6],[]]
 => [6]
 => 1000000 => 1
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
 => [2,1,1,1,1,1]
 => 10111110 => 5
[1,1,1,1,3] => [[3,1,1,1,1],[]]
 => [3,1,1,1,1]
 => 10011110 => 4
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [3,2,1,1,1]
 => 10101110 => 3
[1,1,1,4] => [[4,1,1,1],[]]
 => [4,1,1,1]
 => 10001110 => 3
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => [3,2,2,1,1]
 => 10110110 => 2
[1,1,2,3] => [[4,2,1,1],[1]]
 => [4,2,1,1]
 => 10010110 => 2
[1,1,3,2] => [[4,3,1,1],[2]]
 => [4,3,1,1]
 => 10100110 => 2
[1,1,5] => [[5,1,1],[]]
 => [5,1,1]
 => 10000110 => 2
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
 => [3,2,2,2,1]
 => 10111010 => 3
[1,2,1,3] => [[4,2,2,1],[1,1]]
 => [4,2,2,1]
 => 10011010 => 2
[1,2,2,2] => [[4,3,2,1],[2,1]]
 => [4,3,2,1]
 => 10101010 => 1
[1,2,4] => [[5,2,1],[1]]
 => [5,2,1]
 => 10001010 => 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
 => [4,3,3,1]
 => 10110010 => 2
[1,3,3] => [[5,3,1],[2]]
 => [5,3,1]
 => 10010010 => 1
[1,4,2] => [[5,4,1],[3]]
 => [5,4,1]
 => 10100010 => 1
[1,6] => [[6,1],[]]
 => [6,1]
 => 10000010 => 1
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]]
 => [3,2,2,2,2]
 => 10111100 => 4
[2,1,1,3] => [[4,2,2,2],[1,1,1]]
 => [4,2,2,2]
 => 10011100 => 3

Description

The length of a longest cyclic run of ones of a binary word. Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.

Matching statistic: St000688

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000688: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => []
 => []
 => ? = 1
[2] => [1,1,0,0]
 => []
 => []
 => ? = 1
[1,2] => [1,0,1,1,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[3] => [1,1,1,0,0,0]
 => []
 => []
 => ? = 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1
[4] => [1,1,1,1,0,0,0,0]
 => []
 => []
 => ? = 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? = 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [4,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 3
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 2
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [4,4,3]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 2
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => ? = 1
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 5
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => 4
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => 3
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,3,3,3,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => 3
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,5,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => 2
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [5,5,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => 2
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => 2
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => 3
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => 2
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => 1
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [3,3,3,3,1,1]
 => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [5,5,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => 2
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [4,4,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => 1
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => 1
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,2]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => 4
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,2]
 => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => 3
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,2]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => 2
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [3,3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => 2
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [5,5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => 2
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [4,4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => 1
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [5,5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => 1
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => []
 => []
 => ? = 1

Description

The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. The global dimension is given by [[St000684]] and the dominant dimension is given by [[St000685]]. To every Dyck path there is an LNakayama algebra associated as described in [[St000684]]. Dyck paths for which the global dimension and the dominant dimension of the the LNakayama algebra coincide and both dimensions at least

$2$ correspond to the LNakayama algebras that are higher Auslander algebras in the sense of [1].

Matching statistic: St001026

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001026: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => []
 => []
 => ? = 1
[2] => [1,1,0,0]
 => []
 => []
 => ? = 1
[1,2] => [1,0,1,1,0,0]
 => [1,1]
 => [1,0,1,1,0,0]
 => 1
[3] => [1,1,1,0,0,0]
 => []
 => []
 => ? = 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 1
[4] => [1,1,1,1,0,0,0,0]
 => []
 => []
 => ? = 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => []
 => ? = 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [4,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 3
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 2
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [4,4,3]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 2
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => []
 => []
 => ? = 1
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => 5
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => 4
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => 3
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,3,3,3,2,1]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => 3
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,5,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => 2
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => 2
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [5,5,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => 2
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => 2
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => 3
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => 2
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => 1
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [3,3,3,3,1,1]
 => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [5,5,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => 2
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [4,4,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => 1
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [5,5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => 1
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,2]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => 4
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,2]
 => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => 3
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,2]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => 2
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [3,3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => 2
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [5,5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => 2
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [4,4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => 1
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [5,5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => 1
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [2,2,2,2,2]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => []
 => []
 => ? = 1

