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Your data matches 533 different statistics following compositions of up to 3 maps.
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(click to perform a complete search on your data)
Matching statistic: St000966
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
St000966: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0
[1,0,1,0]
=> 0
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> 0
[1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0]
=> 0
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> 0
[1,1,1,0,0,1,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> 0
Description
Number of peaks minus the global dimension of the corresponding LNakayama algebra.
Matching statistic: St001175
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> 0
[1,0,1,0]
=> [1,2] => [2]
=> 0
[1,1,0,0]
=> [2,1] => [1,1]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [3]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 0
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1]
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,1]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,1]
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,1,1]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,1]
=> 0
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,1,1]
=> 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [4,1]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [4,1]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,2,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,1]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,1]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,1,1,1]
=> 0
Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St000386
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
Description
The number of factors DDU in a Dyck path.
Matching statistic: St001011
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
Description
Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001033
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,2] => [2]
=> [1,0,1,0]
=> 0
[1,1,0,0]
=> [2,1] => [1,1]
=> [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [3]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [1,0,1,1,0,0]
=> 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
Description
The normalized area of the parallelogram polyomino associated with the Dyck path.
The area of the smallest parallelogram polyomino equals the semilength of the Dyck path. This statistic is therefore the area of the parallelogram polyomino minus the semilength of the Dyck path.
The area itself is equidistributed with [[St001034]] and with [[St000395]].
Matching statistic: St001037
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St001596
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001596: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> [[1],[]]
=> 0
[1,0,1,0]
=> [1,2] => [1,1]
=> [[1,1],[]]
=> 0
[1,1,0,0]
=> [2,1] => [2]
=> [[2],[]]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [[1,1,1],[]]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [[2,1],[]]
=> 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [[2,1],[]]
=> 0
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [[2,1],[]]
=> 0
[1,1,1,0,0,0]
=> [3,2,1] => [3]
=> [[3],[]]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [[2,1,1],[]]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [[2,1,1],[]]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> [[2,1,1],[]]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,1]
=> [[3,1],[]]
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [[2,1,1],[]]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [[2,2],[]]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> [[2,1,1],[]]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> [[2,1,1],[]]
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> [[3,1],[]]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1]
=> [[3,1],[]]
=> 0
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [[3,1],[]]
=> 0
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [3,1]
=> [[3,1],[]]
=> 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4]
=> [[4],[]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [[2,2,1],[]]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,1]
=> [[4,1],[]]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [[2,2,1],[]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [[2,2,1],[]]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> [[2,2,1],[]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2]
=> [[3,2],[]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> [[2,2,1],[]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [[3,1,1],[]]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,1]
=> [[4,1],[]]
=> 0
Description
The number of two-by-two squares inside a skew partition.
This is, the number of cells $(i,j)$ in a skew partition for which the box $(i+1,j+1)$ is also a cell inside the skew partition.
Matching statistic: St001840
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001840: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001840: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> {{1}}
=> 0
[1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 1
Description
The number of descents of a set partition.
The Mahonian representation of a set partition $\{B_1,\dots,B_k\}$ of $\{1,\dots,n\}$ is the restricted growth word $w_1\dots w_n\}$ obtained by sorting the blocks of the set partition according to their maximal element, and setting $w_i$ to the index of the block containing $i$.
The word $w$ has a descent at position $i$ if $w_i > w_{i+1}$.
Matching statistic: St000345
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000345: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000345: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> []
=> 1 = 0 + 1
[1,0,1,0]
=> [1,2] => [1,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [2]
=> []
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [3]
=> []
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4]
=> []
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2]
=> [2]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,1]
=> [1]
=> 1 = 0 + 1
Description
The number of refinements of a partition.
A partition $\lambda$ refines a partition $\mu$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.
Matching statistic: St000935
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000935: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000935: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> []
=> 1 = 0 + 1
[1,0,1,0]
=> [1,2] => [1,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [2]
=> []
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [3]
=> []
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4]
=> []
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2]
=> [2]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> [2,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,1]
=> [1]
=> 1 = 0 + 1
Description
The number of ordered refinements of an integer partition.
This is, for an integer partition $\mu = (\mu_1,\ldots,\mu_n)$ the number of integer partition $\lambda = (\lambda_1,\ldots,\lambda_m)$ such that there are indices $1 = a_0 < \ldots < a_n = m$ with $\mu_j = \lambda_{a_{j-1}} + \ldots + \lambda_{a_j-1}$.
