- FindStat

Your data matches 425 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001514
(load all 29 compositions to match this statistic)

Mapping

St001514: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule.

Matching statistic: St000538
(load all 31 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000538: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of even inversions of a permutation. An inversion

$i < j$ of a permutation is even if

$i \equiv j~(\operatorname{mod} 2)$ . See [[St000539]] for odd inversions.

Matching statistic: St000836
(load all 32 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of descents of distance 2 of a permutation. This is,

$\operatorname{des}_2(\pi) = | \{ i : \pi(i) > \pi(i+2) \} |$ .

Matching statistic: St000837
(load all 34 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000837: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of ascents of distance 2 of a permutation. This is,

$\operatorname{asc}_2(\pi) = | \{ i : \pi(i) < \pi(i+2) \} |$ .

Matching statistic: St000292
(load all 2 compositions to match this statistic)

Mapping

Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => 10 => 11 => 11 => 0
[1,0,1,0]
 => 1010 => 1101 => 1110 => 0
[1,1,0,0]
 => 1100 => 1101 => 1110 => 0
[1,0,1,0,1,0]
 => 101010 => 110101 => 111010 => 1
[1,0,1,1,0,0]
 => 101100 => 110101 => 111010 => 1
[1,1,0,0,1,0]
 => 110010 => 110101 => 111010 => 1
[1,1,0,1,0,0]
 => 110100 => 111001 => 111100 => 0
[1,1,1,0,0,0]
 => 111000 => 111001 => 111100 => 0
[1,0,1,0,1,0,1,0]
 => 10101010 => 11010101 => 11101010 => 2
[1,0,1,0,1,1,0,0]
 => 10101100 => 11010101 => 11101010 => 2
[1,0,1,1,0,0,1,0]
 => 10110010 => 11010101 => 11101010 => 2
[1,0,1,1,0,1,0,0]
 => 10110100 => 11011001 => 11101100 => 1
[1,0,1,1,1,0,0,0]
 => 10111000 => 11011001 => 11101100 => 1
[1,1,0,0,1,0,1,0]
 => 11001010 => 11010101 => 11101010 => 2
[1,1,0,0,1,1,0,0]
 => 11001100 => 11010101 => 11101010 => 2
[1,1,0,1,0,0,1,0]
 => 11010010 => 11100101 => 11110010 => 1
[1,1,0,1,0,1,0,0]
 => 11010100 => 11101001 => 11110100 => 1
[1,1,0,1,1,0,0,0]
 => 11011000 => 11101001 => 11110100 => 1
[1,1,1,0,0,0,1,0]
 => 11100010 => 11100101 => 11110010 => 1
[1,1,1,0,0,1,0,0]
 => 11100100 => 11101001 => 11110100 => 1
[1,1,1,0,1,0,0,0]
 => 11101000 => 11110001 => 11111000 => 0
[1,1,1,1,0,0,0,0]
 => 11110000 => 11110001 => 11111000 => 0

Description

The number of ascents of a binary word.

Matching statistic: St000390
(load all 2 compositions to match this statistic)

Mapping

Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => 10 => 11 => 11 => 1 = 0 + 1
[1,0,1,0]
 => 1010 => 1101 => 1110 => 1 = 0 + 1
[1,1,0,0]
 => 1100 => 1101 => 1110 => 1 = 0 + 1
[1,0,1,0,1,0]
 => 101010 => 110101 => 111010 => 2 = 1 + 1
[1,0,1,1,0,0]
 => 101100 => 110101 => 111010 => 2 = 1 + 1
[1,1,0,0,1,0]
 => 110010 => 110101 => 111010 => 2 = 1 + 1
[1,1,0,1,0,0]
 => 110100 => 111001 => 111100 => 1 = 0 + 1
[1,1,1,0,0,0]
 => 111000 => 111001 => 111100 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => 10101010 => 11010101 => 11101010 => 3 = 2 + 1
[1,0,1,0,1,1,0,0]
 => 10101100 => 11010101 => 11101010 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => 10110010 => 11010101 => 11101010 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => 10110100 => 11011001 => 11101100 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => 10111000 => 11011001 => 11101100 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => 11001010 => 11010101 => 11101010 => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
 => 11001100 => 11010101 => 11101010 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
 => 11010010 => 11100101 => 11110010 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => 11010100 => 11101001 => 11110100 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => 11011000 => 11101001 => 11110100 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => 11100010 => 11100101 => 11110010 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => 11100100 => 11101001 => 11110100 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => 11101000 => 11110001 => 11111000 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => 11110000 => 11110001 => 11111000 => 1 = 0 + 1

Description

The number of runs of ones in a binary word.

Matching statistic: St001315

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001315: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The dissociation number of a graph.

Matching statistic: St001557
(load all 9 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00277: Permutations —catalanization⟶ Permutations
St001557: Permutations ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of inversions of the second entry of a permutation. This is, for a permutation

$\pi$ of length

$n$ ,

$\# \{2 < k \leq n \mid \pi(2) > \pi(k)\}.$ The number of inversions of the first entry is [[St000054]] and the number of inversions of the third entry is [[St001556]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.

