- FindStat

Your data matches 1092 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000920
(load all 29 compositions to match this statistic)

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The logarithmic height of a Dyck path. This is the floor of the binary logarithm of the usual height increased by one:

$\lfloor\log_2(1+height(D))\rfloor$

Matching statistic: St000010
(load all 6 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

Matching statistic: St000147
(load all 5 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest part of an integer partition.

Matching statistic: St000007
(load all 74 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
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern

$([1], {(1,1)})$ , i.e., the upper right quadrant is shaded, see [1].

Matching statistic: St000319
(load all 5 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The spin of an integer partition. The Ferrers shape of an integer partition

$\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of

$\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let

$\lambda = (5,5,4,4,2,1)$ . Removing the border strips successively yields the sequence of partitions

$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$ The first strip

$(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses

$4$ times, the second strip

$(4,3,3,1) \setminus (2,2)$ crosses

$3$ times, the strip

$(2,2) \setminus (1)$ crosses

$1$ time, and the remaining strip

$(1) \setminus ()$ does not cross. This yields the spin of

$(5,5,4,4,2,1)$ to be

$4+3+1 = 8$ .

Matching statistic: St000320
(load all 5 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The dinv adjustment of an integer partition. The Ferrers shape of an integer partition

$\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For

$0 \leq j < \lambda_1$ let

$n_j$ be the length of the border strip starting at

$(\lambda_1-j,0)$ . The dinv adjustment is then defined by

$\sum_{j:n_j > 0}(\lambda_1-1-j).$ The following example is taken from Appendix B in [2]: Let

$\lambda=(5,5,4,4,2,1)$ . Removing the border strips successively yields the sequence of partitions

$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$ and we obtain

$(n_0,\ldots,n_4) = (10,7,0,3,1)$ . The dinv adjustment is thus

$4+3+1+0 = 8$ .

Matching statistic: St001918
(load all 5 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The degree of the cyclic sieving polynomial corresponding to an integer partition. Let

$\lambda$ be an integer partition of

$n$ and let

$N$ be the least common multiple of the parts of

$\lambda$ . Fix an arbitrary permutation

$\pi$ of cycle type

$\lambda$ . Then

$\pi$ induces a cyclic action of order

$N$ on

$\{1,\dots,n\}$ . The corresponding character can be identified with the cyclic sieving polynomial

$C_\lambda(q)$ of this action, modulo

$q^N-1$ . Explicitly, it is

$\sum_{p\in\lambda} [p]_{q^{N/p}},$ where

$[p]_q = 1+\dots+q^{p-1}$ is the

$q$ -integer. This statistic records the degree of

$C_\lambda(q)$ . Equivalently, it equals

$\left(1 - \frac{1}{\lambda_1}\right) N,$ where

$\lambda_1$ is the largest part of

$\lambda$ . The statistic is undefined for the empty partition.