The following 523 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001471The magnitude of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St000919The number of maximal left branches of a binary tree. St000257The number of distinct parts of a partition that occur at least twice. St000662The staircase size of the code of a permutation. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000353The number of inner valleys of a permutation. St000354The number of recoils of a permutation. St000619The number of cyclic descents of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001469The holeyness of a permutation. St001665The number of pure excedances of a permutation. St000647The number of big descents of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000996The number of exclusive left-to-right maxima of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000306The bounce count of a Dyck path. St000862The number of parts of the shifted shape of a permutation. St001624The breadth of a lattice. St000035The number of left outer peaks of a permutation. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St001657The number of twos in an integer partition. St000052The number of valleys of a Dyck path not on the x-axis. St000225Difference between largest and smallest parts in a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000150The floored half-sum of the multiplicities of a partition. St000779The tier of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000834The number of right outer peaks of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St000711The number of big exceedences of a permutation. St000871The number of very big ascents of a permutation. St000872The number of very big descents of a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000245The number of ascents of a permutation. St000360The number of occurrences of the pattern 32-1. St000646The number of big ascents of a permutation. St001280The number of parts of an integer partition that are at least two. St001587Half of the largest even part of an integer partition. St001732The number of peaks visible from the left. St001394The genus of a permutation. St000534The number of 2-rises of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001096The size of the overlap set of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St000455The second largest eigenvalue of a graph if it is integral. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000661The number of rises of length 3 of a Dyck path. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000929The constant term of the character polynomial of an integer partition. St000117The number of centered tunnels of a Dyck path. St000292The number of ascents of a binary word. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001092The number of distinct even parts of a partition. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001163The number of simple modules with dominant dimension at least three 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. 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. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. 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. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001484The number of singletons of an integer partition. St001524The degree of symmetry of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St000664The number of right ropes of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St001549The number of restricted non-inversions between exceedances. St000358The number of occurrences of the pattern 31-2. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. 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. St001728The number of invisible descents of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001523The degree of symmetry of a Dyck path. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001513The number of nested exceedences of a permutation. St000356The number of occurrences of the pattern 13-2. St000658The number of rises of length 2 of a Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001498The normalised height of a Nakayama algebra with magnitude 1. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. 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. St001525The number of symmetric hooks on the diagonal of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000153The number of adjacent cycles of a permutation. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000731The number of double exceedences of a permutation. St001621The number of atoms of a lattice. St000714The number of semistandard Young tableau of given shape, with entries at most 2. 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. St001095The number of non-isomorphic posets with precisely one further covering relation. St000137The Grundy value of an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000618The number of self-evacuating tableaux of given shape. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. 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. St000781The number of proper colouring schemes of a Ferrers diagram. St000941The number of characters of the symmetric group whose value on the partition is even. St000944The 3-degree of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001128The exponens consonantiae of a partition. St001176The size of a partition minus its first part. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001383The BG-rank of an integer partition. St001432The order dimension of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001561The value of the elementary symmetric function evaluated at 1. St001568The smallest positive integer that does not appear twice in the partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001780The order of promotion on the set of standard tableaux of given shape. St001901The largest multiplicity of an irreducible representation 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. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001877Number of indecomposable injective modules with projective dimension 2. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St000761The number of ascents in an integer composition. St000666The number of right tethers of a permutation. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St000516The number of stretching pairs of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000252The number of nodes of degree 3 of a binary tree. 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. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001470The cyclic holeyness of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000325The width of the tree associated to a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St000039The number of crossings of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000317The cycle descent number of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. 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. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. 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. St001298The number of repeated entries in the Lehmer code of a permutation. St001427The number of descents of a signed permutation. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001577The minimal number of edges to add or remove to make a graph a cograph. St001871The number of triconnected components of a graph. St000015The number of peaks of a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. 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:
St001235The global dimension of the corresponding Comp-Nakayama algebra. St001530The depth of a Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000649The number of 3-excedences of a permutation. St000535The rank-width of a graph. St000544The cop number of a graph. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St000629The defect of a binary word. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001393The induced matching number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St000331The number of upper interactions of a Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. 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. 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. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000408The number of occurrences of the pattern 4231 in a permutation. St000842The breadth of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001489The maximum of the number of descents and the number of inverse descents. St001353The number of prime nodes in the modular decomposition of a graph. St000007The number of saliances of the permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000017The number of inversions of a standard tableau. St000864The number of circled entries of the shifted recording tableau of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of 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. