- FindStat

Your data matches 113 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000053
(load all 2 compositions to match this statistic)

Mapping

Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of valleys of the Dyck path.

Matching statistic: St001197
(load all 2 compositions to match this statistic)

Mapping

Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001197: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.

Matching statistic: St001506
(load all 2 compositions to match this statistic)

Mapping

Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001506: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra.

Matching statistic: St000010
(load all 3 compositions to match this statistic)

Mapping

Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

Matching statistic: St000011
(load all 2 compositions to match this statistic)

Mapping

Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.

Matching statistic: St000097
(load all 7 compositions to match this statistic)

Mapping

Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The order of the largest clique of the graph. A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.

Matching statistic: St000098
(load all 7 compositions to match this statistic)

Mapping

Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The chromatic number of a graph. The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.

Matching statistic: St000288
(load all 13 compositions to match this statistic)

Mapping

Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 1 => 1 = 0 + 1
[1,2] => [2] => 10 => 1 = 0 + 1
[2,1] => [2] => 10 => 1 = 0 + 1
[1,2,3] => [3] => 100 => 1 = 0 + 1
[1,3,2] => [1,2] => 110 => 2 = 1 + 1
[2,1,3] => [3] => 100 => 1 = 0 + 1
[2,3,1] => [3] => 100 => 1 = 0 + 1
[3,1,2] => [3] => 100 => 1 = 0 + 1
[3,2,1] => [2,1] => 101 => 2 = 1 + 1
[1,2,3,4] => [4] => 1000 => 1 = 0 + 1
[1,2,4,3] => [2,2] => 1010 => 2 = 1 + 1
[1,3,2,4] => [1,3] => 1100 => 2 = 1 + 1
[1,3,4,2] => [1,3] => 1100 => 2 = 1 + 1
[1,4,2,3] => [1,3] => 1100 => 2 = 1 + 1
[1,4,3,2] => [1,2,1] => 1101 => 3 = 2 + 1
[2,1,3,4] => [4] => 1000 => 1 = 0 + 1
[2,1,4,3] => [2,2] => 1010 => 2 = 1 + 1
[2,3,1,4] => [4] => 1000 => 1 = 0 + 1
[2,3,4,1] => [4] => 1000 => 1 = 0 + 1
[2,4,1,3] => [4] => 1000 => 1 = 0 + 1
[2,4,3,1] => [3,1] => 1001 => 2 = 1 + 1
[3,1,2,4] => [4] => 1000 => 1 = 0 + 1
[3,1,4,2] => [2,2] => 1010 => 2 = 1 + 1
[3,2,1,4] => [2,2] => 1010 => 2 = 1 + 1
[3,2,4,1] => [2,2] => 1010 => 2 = 1 + 1
[3,4,1,2] => [4] => 1000 => 1 = 0 + 1
[3,4,2,1] => [3,1] => 1001 => 2 = 1 + 1
[4,1,2,3] => [4] => 1000 => 1 = 0 + 1
[4,1,3,2] => [3,1] => 1001 => 2 = 1 + 1
[4,2,1,3] => [2,2] => 1010 => 2 = 1 + 1
[4,2,3,1] => [3,1] => 1001 => 2 = 1 + 1
[4,3,1,2] => [1,3] => 1100 => 2 = 1 + 1
[4,3,2,1] => [1,2,1] => 1101 => 3 = 2 + 1
[1,2,3,4,5] => [5] => 10000 => 1 = 0 + 1
[1,2,3,5,4] => [3,2] => 10010 => 2 = 1 + 1
[1,2,4,3,5] => [2,3] => 10100 => 2 = 1 + 1
[1,2,4,5,3] => [2,3] => 10100 => 2 = 1 + 1
[1,2,5,3,4] => [2,3] => 10100 => 2 = 1 + 1
[1,2,5,4,3] => [2,2,1] => 10101 => 3 = 2 + 1
[1,3,2,4,5] => [1,4] => 11000 => 2 = 1 + 1
[1,3,2,5,4] => [1,2,2] => 11010 => 3 = 2 + 1
[1,3,4,2,5] => [1,4] => 11000 => 2 = 1 + 1
[1,3,4,5,2] => [1,4] => 11000 => 2 = 1 + 1
[1,3,5,2,4] => [1,4] => 11000 => 2 = 1 + 1
[1,3,5,4,2] => [1,3,1] => 11001 => 3 = 2 + 1
[1,4,2,3,5] => [1,4] => 11000 => 2 = 1 + 1
[1,4,2,5,3] => [1,2,2] => 11010 => 3 = 2 + 1
[1,4,3,2,5] => [1,2,2] => 11010 => 3 = 2 + 1
[1,4,3,5,2] => [1,2,2] => 11010 => 3 = 2 + 1
[1,4,5,2,3] => [1,4] => 11000 => 2 = 1 + 1

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St001068
(load all 2 compositions to match this statistic)

Mapping

Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of torsionless simple modules in the corresponding Nakayama algebra.

Matching statistic: St001203
(load all 2 compositions to match this statistic)

Mapping

Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001203: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

We associate to 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 Dyck path as follows: In the list $L$ delete the first entry $c_0$ and substract from all other entries $n-1$ and then append the last element 1 (this was suggested by Christian Stump). The result is a Kupisch series of an LNakayama algebra. Example: [5,6,6,6,6] goes into [2,2,2,2,1]. Now associate to the CNakayama algebra with the above properties the Dyck path corresponding to the Kupisch series of the LNakayama algebra. The statistic return the global dimension of the CNakayama algebra divided by 2.

The following 103 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001494The Alon-Tarsi number of a graph. St001581The achromatic number of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000024The number of double up and double down steps of a Dyck path. St000157The number of descents of a standard tableau. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000228The size of a partition. St000733The row containing the largest entry of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000691The number of changes of a binary word. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000306The bounce count of a Dyck path. St000312The number of leaves in a graph. St000711The number of big exceedences of a permutation. St000710The number of big deficiencies of a permutation. St000619The number of cyclic descents of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000822The Hadwiger number of the graph. 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000454The largest eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001270The bandwidth of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001644The dimension of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001812The biclique partition number of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000087The number of induced subgraphs. St000286The number of connected components of the complement of a graph. St000363The number of minimal vertex covers of a graph. St000443The number of long tunnels of a Dyck path. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000926The clique-coclique number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000445The number of rises of length 1 of a Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001427The number of descents of a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001964The interval resolution global dimension of a poset. St000307The number of rowmotion orbits 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. St001896The number of right descents of a signed permutations.