- FindStat

Your data matches 108 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000765
(load all 2 compositions to match this statistic)

Mapping

St000765: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of weak records in an integer composition. A weak record is an element

$a_i$ such that

$a_i \geq a_j$ for all

$j < i$ .

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

Mapping

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

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

Mapping

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 = 2 - 1
[1,1] => 11 => 2 = 3 - 1
[2] => 10 => 1 = 2 - 1
[1,1,1] => 111 => 3 = 4 - 1
[1,2] => 110 => 2 = 3 - 1
[2,1] => 101 => 1 = 2 - 1
[3] => 100 => 1 = 2 - 1
[1,1,1,1] => 1111 => 4 = 5 - 1
[1,1,2] => 1110 => 3 = 4 - 1
[1,2,1] => 1101 => 2 = 3 - 1
[1,3] => 1100 => 2 = 3 - 1
[2,1,1] => 1011 => 2 = 3 - 1
[2,2] => 1010 => 1 = 2 - 1
[3,1] => 1001 => 1 = 2 - 1
[4] => 1000 => 1 = 2 - 1
[1,1,1,1,1] => 11111 => 5 = 6 - 1
[1,1,1,2] => 11110 => 4 = 5 - 1
[1,1,2,1] => 11101 => 3 = 4 - 1
[1,1,3] => 11100 => 3 = 4 - 1
[1,2,1,1] => 11011 => 2 = 3 - 1
[1,2,2] => 11010 => 2 = 3 - 1
[1,3,1] => 11001 => 2 = 3 - 1
[1,4] => 11000 => 2 = 3 - 1
[2,1,1,1] => 10111 => 3 = 4 - 1
[2,1,2] => 10110 => 2 = 3 - 1
[2,2,1] => 10101 => 1 = 2 - 1
[2,3] => 10100 => 1 = 2 - 1
[3,1,1] => 10011 => 2 = 3 - 1
[3,2] => 10010 => 1 = 2 - 1
[4,1] => 10001 => 1 = 2 - 1
[5] => 10000 => 1 = 2 - 1
[1,1,1,1,1,1] => 111111 => 6 = 7 - 1
[1,1,1,1,2] => 111110 => 5 = 6 - 1
[1,1,1,2,1] => 111101 => 4 = 5 - 1
[1,1,1,3] => 111100 => 4 = 5 - 1
[1,1,2,1,1] => 111011 => 3 = 4 - 1
[1,1,2,2] => 111010 => 3 = 4 - 1
[1,1,3,1] => 111001 => 3 = 4 - 1
[1,1,4] => 111000 => 3 = 4 - 1
[1,2,1,1,1] => 110111 => 3 = 4 - 1
[1,2,1,2] => 110110 => 2 = 3 - 1
[1,2,2,1] => 110101 => 2 = 3 - 1
[1,2,3] => 110100 => 2 = 3 - 1
[1,3,1,1] => 110011 => 2 = 3 - 1
[1,3,2] => 110010 => 2 = 3 - 1
[1,4,1] => 110001 => 2 = 3 - 1
[1,5] => 110000 => 2 = 3 - 1
[2,1,1,1,1] => 101111 => 4 = 5 - 1
[2,1,1,2] => 101110 => 3 = 4 - 1
[2,1,2,1] => 101101 => 2 = 3 - 1

Description

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

Matching statistic: St000723
(load all 2 compositions to match this statistic)

Mapping

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

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: St001024
(load all 3 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001210
(load all 8 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001210: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Gives 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.

Matching statistic: St001733
(load all 8 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001733: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of weak left to right maxima of a Dyck path. A weak left to right maximum is a peak whose height is larger than or equal to the height of all peaks to its left.

Matching statistic: St000444
(load all 4 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the maximal rise of a Dyck path.

Matching statistic: St000469
(load all 3 compositions to match this statistic)

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00259: Graphs —vertex addition⟶ Graphs
St000469: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The distinguishing number of a graph. This is the minimal number of colours needed to colour the vertices of a graph, such that only the trivial automorphism of the graph preserves the colouring. For connected graphs, this statistic is at most one plus the maximal degree of the graph, with equality attained for complete graphs, complete bipartite graphs and the cycle with five vertices, see Theorem 4.2 of [2].

Matching statistic: St000774
(load all 2 compositions to match this statistic)

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00259: Graphs —vertex addition⟶ Graphs
St000774: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal multiplicity of a Laplacian eigenvalue in a graph.

The following 98 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. 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. St001933The largest multiplicity of a part in an integer partition. St000013The height of a Dyck path. St000381The largest part of an integer composition. St000439The position of the first down step of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000982The length of the longest constant subword. St001062The maximal size of a block of a set partition. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000025The number of initial rises of a Dyck path. St000314The number of left-to-right-maxima of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000542The number of left-to-right-minima of a permutation. St000700The protection number of an ordered tree. St000838The number of terminal right-hand endpoints when the vertices are written in order. St001372The length of a longest cyclic run of ones of a binary word. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001274The number of indecomposable injective modules with projective dimension equal to two. St001366The maximal multiplicity of a degree of a vertex of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000654The first descent of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000989The number of final rises of a permutation. St001330The hat guessing number of a graph. St000308The height of the tree associated to a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. St000642The size of the smallest orbit of antichains under Panyushev complementation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000485The length of the longest cycle of a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . 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. 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. St000873The aix statistic of a permutation. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001060The distinguishing index of a graph. St001948The number of augmented double ascents of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001118The acyclic chromatic index of a graph. St000454The largest eigenvalue of a graph if it is integral. St001615The number of join prime elements of a lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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

$eA$ . St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000264The girth of a graph, which is not a tree. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St001267The length of the Lyndon factorization of the binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001557The number of inversions of the second entry of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001875The number of simple modules with projective dimension at most 1. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000054The first entry of the permutation. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St000352The Elizalde-Pak rank of a permutation. St001096The size of the overlap set of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000383The last part of an integer composition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000090The variation of a composition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St000663The number of right floats of a permutation. St001556The number of inversions of the third entry of a permutation. 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. 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.