Matching statistic: St000288
(load all 3 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [2,1] => [1,1]
 => 110 => 2
[1,0,1,0]
 => [3,1,2] => [2,1]
 => 1010 => 2
[1,1,0,0]
 => [2,3,1] => [2,1]
 => 1010 => 2
[1,0,1,0,1,0]
 => [4,1,2,3] => [3,1]
 => 10010 => 2
[1,0,1,1,0,0]
 => [3,1,4,2] => [2,2]
 => 1100 => 2
[1,1,0,0,1,0]
 => [2,4,1,3] => [2,2]
 => 1100 => 2
[1,1,0,1,0,0]
 => [4,3,1,2] => [2,1,1]
 => 10110 => 3
[1,1,1,0,0,0]
 => [2,3,4,1] => [3,1]
 => 10010 => 2
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [4,1]
 => 100010 => 2
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [3,2]
 => 10100 => 2
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [3,2]
 => 10100 => 2
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [3,2]
 => 10100 => 2
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [3,2]
 => 10100 => 2
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [3,2]
 => 10100 => 2
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [3,2]
 => 10100 => 2
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [4,1]
 => 100010 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [5,1]
 => 1000010 => 2
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [4,2]
 => 100100 => 2
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [4,2]
 => 100100 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [4,2]
 => 100100 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [4,2]
 => 100100 => 2
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [3,3]
 => 11000 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [3,3]
 => 11000 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [4,2]
 => 100100 => 2
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [4,2]
 => 100100 => 2
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [3,3]
 => 11000 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [3,3]
 => 11000 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [4,2]
 => 100100 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => [4,2]
 => 100100 => 2
[1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => [4,2]
 => 100100 => 2
[1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => [4,2]
 => 100100 => 2
[1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [5,1]
 => 1000010 => 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => [6,1]
 => 10000010 => 2
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => [5,2]
 => 1000100 => 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => [5,2]
 => 1000100 => 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => [5,2]
 => 1000100 => 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => [5,2]
 => 1000100 => 2
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,1,2,6,3,7,5] => [4,3]
 => 101000 => 2
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => [4,3]
 => 101000 => 2
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => [5,2]
 => 1000100 => 2
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => [5,2]
 => 1000100 => 2
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [3,1,6,2,4,7,5] => [4,3]
 => 101000 => 2
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => [4,3]
 => 101000 => 2
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,1,5,2,6,7,4] => [4,3]
 => 101000 => 2
[1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => [4,3]
 => 101000 => 2
[1,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5] => [4,3]
 => 101000 => 2
[1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => [4,3]
 => 101000 => 2
[1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => [5,2]
 => 1000100 => 2
[1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,7,1,3,4,5,6] => [5,2]
 => 1000100 => 2
[1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,6,1,3,4,7,5] => [4,3]
 => 101000 => 2

Description

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

Matching statistic: St000318
(load all 6 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
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of addable cells of the Ferrers diagram of an integer partition.

Matching statistic: St000378
(load all 2 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The diagonal inversion number of an integer partition. The dinv of a partition is the number of cells

$c$ in the diagram of an integer partition

$\lambda$ for which

$\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$ . See also exercise 3.19 of [2]. This statistic is equidistributed with the length of the partition, see [3].

