- FindStat

Your data matches 422 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001483
(load all 23 compositions to match this statistic)

Mapping

Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001483: 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]
 => [1,1,0,0]
 => 1
[1,1,0,0]
 => [1,0,1,0]
 => 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 2
[1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 2
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 1
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 2
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 2
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => 1

Description

The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module.

Matching statistic: St000648
(load all 27 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000648: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of 2-excedences of a permutation. This is the number of positions

$1\leq i\leq n$ such that

$\sigma(i)=i+2$ .

Matching statistic: St000392
(load all 5 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [2,1] => 1 => 1
[1,0,1,0]
 => [3,1,2] => 10 => 1
[1,1,0,0]
 => [2,3,1] => 01 => 1
[1,0,1,0,1,0]
 => [4,1,2,3] => 100 => 1
[1,0,1,1,0,0]
 => [3,1,4,2] => 101 => 1
[1,1,0,0,1,0]
 => [2,4,1,3] => 010 => 1
[1,1,0,1,0,0]
 => [4,3,1,2] => 110 => 2
[1,1,1,0,0,0]
 => [2,3,4,1] => 001 => 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1001 => 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 1010 => 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 1001 => 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0100 => 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 0101 => 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => 1101 => 2
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 0010 => 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => 0110 => 2
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => 1010 => 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0001 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => 10101 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => 10010 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => 10110 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 10001 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 01010 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 01001 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => 11010 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => 11001 => 2
[1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 00101 => 1
[1,1,1,0,0,1,1,0,0,0]
 => [2,5,4,1,6,3] => 01101 => 2
[1,1,1,0,1,1,0,0,0,0]
 => [5,3,4,1,6,2] => 10101 => 1
[1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 00010 => 1
[1,1,1,1,0,0,0,1,0,0]
 => [2,3,6,5,1,4] => 00110 => 2
[1,1,1,1,0,0,1,0,0,0]
 => [2,6,4,5,1,3] => 01010 => 1
[1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 00001 => 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,1,5,2,6,7,4] => 101001 => 1
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5] => 100101 => 1
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => 100010 => 1
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => 100001 => 1
[1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => 010101 => 1
[1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => 010010 => 1
[1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => 010001 => 1
[1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,3,5,1,7,4,6] => 001010 => 1
[1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => 001001 => 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,5,4,1,6,7,3] => 011001 => 2
[1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,3,4,6,1,7,5] => 000101 => 1
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [2,3,6,5,1,7,4] => 001101 => 2
[1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => 000010 => 1
[1,1,1,1,1,0,0,0,0,1,0,0]
 => [2,3,4,7,6,1,5] => 000110 => 2
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => 000001 => 1
[1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [2,3,5,1,6,7,8,4] => 0010001 => 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [2,3,4,5,7,1,8,6] => 0000101 => 1

Description

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

Matching statistic: St001385
(load all 4 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001385: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. Equivalently, given an integer partition

$\lambda$ , this is the number of molecular combinatorial species that decompose into a product of atomic species of sizes

$\lambda_1,\lambda_2,\dots$ . In particular, the value on the partition

$(n)$ is the number of atomic species of degree

$n$ , [2].

Matching statistic: St001934
(load all 4 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of monotone factorisations of genus zero of a permutation of given cycle type. A monotone factorisation of genus zero of a permutation

$\pi\in\mathfrak S_n$ with

$\ell$ cycles, including fixed points, is a tuple of

$r = n - \ell$ transpositions

$(a_1, b_1),\dots,(a_r, b_r)$ with

$b_1 \leq \dots \leq b_r$ and

$a_i < b_i$ for all

$i$ , whose product, in this order, is

$\pi$ . For example, the cycle

$(2,3,1)$ has the two factorizations

$(2,3)(1,3)$ and

$(1,2)(2,3)$ .

Matching statistic: St000052
(load all 4 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of valleys of a Dyck path not on the x-axis. That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.

Matching statistic: St000237
(load all 5 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of small exceedances. This is the number of indices

$i$ such that

$\pi_i=i+1$ .

Matching statistic: St000366
(load all 46 compositions to match this statistic)

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000366: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of double descents of a permutation. A double descent of a permutation

$\pi$ is a position

$i$ such that

$\pi(i) > \pi(i+1) > \pi(i+2)$ .