Matching statistic: St001960
(load all 32 compositions to match this statistic)

Mapping

Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [.,.]
 => [1] => [1] => ? = 0
[1,0,1,0]
 => [.,[.,.]]
 => [2,1] => [2,1] => 0
[1,1,0,0]
 => [[.,.],.]
 => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => [3,2,1] => [3,2,1] => 1
[1,0,1,1,0,0]
 => [.,[[.,.],.]]
 => [2,3,1] => [3,2,1] => 1
[1,1,0,0,1,0]
 => [[.,.],[.,.]]
 => [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0]
 => [[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => 0
[1,1,1,0,0,0]
 => [[[.,.],.],.]
 => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,3,2,1] => 2
[1,0,1,0,1,1,0,0]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,3,2,1] => 2
[1,0,1,1,0,0,1,0]
 => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => [4,3,2,1] => 2
[1,0,1,1,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,2,3,1] => 1
[1,0,1,1,1,0,0,0]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,2,3,1] => 1
[1,1,0,0,1,0,1,0]
 => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,4,3,2] => 2
[1,1,0,0,1,1,0,0]
 => [[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,4,3,2] => 2
[1,1,0,1,0,0,1,0]
 => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,1,4,3] => 1
[1,1,0,1,0,1,0,0]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,2,1,4] => 1
[1,1,0,1,1,0,0,0]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,2,1,4] => 1
[1,1,1,0,0,0,1,0]
 => [[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,0,0,1,0,0]
 => [[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,3,2,4] => 1
[1,1,1,0,1,0,0,0]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => 0
[1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => 0

Description

The number of descents of a permutation minus one if its first entry is not one. This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].

Matching statistic: St001424
(load all 5 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001424: Binary words ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => []
 =>  => ? = 0
[1,0,1,0]
 => [1]
 => 10 => 0
[1,1,0,0]
 => []
 =>  => ? = 0
[1,0,1,0,1,0]
 => [2,1]
 => 1010 => 1
[1,0,1,1,0,0]
 => [1,1]
 => 110 => 1
[1,1,0,0,1,0]
 => [2]
 => 100 => 1
[1,1,0,1,0,0]
 => [1]
 => 10 => 0
[1,1,1,0,0,0]
 => []
 =>  => ? = 0
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 101010 => 2
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 11010 => 2
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 100110 => 2
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 10110 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1110 => 1
[1,1,0,0,1,0,1,0]
 => [3,2]
 => 10100 => 2
[1,1,0,0,1,1,0,0]
 => [2,2]
 => 1100 => 2
[1,1,0,1,0,0,1,0]
 => [3,1]
 => 10010 => 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => 1010 => 1
[1,1,0,1,1,0,0,0]
 => [1,1]
 => 110 => 1
[1,1,1,0,0,0,1,0]
 => [3]
 => 1000 => 1
[1,1,1,0,0,1,0,0]
 => [2]
 => 100 => 1
[1,1,1,0,1,0,0,0]
 => [1]
 => 10 => 0
[1,1,1,1,0,0,0,0]
 => []
 =>  => ? = 0

Description

The number of distinct squares in a binary word. A factor of a word is a sequence of consecutive letters. This statistic records the number of distinct non-empty words

$u$ such that

$uu$ is a factor of the word. Note that every word of length at least four contains a square.

The following 415 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001488The number of corners of a skew partition. St001060The distinguishing index of a graph. St000562The number of internal points of a set partition. St000871The number of very big ascents of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000091The descent variation of a composition. St000486The number of cycles of length at least 3 of a permutation. St000650The number of 3-rises of a permutation. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001822The number of alignments of a signed permutation. St001857The number of edges in the reduced word graph of a signed permutation. St000291The number of descents of a binary word. St000886The number of permutations with the same antidiagonal sums. St000905The number of different multiplicities of parts of an integer composition. St000955Number of times one has

$Ext^i(D(A),A)>0$ for

$i>0$ for the corresponding LNakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001405The number of bonds in a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001569The maximal modular displacement of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000630The length of the shortest palindromic decomposition of a binary word. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001642The Prague dimension of a graph. St001672The restrained domination number of a graph. St000735The last entry on the main diagonal of a standard tableau. St000460The hook length of the last cell along the main diagonal of an integer partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001964The interval resolution global dimension of a poset. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001487The number of inner corners of a skew partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001516The number of cyclic bonds of a permutation. St000782The indicator function of whether a given perfect matching is an L & P matching. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St000124The cardinality of the preimage of the Simion-Schmidt map. St000477The weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000997The even-odd crank of an integer partition. St001128The exponens consonantiae of a partition. 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

$eA$ . 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

$eA$ . 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

$eA$ . St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000260The radius of a connected graph. St000284The Plancherel distribution on integer partitions. St000454The largest eigenvalue of a graph if it is integral. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000706The product of the factorials of the multiplicities of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001568The smallest positive integer that does not appear twice in the partition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000928The sum of the coefficients of the character polynomial of an integer partition. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001330The hat guessing number of a graph. St001720The minimal length of a chain of small intervals in a lattice. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000102The charge of a semistandard tableau. St001556The number of inversions of the third entry of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001948The number of augmented double ascents of a permutation. St001208The number of connected components of the quiver of