The following 1082 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000381The largest part of an integer composition. St000676The number of odd rises of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000759The smallest missing part in an integer partition. St001330The hat guessing number of a graph. St000013The height of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000069The number of maximal elements of a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000153The number of adjacent cycles of a permutation. St000157The number of descents of a standard tableau. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000519The largest length of a factor maximising the subword complexity. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000548The number of different non-empty partial sums of 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. St001280The number of parts of an integer partition that are at least two. St001484The number of singletons of an integer partition. 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. St001172The number of 1-rises at odd height of a Dyck path. St001584The area statistic between a Dyck path and its bounce path. St000679The pruning number of an ordered tree. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001732The number of peaks visible from the left. St000065The number of entries equal to -1 in an alternating sign matrix. St000386The number of factors DDU in a Dyck path. St000496The rcs statistic of a set partition. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001781The interlacing number of a set partition. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001843The Z-index of a set partition. St000668The least common multiple of the parts of the partition. St000040The number of regions of the inversion arrangement of a permutation. St000047The number of standard immaculate tableaux of a given shape. St000058The order of a permutation. St000071The number of maximal chains in a poset. St000078The number of alternating sign matrices whose left key is the permutation. St000110The number of permutations less than or equal to a permutation in left weak order. St000201The number of leaf nodes in a binary tree. St000255The number of reduced Kogan faces with the permutation as type. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000392The length of the longest run of ones in a binary word. St000396The register function (or Horton-Strahler number) of a binary tree. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000527The width of the poset. St000669The number of permutations obtained by switching ascents or descents of size 2. St000701The protection number of a binary tree. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000862The number of parts of the shifted shape of a permutation. St000883The number of longest increasing subsequences of a permutation. St000909The number of maximal chains of maximal size in a poset. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001597The Frobenius rank of a skew partition. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000091The descent variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000141The maximum drop size of a permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000209Maximum difference of elements in cycles. St000214The number of adjacencies of a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000237The number of small exceedances. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000356The number of occurrences of the pattern 13-2. St000374The number of exclusive right-to-left minima of a permutation. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000446The disorder of a permutation. St000457The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000463The number of admissible inversions of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000534The number of 2-rises of a permutation. St000632The jump number of the poset. St000662The staircase size of the code of a permutation. St000663The number of right floats of a permutation. St000670The reversal length of a permutation. St000703The number of deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000761The number of ascents in an integer composition. St000766The number of inversions of an integer composition. St000769The major index of a composition regarded as a word. St000868The aid statistic in the sense of Shareshian-Wachs. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001115The number of even descents of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001377The major index minus the number of inversions of a permutation. St001403The number of vertical separators in a permutation. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001469The holeyness of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001639The number of alternating subsets such that applying the permutation does not yield an alternating subset. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001665The number of pure excedances of 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. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001911A descent variant minus the number of inversions. St001928The number of non-overlapping descents in a permutation. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000001The number of reduced words for a permutation. St000003The number of standard Young tableaux of the partition. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. St000054The first entry of the permutation. St000068The number of minimal elements in a poset. St000075The orbit size of a standard tableau under promotion. St000124The cardinality of the preimage of the Simion-Schmidt map. St000172The Grundy number of a graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000273The domination number of a graph. St000277The number of ribbon shaped standard tableaux. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000297The number of leading ones in a binary word. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000363The number of minimal vertex covers of a graph. St000382The first part of an integer composition. St000383The last part of an integer composition. St000388The number of orbits of vertices of a graph under automorphisms. St000390The number of runs of ones in a binary word. St000413The number of ordered trees with the same underlying unordered tree. St000418The number of Dyck paths that are weakly below a Dyck path. St000444The length of the maximal rise of a Dyck path. St000453The number of distinct Laplacian eigenvalues of a graph. St000468The Hosoya index of a graph. St000482The (zero)-forcing number of a graph. St000522The number of 1-protected nodes of a rooted tree. St000542The number of left-to-right-minima of a permutation. St000544The cop number of a graph. St000553The number of blocks of a graph. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000706The product of the factorials of the multiplicities of an integer partition. St000729The minimal arc length of a set partition. 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. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000767The number of runs in an integer composition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000820The number of compositions obtained by rotating the composition. St000847The number of standard Young tableaux whose descent set is the binary word. St000903The number of different parts of an integer composition. St000916The packing number of a graph. St000935The number of ordered refinements of an integer partition. St000972The composition number of a graph. St000993The multiplicity of the largest part of an integer partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001093The detour number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001285The number of primes in the column sums of the two line notation of a permutation. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001313The number of Dyck paths above the lattice path given by a binary word. St001352The number of internal nodes in the modular decomposition of a graph. 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. St001372The length of a longest cyclic run of ones of a binary word. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001494The Alon-Tarsi number of a graph. St001528The number of permutations such that the product with the permutation has the same number of fixed points. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001593This is the number of standard Young tableaux of the given shifted shape. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001725The harmonious chromatic number 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. St001780The order of promotion on the set of standard tableaux of given shape. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001829The common independence number of a graph. St001883The mutual visibility number of a graph. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001963The tree-depth of a graph. St000008The major index of the composition. St000012The area of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000057The Shynar inversion number of a standard tableau. St000081The number of edges of a graph. St000089The absolute variation of a composition. St000090The variation of a composition. 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. St000142The number of even parts of a partition. St000169The cocharge of a standard tableau. St000171The degree of the graph. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000185The weighted size of a partition. St000204The number of internal nodes of a binary tree. St000211The rank of the set partition. St000218The number of occurrences of the pattern 213 in a permutation. St000223The number of nestings in the permutation. St000225Difference between largest and smallest parts in a partition. St000234The number of global ascents of a permutation. St000245The number of ascents of a permutation. St000246The number of non-inversions of a permutation. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000272The treewidth of a graph. St000292The number of ascents of a binary word. St000336The leg major index of a standard tableau. St000340The number of non-final maximal constant sub-paths of length greater than one. St000348The non-inversion sum of a binary word. St000352The Elizalde-Pak rank of a permutation. St000358The number of occurrences of the pattern 31-2. St000359The number of occurrences of the pattern 23-1. St000361The second Zagreb index of a graph. St000362The size of a minimal vertex cover of a graph. St000366The number of double descents of a permutation. St000369The dinv deficit of a Dyck path. St000371The number of mid points of decreasing 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