Matching statistic: St000373
(load all 14 compositions to match this statistic)

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000373: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . Given a permutation

$\pi = [\pi_1,\ldots,\pi_n]$ , this statistic counts the number of position

$j$ such that

$\pi_j \geq j$ and there exist indices

$i,k$ with

$i < j < k$ and

$\pi_i > \pi_j > \pi_k$ . See also [[St000213]] and [[St000119]].

Matching statistic: St000496
(load all 4 compositions to match this statistic)

Mapping

Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00215: Set partitions —Wachs-White⟶ Set partitions
St000496: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The rcs statistic of a set partition. Let

$S = B_1,\ldots,B_k$ be a set partition with ordered blocks

$B_i$ and with

$\operatorname{min} B_a < \operatorname{min} B_b$ for

$a < b$ . According to [1, Definition 3], a '''rcs''' (right-closer-smaller) of

$S$ is given by a pair

$i > j$ such that

$j = \operatorname{max} B_b$ and

$i \in B_a$ for

$a < b$ .

The following 412 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000660The number of rises of length at least 3 of a Dyck path. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000884The number of isolated descents of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001728The number of invisible descents of a permutation. St001781The interlacing number of a set partition. St001801Half the number of preimage-image pairs of different parity in a permutation. St000010The length of the partition. St000146The Andrews-Garvan crank of a partition. St000183The side length of the Durfee square of an integer partition. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000346The number of coarsenings of a partition. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000679The pruning number of an ordered tree. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000701The protection number of a binary tree. St000730The maximal arc length of a set partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000883The number of longest increasing subsequences of a permutation. St000920The logarithmic height of a Dyck path. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001367The smallest number which does not occur as degree of a vertex in a graph. St001432The order dimension of the partition. St001484The number of singletons of an integer partition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001732The number of peaks visible from the left. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000065The number of entries equal to -1 in an alternating sign matrix. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000142The number of even parts of a partition. St000214The number of adjacencies of a permutation. St000232The number of crossings of a set partition. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000381The largest part of an integer composition. St000386The number of factors DDU in a Dyck path. St000439The position of the first down step of a Dyck path. St000481The number of upper covers of a partition in dominance order. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000534The number of 2-rises of a permutation. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000565The major index of a set partition. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) 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. St000647The number of big descents of a permutation. St000661The number of rises of length 3 of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000731The number of double exceedences of a permutation. St000759The smallest missing part in an integer partition. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001071The beta invariant of the graph. St001092The number of distinct even parts of a partition. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001252Half the sum of the even parts of a partition. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001323The independence gap of a graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001354The number of series nodes in the modular decomposition of a graph. St001394The genus of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001479The number of bridges of a graph. St001584The area statistic between a Dyck path and its bounce path. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001657The number of twos in an integer partition. St001689The number of celebrities in a graph. St001727The number of invisible inversions of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001797The number of overfull subgraphs of a graph. St001826The maximal number of leaves on a vertex of a graph. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001843The Z-index of a set partition. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000552The number of cut vertices of a graph. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000732The number of double deficiencies of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St001552The number of inversions between excedances and fixed points of a permutation. St000124The cardinality of the preimage of the Simion-Schmidt map. St000886The number of permutations with the same antidiagonal sums. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000248The number of anti-singletons of a set partition. St000359The number of occurrences of the pattern 23-1. St000389The number of runs of ones of odd length in a binary word. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000502The number of successions of a set partitions. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 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. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000658The number of rises of length 2 of a Dyck path. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St000932The number of occurrences of the pattern UDU in a Dyck path. St001735The number of permutations with the same set of runs. St000002The number of occurrences of the pattern 123 in a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000355The number of occurrences of the pattern 21-3. St000454The largest eigenvalue of a graph if it is integral. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001083The number of boxed occurrences of 132 in a permutation. St001377The major index minus the number of inversions of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St000039The number of crossings of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St000317The cycle descent number of a permutation. 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. 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. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. 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. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St000025The number of initial rises of a Dyck path. St000092The number of outer peaks of a permutation. St000105The number of blocks in the set partition. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001733The number of weak left to right maxima of a Dyck path. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000095The number of triangles of a graph. St000217The number of occurrences of the pattern 312 in a permutation. St000356The number of occurrences of the pattern 13-2. St000357The number of occurrences of the pattern 12-3. 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. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000646The number of big ascents of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001062The maximal size of a block of a set partition. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001396Number of triples of incomparable elements in a finite poset. 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. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St001130The number of two successive successions in a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000353The number of inner valleys of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St001082The number of boxed occurrences of 123 in 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