Description

The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001118

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001118: Graphs ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => [1] => ([],1)
 => ? = 1
[2] => [1,1,0,0]
 => [1,2] => ([],2)
 => ? = 1
[1,2] => [1,0,1,1,0,0]
 => [2,1,3] => ([(1,2)],3)
 => 1
[3] => [1,1,1,0,0,0]
 => [1,2,3] => ([],3)
 => ? = 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 2
[1,3] => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => ([(2,3)],4)
 => 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => ([(2,3)],4)
 => 1
[4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([],4)
 => ? = 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([],5)
 => ? = 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6)
 => 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,3,1,5,4,6] => ([(1,2),(3,5),(4,5)],6)
 => 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
 => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,1,4,5,3,6] => ([(1,2),(3,5),(4,5)],6)
 => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
 => 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
 => 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => ([(4,5)],6)
 => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
 => 3
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
 => 2
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
 => 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,3,2,4,5,6] => ([(4,5)],6)
 => 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
 => 2
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,2,4,3,5,6] => ([(4,5)],6)
 => 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,2,3,5,4,6] => ([(4,5)],6)
 => 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([],6)
 => ? = 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] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 5
[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] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 4
[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] => ([(1,2),(3,6),(4,6),(5,6)],7)
 => 3
[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] => ([(3,6),(4,6),(5,6)],7)
 => 3
[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] => ([(1,6),(2,6),(3,5),(4,5)],7)
 => 2
[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] => ([(2,3),(4,6),(5,6)],7)
 => 2
[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] => ([(2,3),(4,6),(5,6)],7)
 => 2
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,1,4,5,6,7] => ([(4,6),(5,6)],7)
 => 2
[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] => ([(1,2),(3,6),(4,6),(5,6)],7)
 => 3
[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] => ([(2,3),(4,6),(5,6)],7)
 => 2
[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] => ([(1,6),(2,5),(3,4)],7)
 => 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] => ([(3,6),(4,5)],7)
 => 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] => ([(2,3),(4,6),(5,6)],7)
 => 2
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => ([(3,6),(4,5)],7)
 => 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] => ([(3,6),(4,5)],7)
 => 1
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => ([(5,6)],7)
 => 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] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 4
[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] => ([(3,6),(4,6),(5,6)],7)
 => 3
[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] => ([(2,3),(4,6),(5,6)],7)
 => 2
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,3,4,2,5,6,7] => ([(4,6),(5,6)],7)
 => 2
[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] => ([(2,3),(4,6),(5,6)],7)
 => 2
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,5,4,6,7] => ([(3,6),(4,5)],7)
 => 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] => ([(3,6),(4,5)],7)
 => 1
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,3,2,4,5,6,7] => ([(5,6)],7)
 => 1
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7] => ([],7)
 => ? = 1

Description

The acyclic chromatic index of a graph. An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest. The smallest number of colours such that such a colouring exists is the acyclic chromatic index.

The following 158 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000731The number of double exceedences of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000358The number of occurrences of the pattern 31-2. St001411The number of patterns 321 or 3412 in a permutation. St001727The number of invisible inversions of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001532The leading coefficient of the Poincare polynomial of the poset cone. St000065The number of entries equal to -1 in an alternating sign matrix. St000538The number of even inversions of a permutation. St000732The number of double deficiencies of a permutation. St000836The number of descents of distance 2 of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St000214The number of adjacencies of a permutation. St000648The number of 2-excedences of a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001596The number of two-by-two squares inside a skew partition. St000359The number of occurrences of the pattern 23-1. St000441The number of successions of a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000711The number of big exceedences of a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000356The number of occurrences of the pattern 13-2. St001083The number of boxed occurrences of 132 in a permutation. St001172The number of 1-rises at odd height of a Dyck path. St000237The number of small exceedances. 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. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St000039The number of crossings of a permutation. 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000217The number of occurrences of the pattern 312 in a permutation. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000444The length of the maximal rise of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001549The number of restricted non-inversions between exceedances. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000837The number of ascents of distance 2 of a permutation. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. 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. St000215The number of adjacencies of a permutation, zero appended. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001530The depth of a Dyck path. St001845The number of join irreducibles minus the rank of a lattice. St000054The first entry of the permutation. St001271The competition number of a graph. St001330The hat guessing number of a graph. St001651The Frankl number of a lattice. St000455The second largest eigenvalue of a graph if it is integral. 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. 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. St000485The length of the longest cycle of a permutation. 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. 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. 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. St000772The multiplicity of the largest 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

$eA$ . St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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. 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.