$3$ . St000376The bounce deficit of a Dyck path. St000387The matching number of a graph. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000441The number of successions of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000448The number of pairs of vertices of a graph with distance 2. St000454The largest eigenvalue of a graph if it is integral. St000478Another weight of a partition according to Alladi. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000491The number of inversions of a set partition. St000492The rob statistic of a set partition. St000497The lcb statistic of a set partition. St000498The lcs statistic of a set partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000546The number of global descents of a permutation. St000552The number of cut vertices of a graph. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000556The 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. 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. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (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. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) 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. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is 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. St000648The number of 2-excedences of a permutation. St000651The maximal size of a rise in a permutation. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000665The number of rafts of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000682The Grundy value of Welter's game on a binary word. 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. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000692Babson and Steingrímsson's statistic of a permutation. St000731The number of double exceedences of a permutation. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000834The number of right outer peaks of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000877The depth of the binary word interpreted as a path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000944The 3-degree of an integer partition. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000984The number of boxes below precisely one peak. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001071The beta invariant of the graph. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001092The number of distinct even parts of a partition. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001120The length of a longest path in a graph. 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. St001176The size of a partition minus its first part. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001214The aft of an integer partition. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001305The number of induced cycles on four vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001333The cardinality of a minimal edge-isolating set of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001341The number of edges in the center of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001393The induced matching number of a graph. St001394The genus of a permutation. St001402The number of separators in a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001521Half the total irregularity of a graph. St001586The number of odd parts smaller than the largest even part in an integer partition. 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. St001638The book thickness of a graph. St001644The dimension of a graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001657The number of twos in an integer partition. St001673The degree of asymmetry of an integer composition. St001689The number of celebrities in a graph. St001695The natural comajor index of a standard Young tableau. St001697The shifted natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001727The number of invisible inversions of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001736The total number of cycles in a graph. St001743The discrepancy of a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001764The number of non-convex subsets of vertices in a graph. St001777The number of weak descents in an integer composition. St001792The arboricity of a graph. St001797The number of overfull subgraphs of a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001826The maximal number of leaves on a vertex of a graph. St001931The weak major index of an integer composition regarded as a word. St001961The sum of the greatest common divisors of all pairs of parts. St001962The proper pathwidth of a graph. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000572The dimension exponent of a set partition. 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. 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. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000100The number of linear extensions of a poset. St000402Half the size of the symmetry class of a permutation. St000485The length of the longest cycle of a permutation. St000529The number of permutations whose descent word is the given binary word. St000530The number of permutations with the same descent word as the given permutation. St000543The size of the conjugacy class of a binary word. St000619The number of cyclic descents of a permutation. St000626The minimal period of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000652The maximal difference between successive positions of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000983The length of the longest alternating subword. St000988The orbit size of a permutation under Foata's bijection. St001081The number of minimal length factorizations of a permutation into star transpositions. St001246The maximal difference between two consecutive entries of a permutation. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000354The number of recoils of a permutation. St000462The major index minus the number of excedences of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000539The number of odd inversions of a permutation. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. 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. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000624The normalized sum of the minimal distances to a greater element. St000628The balance of a binary word. St000646The number of big ascents of a permutation. St000691The number of changes of a binary word. St000795The mad of a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000833The comajor index of a permutation. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000287The number of connected components of a graph. St000456The monochromatic index of a connected graph. St000568The hook number of a binary tree. St000627The exponent of a binary word. St000690The size of the conjugacy class of a permutation. 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. St000886The number of permutations with the same antidiagonal sums. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000933The number of multipartitions of sizes given by an integer partition. St000982The length of the longest constant subword. St000990The first ascent of a permutation. St001052The length of the exterior of a permutation. St001062The maximal size of a block of a set partition. St001128The exponens consonantiae of a partition. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001312Number of parabolic noncrossing partitions indexed by the composition. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001415The length of the longest palindromic prefix of a binary word. St001463The number of distinct columns in the nullspace of a graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001592The maximal number of simple paths between any two different vertices of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001642The Prague dimension of a graph. St001675The number of parts equal to the part in the reversed composition. St001734The lettericity of a graph. St001884The number of borders of a binary word. St000248The number of anti-singletons of a set partition. St000254The nesting number of a set partition. St000289The decimal representation of a binary word. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000295The length of the border of a binary word. St000347The inversion sum of a binary word. St000353The number of inner valleys of a permutation. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000379The number of Hamiltonian cycles in a graph. 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. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 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. St000503The maximal difference between two elements in a common block. St000538The number of even inversions 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. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is 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. 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. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000653The last descent of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000728The dimension of a set partition. St000732The number of double deficiencies of a permutation. St000753The Grundy value for the game of Kayles on a binary word. St000779The tier of a permutation. St000794The mak of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of 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. St000836The number of descents of distance 2 of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000919The number of maximal left branches of a binary tree. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St001114The number of odd descents of a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001721The degree of a binary word. St001728The number of invisible descents of a permutation. St001742The difference of the maximal and the minimal degree in a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001308The number of induced paths on three vertices in a graph. St001323The independence gap of a graph. St001350Half of the Albertson index of a graph. St001552The number of inversions between excedances and fixed points of a permutation. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000781The number of proper colouring schemes of a Ferrers diagram. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001718The number of non-empty open intervals in a poset. St001877Number of indecomposable injective modules with projective dimension 2. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000439The position of the first down step of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000026The position of the first return of a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St000355The number of occurrences of the pattern 21-3. St000699The toughness times the least common multiple of 1,. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000219The number of occurrences of the pattern 231 in a permutation. St000264The girth of a graph, which is not a tree. St000306The bounce count of a Dyck path. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. 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