$K[x]/(x^n)$ . St000542The number of left-to-right-minima of a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000352The Elizalde-Pak rank of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001399The distinguishing number of a poset. St001403The number of vertical separators in a permutation. St000633The size of the automorphism group of a poset. St000649The number of 3-excedences of a permutation. St000850The number of 1/2-balanced pairs in a poset. St000031The number of cycles in the cycle decomposition of a permutation. St001568The smallest positive integer that does not appear twice in the partition. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000664The number of right ropes of a permutation. St001330The hat guessing number of a graph. St000028The number of stack-sorts needed to sort a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000022The number of fixed points of a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000441The number of successions of a permutation. St000451The length of the longest pattern of the form k 1 2. St000665The number of rafts of a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St000996The number of exclusive left-to-right maxima of a permutation. St001624The breadth of a lattice. St000877The depth of the binary word interpreted as a path. St000806The semiperimeter of the associated bargraph. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000485The length of the longest cycle of a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. 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. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000054The first entry of the permutation. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000864The number of circled entries of the shifted recording tableau of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. 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. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000951The dimension of

$Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000989The number of final rises of a permutation. St000991The number of right-to-left minima of a permutation. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. 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. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001513The number of nested exceedences of a permutation. St001530The depth of a Dyck path. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001847The number of occurrences of the pattern 1432 in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000456The monochromatic index of a connected graph. St001545The second Elser number of a connected graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001271The competition number of a graph. St001862The number of crossings of a signed 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

$eA$ . St001846The number of elements which do not have a complement in the lattice. St001866The nesting alignments of a signed permutation. St001864The number of excedances of a signed permutation. St000764The number of strong records in an integer composition. St000058The order of a permutation. St000768The number of peaks in an integer composition. St000669The number of permutations obtained by switching ascents or descents of size 2. St000264The girth of a graph, which is not a tree. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. 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

$K[x]/(x^n)$ . St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000488The number of cycles of a permutation of length at most 2. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001095The number of non-isomorphic posets with precisely one further covering relation. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001550The number of inversions between exceedances where the greater exceedance is linked. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St000254The nesting number of a set partition. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001490The number of connected components of a skew partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001890The maximum magnitude of the Möbius function of a poset. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000007The number of saliances of the permutation. St000842The breadth of a permutation. St001115The number of even descents of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St000455The second largest eigenvalue of a graph if it is integral. St000068The number of minimal elements in a poset. St000891The number of distinct diagonal sums of a permutation matrix. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000405The number of occurrences of the pattern 1324 in a permutation. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001868The number of alignments of type NE of a signed permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St000338The number of pixed points of a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000239The number of small weak excedances. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000535The rank-width of a graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000805The number of peaks of the associated bargraph. St000942The number of critical left to right maxima of the parking functions. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001413Half the length of the longest even length palindromic prefix 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. St001743The discrepancy of a graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001904The length of the initial strictly increasing segment of a parking function. St001946The number of descents in a parking function. St000090The variation of a composition. St000091The descent variation of a composition. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000406The number of occurrences of the pattern 3241 in a permutation. St000487The length of the shortest cycle of a permutation. St000516The number of stretching pairs of a permutation. St000650The number of 3-rises 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. St000879The number of long braid edges in the graph of braid moves of a permutation. St000962The 3-shifted major index of a permutation. St001093The detour number of a graph. St001114The number of odd descents of a permutation. St001153The number of blocks with even minimum in a set partition. St001193The dimension of

$Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra

$A$ such that

$eA$ is a minimal faithful projective-injective module. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001353The number of prime nodes in the modular decomposition of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001402The number of separators in a permutation. St001470The cyclic holeyness of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. 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. St001674The number of vertices of the largest induced star graph in the graph. St001691The number of kings in a graph. St001715The number of non-records in a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001857The number of edges in the reduced word graph of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001884The number of borders of a binary word. St000741The Colin de Verdière graph invariant. St000908The length of the shortest maximal antichain in a poset. St000911The number of maximal antichains of maximal size in a poset. St000914The sum of the values of the Möbius function of a poset. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001487The number of inner corners of a skew partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000298The order dimension or Dushnik-Miller dimension of a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000872The number of very big descents of a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St000907The number of maximal antichains of minimal length in a poset. St001301The first Betti number of the order complex associated with the poset. St001434The number of negative sum pairs of a signed permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001631The number of simple modules

$S$ with

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

$A$ of the poset. St001845The number of join irreducibles minus the rank of a lattice. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St000422The energy of a graph, if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001260The permanent of an alternating sign matrix. St000894The trace of an alternating sign matrix. 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

$eA$ .