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001236The dominant dimension of the corresponding Comp-Nakayama algebra. 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. 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. St000022The number of fixed points of a permutation. St000039The number of crossings of a permutation. St000089The absolute variation of a composition. St000090The variation of a composition. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000101The cocharge of a semistandard tableau. St000117The number of centered tunnels of a Dyck path. St000133The "bounce" of a permutation. St000153The number of adjacent cycles of a permutation. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000221The number of strong fixed points of a permutation. St000233The number of nestings of a set partition. St000234The number of global ascents of a permutation. St000236The number of cyclical small weak excedances. St000237The number of small exceedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000247The number of singleton blocks of a set partition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000295The length of the border of a binary word. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by

$4$ . St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St000317The cycle descent number of a permutation. St000322The skewness of a graph. St000355The number of occurrences of the pattern 21-3. St000357The number of occurrences of the pattern 12-3. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000405The number of occurrences of the pattern 1324 in a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000455The second largest eigenvalue of a graph if it is integral. St000461The rix statistic of a permutation. St000462The major index minus the number of excedences of a permutation. St000496The rcs statistic of a set partition. St000516The number of stretching pairs of a permutation. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000623The number of occurrences of the pattern 52341 in a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000709The number of occurrences of 14-2-3 or 14-3-2. St000731The number of double exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000873The aix statistic of a permutation. St000894The trace of an alternating sign matrix. St000943The number of spots the most unlucky car had to go further in a parking function. St000962The 3-shifted major index of a permutation. St000989The number of final rises of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001082The number of boxed occurrences of 123 in a permutation. 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. St001130The number of two successive successions in a permutation. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001381The fertility of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001429The number of negative entries in a signed permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001513The number of nested exceedences of a permutation. St001520The number of strict 3-descents. St001524The degree of symmetry of a binary word. St001536The number of cyclic misalignments of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001631The number of simple modules

$S$ with

$dim Ext^1(S,A)=1$ in the incidence algebra

$A$ of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001781The interlacing number of a set partition. St001783The number of odd automorphisms of a graph. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001851The number of Hecke atoms of a signed permutation. St001856The number of edges in the reduced word graph of a permutation. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001871The number of triconnected components of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001895The oddness of a signed permutation. St001903The number of fixed points of a parking function. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000056The decomposition (or block) number of a permutation. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000309The number of vertices with even degree. St000314The number of left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000353The number of inner valleys of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000450The number of edges minus the number of vertices plus 2 of a graph. St000456The monochromatic index of a connected graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000570The Edelman-Greene number of a permutation. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000654The first descent of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000729The minimal arc length of a set partition. St000739The first entry in the last row of a semistandard tableau. St000740The last entry of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000872The number of very big descents of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000883The number of longest increasing subsequences of a permutation. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000958The number of Bruhat factorizations of a permutation. St000968We 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

$c_{n−1}$ . St000990The first ascent of a permutation. St000991The number of right-to-left minima of a permutation. St001050The number of terminal closers of a set partition. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001162The minimum jump of a permutation. St001192The maximal dimension of

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

$S$ over the corresponding Nakayama algebra

$A$ . St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001204Call 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. St001256Number of simple reflexive modules that are 2-stable reflexive. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001260The permanent of an alternating sign matrix. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001344The neighbouring number of a permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001410The minimal entry of a semistandard tableau. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001481The minimal height of a peak of a Dyck path. St001490The number of connected components of a skew partition. St001518The number of graphs with the same ordinary spectrum as the given graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001806The upper middle entry of a permutation. St001828The Euler characteristic of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001884The number of borders of a binary word. St001889The size of the connectivity set of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001928The number of non-overlapping descents in a permutation. St001937The size of the center of a parking function. St001946The number of descents in a parking function. St000058The order of a permutation. St000075The orbit size of a standard tableau under promotion. St000084The number of subtrees. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000105The number of blocks in the set partition. St000134The size of the orbit of an alternating sign matrix under gyration. St000251The number of nonsingleton blocks of a set partition. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000401The size of the symmetry class of a permutation. St000402Half the size of the symmetry class of a permutation. St000417The size of the automorphism group of the ordered tree. St000422The energy of a graph, if it is integral. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St000679The pruning number of an ordered tree. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000822The Hadwiger number of the graph. St000823The number of unsplittable factors of the set partition. St000893The number of distinct diagonal sums of an alternating sign matrix. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001058The breadth of the ordered tree. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001423The number of distinct cubes in a binary word. St001517The length of a longest pair of twins in a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001555The order of a signed permutation. St001597The Frobenius rank of a skew partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001624The breadth of a lattice. St001644The dimension of a graph. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001734The lettericity of a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001893The flag descent of a signed permutation. St000264The girth of a graph, which is not a tree. St000495The number of inversions of distance at most 2 of a permutation. St000638The number of up-down runs of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one.