$eA$ . St001568The smallest positive integer that does not appear twice in the partition. St000150The floored half-sum of the multiplicities of a partition. St000257The number of distinct parts of a partition that occur at least twice. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000567The sum of the products of all pairs of parts. St000929The constant term of the character polynomial of an integer partition. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000667The greatest common divisor of the parts of the partition. St001192The maximal dimension of

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

$S$ over the corresponding Nakayama algebra

$A$ . 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. St001722The number of minimal chains with small intervals between a binary word and the top element. St000296The length of the symmetric border of a binary word. St000629The defect of a binary word. 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. 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. St000961The shifted major index of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001396Number of triples of incomparable elements in a finite poset. St001091The number of parts in an integer partition whose next smaller part has the same size. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001809The index of the step at the first peak of maximal height in a Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. 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. St000284The Plancherel distribution on integer partitions. St000655The length of the minimal rise of a Dyck path. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. 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. St000934The 2-degree of an integer partition. St001571The Cartan determinant of the integer partition. St000143The largest repeated part of a partition. St000941The number of characters of the symmetric group whose value on the partition is even. 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. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. 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. St000842The breadth of a permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St000678The number of up steps after the last double rise of a Dyck path. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St001130The number of two successive successions in a permutation. St000011The number of touch points (or returns) of a Dyck path. St000256The number of parts from which one can substract 2 and still get an integer partition. St001184Number of indecomposable injective modules with grade at least 1 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. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000365The number of double ascents of a permutation. St000932The number of occurrences of the pattern UDU in a Dyck path. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000117The number of centered tunnels of a Dyck path. St001082The number of boxed occurrences of 123 in a permutation. St001530The depth of a Dyck path. 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. 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. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 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. St000233The number of nestings of a set partition. St000617The number of global maxima of a Dyck path. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St001060The distinguishing index of a graph. St001316The domatic number of a graph. St000002The number of occurrences of the pattern 123 in a permutation. St000913The number of ways to refine the partition into singletons. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001075The minimal size of a block of a set partition. 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. St001196The global dimension of