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001520The number of strict 3-descents. St000223The number of nestings in the permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000542The number of left-to-right-minima of a permutation. St000232The number of crossings of a set partition. St000233The number of nestings of a set partition. St000496The rcs statistic of a set partition. 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. St000091The descent variation of a composition. St001781The interlacing number of a set partition. St001839The number of excedances 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. St001330The hat guessing number of a graph. St000787The number of flips required to make a perfect matching noncrossing. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} 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. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (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. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. 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. St000748The major index of the permutation obtained by flattening the set partition. St000627The exponent of a binary word. St001162The minimum jump of a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000920The logarithmic height of a Dyck path. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St000069The number of maximal elements of a poset. St000031The number of cycles in the cycle decomposition of a permutation. St000883The number of longest increasing subsequences of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St000355The number of occurrences of the pattern 21-3. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000648The number of 2-excedences of a permutation. St000663The number of right floats of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001550The number of inversions between exceedances where the greater exceedance is linked. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001715The number of non-records in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001344The neighbouring number of a permutation. St000732The number of double deficiencies of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001720The minimal length of a chain of small intervals in a lattice. St000650The number of 3-rises of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001846The number of elements which do not have a complement in the lattice. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000768The number of peaks in an integer composition. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001695The 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. St001964The interval resolution global dimension of a poset. St001875The number of simple modules with projective dimension at most 1. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000068The number of minimal elements in a poset. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000002The number of occurrences of the pattern 123 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001570The minimal number of edges to add to make a graph Hamiltonian. St000058The order of a permutation. St000260The radius of a connected graph. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000314The number of left-to-right-maxima of a permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000022The number of fixed points of a permutation. 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. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000359The number of occurrences of the pattern 23-1. St000366The number of double descents of a permutation. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St001130The number of two successive successions in a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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. 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). St001896The number of right descents of a signed permutations. St000570The Edelman-Greene number of a permutation. St000669The number of permutations obtained by switching ascents or descents of size 2. St000702The number of weak deficiencies of a permutation. St000991The number of right-to-left minima of a permutation. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001735The number of permutations with the same set of runs. St001778The largest greatest common divisor of an element and its image in a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001487The number of inner corners of a skew partition. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St001557The number of inversions of the second entry of a permutation. St000527The width of the poset. St001569The maximal modular displacement of a permutation. St000258The burning number of a graph. 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 $eA$. St001371The length of the longest Yamanouchi prefix of a binary word. St000660The number of rises of length at least 3 of a Dyck path. St000181The number of connected components of the Hasse diagram for the poset. St000632The jump number of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001399The distinguishing number of a poset. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001271The competition number of a graph. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St000914The sum of the values of the Möbius function of a poset. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001862The number of crossings of a signed permutation. St001866The nesting alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000741The Colin de Verdière graph invariant. St000383The last part of an integer composition. St001867The number of alignments of type EN of a signed permutation. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St000805The number of peaks of the associated bargraph. St000900The minimal number of repetitions of a part in an integer composition. 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 $c_{n−1}$. St000291The number of descents of a binary word. St000338The number of pixed points of a permutation. St000348The non-inversion sum of a binary word. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000461The rix statistic of a permutation. St000488The number of cycles of a permutation of length at most 2. St000546The number of global descents of a permutation. St000682The Grundy value of Welter's game on a binary word. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000877The depth of the binary word interpreted as a path. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. 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. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001705The number of occurrences of the pattern 2413 in a permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001821The sorting index of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001856The number of edges in the reduced word graph of a permutation. St001948The number of augmented double ascents of a permutation. St000335The difference of lower and upper interactions. St000390The number of runs of ones in a binary word. St000654The first descent of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000902 The minimal number of repetitions of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000983The length of the longest alternating subword. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001313The number of Dyck paths above the lattice path given by a binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001481The minimal height of a peak of a Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001768The number of reduced words of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001884The number of borders of a binary word. St001889The size of the connectivity set of a signed permutation. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000990The first ascent of a permutation. 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. St001488The number of corners of a skew partition. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000498The lcs statistic of a set partition. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. 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. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. 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. 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. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000011The number of touch points (or returns) of a Dyck path. St000729The minimal arc length of a set partition. St000889The number of alternating sign matrices with the same antidiagonal sums. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set 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. St000326The position of the first one in a binary word after appending a 1 at the end. St000352The Elizalde-Pak rank of a permutation. St000054The first entry of the permutation. St000635The number of strictly order preserving maps of a poset into itself.
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