$A$ minus the global dimension of

$eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000640The rank of the largest boolean interval in a poset. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St000025The number of initial rises of a Dyck path. St000105The number of blocks in the set partition. St000258The burning number of a graph. St000808The number of up steps of the associated bargraph. St000918The 2-limited packing number of a graph. St000971The smallest closer of a set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000053The number of valleys of the Dyck path. St000260The radius of a connected graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001271The competition number of a graph. St001481The minimal height of a peak of a Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000317The cycle descent number of a permutation. 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. St000020The rank of the permutation. St000079The number of alternating sign matrices for a given Dyck path. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000109The number of elements less than or equal to the given element in Bruhat order. St000325The width of the tree associated to a permutation. St000638The number of up-down runs of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001779The order of promotion on the set of linear extensions of a poset. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000055The inversion sum of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000222The number of alignments in the permutation. St000224The sorting index of a permutation. St000238The number of indices that are not small weak excedances. St000242The number of indices that are not cyclical small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000357The number of occurrences of the pattern 12-3. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001397Number of pairs of incomparable elements in a finite poset. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001874Lusztig's a-function for the symmetric group. St000397The Strahler number of a rooted tree. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St000981The length of the longest zigzag subpath. 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. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000086The number of subgraphs. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000299The number of nonisomorphic vertex-induced subtrees. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000343The number of spanning subgraphs of a graph. St000420The number of Dyck paths that are weakly above a Dyck path. St000443The number of long tunnels of a Dyck path. St000450The number of edges minus the number of vertices plus 2 of a graph. St000487The length of the shortest cycle of a permutation. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000700The protection number of an ordered tree. St000740The last entry of a permutation. St000806The semiperimeter of the associated bargraph. St000822The Hadwiger number of the graph. St000876The number of factors in the Catalan decomposition of a binary word. St000955Number of times one has

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

$i>0$ for the corresponding LNakayama algebra. St000959The number of strong Bruhat factorizations of a permutation. St000991The number of right-to-left minima of a permutation. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001624The breadth of a lattice. St001652The length of a longest interval of consecutive numbers. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001662The length of the longest factor of consecutive numbers in a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001808The box weight or horizontal decoration of a Dyck path. St000004The major index of a permutation. St000067The inversion number of the alternating sign matrix. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000095The number of triangles of a graph. St000133The "bounce" of a permutation. St000154The sum of the descent bottoms of a permutation. St000156The Denert index of a permutation. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by

$4$ . St000304The load of a permutation. St000305The inverse major index of a permutation. St000332The positive inversions of an alternating sign matrix. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000341The non-inversion sum of a permutation. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000523The number of 2-protected nodes of a rooted tree. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000674The number of hills of a Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000947The major index east count of a Dyck path. St001117The game chromatic index of a graph. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001298The number of repeated entries in the Lehmer code of a permutation. St001427The number of descents of a signed permutation. St001428The number of B-inversions of a signed permutation. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. 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. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001649The length of a longest trail in a graph. St001783The number of odd automorphisms of a graph. St001812The biclique partition number of a graph. St001869The maximum cut size of a graph. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between

$e_i J$ and

$e_j J$ (the radical of the indecomposable projective modules). St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. 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. St001174The Gorenstein dimension of the algebra

$A/I$ when

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

$K[x]/(x^n)$ . St000633The size of the automorphism group of a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000216The absolute length of a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001388The number of non-attacking neighbors of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000654The first descent of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000702The number of weak deficiencies of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001220The width of a permutation. St001346The number of parking functions that give the same permutation. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001531Number of partial orders contained in the poset determined by the Dyck path. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001959The product of the heights of the peaks of a Dyck path. St000461The rix statistic of a permutation. St000472The sum of the ascent bottoms of a permutation. St000837The number of ascents of distance 2 of a permutation. St000873The aix statistic of a permutation. St000989The number of final rises of a permutation. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001480The number of simple summands of the module J^2/J^3. St001731The factorization defect of a permutation. 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. St000045The number of linear extensions of a binary tree. St000145The Dyson rank of a partition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001162The minimum jump of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St000906The length of the shortest maximal chain in a poset. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000528The height of a poset. St001343The dimension of the reduced incidence algebra of a poset. St001717The largest size of an interval in a poset. St000750The number of occurrences of the pattern 4213 in a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000741The Colin de Verdière graph invariant. St000914The sum of the values of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001545The second Elser number of a connected graph. St001890The maximum magnitude of the Möbius function of a poset. St000516The number of stretching pairs of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001344The neighbouring number of a permutation. St001846The number of elements which do not have a complement in the lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St001964The interval resolution global dimension of a poset. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000908The length of the shortest maximal antichain in a poset. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000570The Edelman-Greene number of a permutation. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St000015The number of peaks of a Dyck path. St000084The number of subtrees. St000087The number of induced subgraphs. St000166The depth minus 1 of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000286The number of connected components of the complement of a graph. St000328The maximum number of child nodes in a tree. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000926The clique-coclique number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra 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. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000080The rank of the poset. St000094The depth of an ordered tree. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000331The number of upper interactions of a Dyck path. St000778The metric dimension of a graph. St000789The number of crossing-similar perfect matchings of a perfect matching. St001119The length of a shortest maximal path in a graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001340The cardinality of a minimal non-edge isolating set of a graph. St001345The Hamming dimension of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001949The rigidity index of a graph. St000664The number of right ropes of a permutation. St000061The number of nodes on the left branch of a binary tree. St001684The reduced word complexity of a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000545The number of parabolic double cosets with minimal element being the given permutation. 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

$c_{n−1}$ . St001281The normalized isoperimetric number of a graph. St001555The order of a signed permutation. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St000252The number of nodes of degree 3 of a binary tree. St000406The number of occurrences of the pattern 3241 in a permutation. St000872The number of very big descents of a permutation. St000963The 2-shifted major index of a permutation. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. 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. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001561The value of the elementary symmetric function evaluated at 1. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001769The reflection length of a signed permutation. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001835The number of occurrences of a 231 pattern in the restricted growth word of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001848The atomic length of a signed permutation. St001856The number of edges in the reduced word graph of a permutation. St001861The number of Bruhat lower covers of a permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001894The depth of a signed permutation. St001896The number of right descents of a signed permutations. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001557The number of inversions of the second entry of a permutation. St001569The maximal modular displacement of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001948The number of augmented double ascents of a permutation. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. 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

$eA$ . St000509The diagonal index (content) of a partition. St000477The weight of a partition according to Alladi. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000768The number of peaks in an integer composition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001770The number of facets of a certain subword complex associated with the signed permutation. St001866The nesting alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000748The major index of the permutation obtained by flattening the set partition. 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. St000805The number of peaks of the associated bargraph. 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

$K[x]/(x^n)$ . 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

$K[x]/(x^n)$ . St000074The number of special entries. St001556The number of inversions of the third entry of a permutation. St001821The sorting index of a signed permutation. St001822The number of alignments of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000022The number of fixed points of a permutation. St000635The number of strictly order preserving maps of a poset into itself. St000405The number of occurrences of the pattern 1324 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St001651The Frankl number of a lattice.