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Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St000943
St000943: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,1] => 1
[1,2] => 0
[2,1] => 0
[1,1,1] => 2
[1,1,2] => 1
[1,2,1] => 2
[2,1,1] => 2
[1,1,3] => 1
[1,3,1] => 1
[3,1,1] => 1
[1,2,2] => 1
[2,1,2] => 1
[2,2,1] => 1
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,1,1,1] => 3
[1,1,1,2] => 2
[1,1,2,1] => 3
[1,2,1,1] => 3
[2,1,1,1] => 3
[1,1,1,3] => 2
[1,1,3,1] => 3
[1,3,1,1] => 3
[3,1,1,1] => 3
[1,1,1,4] => 2
[1,1,4,1] => 2
[1,4,1,1] => 2
[4,1,1,1] => 2
[1,1,2,2] => 2
[1,2,1,2] => 2
[1,2,2,1] => 3
[2,1,1,2] => 2
[2,1,2,1] => 3
[2,2,1,1] => 3
[1,1,2,3] => 1
[1,1,3,2] => 2
[1,2,1,3] => 2
[1,2,3,1] => 3
[1,3,1,2] => 2
[1,3,2,1] => 3
[2,1,1,3] => 2
[2,1,3,1] => 3
[2,3,1,1] => 3
[3,1,1,2] => 2
[3,1,2,1] => 3
Description
The number of spots the most unlucky car had to go further in a parking function.
Matching statistic: St001630
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 20%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => [[1],[]]
=> ([],1)
=> ? = 0
[1,1] => [2] => [[2],[]]
=> ([],1)
=> ? = 1
[1,2] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? = 0
[2,1] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? = 0
[1,1,1] => [3] => [[3],[]]
=> ([],1)
=> ? = 2
[1,1,2] => [3] => [[3],[]]
=> ([],1)
=> ? = 1
[1,2,1] => [3] => [[3],[]]
=> ([],1)
=> ? = 2
[2,1,1] => [3] => [[3],[]]
=> ([],1)
=> ? = 2
[1,1,3] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? = 1
[1,3,1] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? = 1
[3,1,1] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? = 1
[1,2,2] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? = 1
[2,1,2] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? = 1
[2,2,1] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? = 1
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0
[2,1,3] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0
[2,3,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0
[3,1,2] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0
[3,2,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0
[1,1,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[1,1,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2
[1,1,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[1,2,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[2,1,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[1,1,1,3] => [4] => [[4],[]]
=> ([],1)
=> ? = 2
[1,1,3,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[1,3,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[3,1,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2
[1,1,4,1] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2
[1,4,1,1] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2
[4,1,1,1] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2
[1,1,2,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2
[1,2,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2
[1,2,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[2,1,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2
[2,1,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[2,2,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[1,1,2,3] => [4] => [[4],[]]
=> ([],1)
=> ? = 1
[1,1,3,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2
[1,2,1,3] => [4] => [[4],[]]
=> ([],1)
=> ? = 2
[1,2,3,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[1,3,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2
[1,3,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[2,1,1,3] => [4] => [[4],[]]
=> ([],1)
=> ? = 2
[2,1,3,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[2,3,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[3,1,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2
[3,1,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[1,1,3,3,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,3,5,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,5,3,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,1,3,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,1,5,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,3,1,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,3,5,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,5,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,5,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,5,1,3,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,5,3,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,5,3,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,1,3,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,1,5,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,3,1,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,3,5,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,5,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,5,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,1,1,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,1,5,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,5,1,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,5,1,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,5,1,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,5,3,1,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,1,1,3,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,1,3,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,1,3,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,3,1,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,3,1,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,3,3,1,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,4,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,4,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,4,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,4,2,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,4,2,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,4,4,2,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,2,4,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,4,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,4,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1,4,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,4,1,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,4,4,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,4,1,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,4,1,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,4,2,1,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,4,2,4,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,4,4,1,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,4,4,2,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[4,1,2,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[4,1,2,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001878
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 20%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => [[1],[]]
=> ([],1)
=> ? = 0
[1,1] => [2] => [[2],[]]
=> ([],1)
=> ? = 1
[1,2] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? = 0
[2,1] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? = 0
[1,1,1] => [3] => [[3],[]]
=> ([],1)
=> ? = 2
[1,1,2] => [3] => [[3],[]]
=> ([],1)
=> ? = 1
[1,2,1] => [3] => [[3],[]]
=> ([],1)
=> ? = 2
[2,1,1] => [3] => [[3],[]]
=> ([],1)
=> ? = 2
[1,1,3] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? = 1
[1,3,1] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? = 1
[3,1,1] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? = 1
[1,2,2] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? = 1
[2,1,2] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? = 1
[2,2,1] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? = 1
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0
[2,1,3] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0
[2,3,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0
[3,1,2] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0
[3,2,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0
[1,1,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[1,1,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2
[1,1,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[1,2,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[2,1,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[1,1,1,3] => [4] => [[4],[]]
=> ([],1)
=> ? = 2
[1,1,3,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[1,3,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[3,1,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2
[1,1,4,1] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2
[1,4,1,1] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2
[4,1,1,1] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2
[1,1,2,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2
[1,2,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2
[1,2,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[2,1,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2
[2,1,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[2,2,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[1,1,2,3] => [4] => [[4],[]]
=> ([],1)
=> ? = 1
[1,1,3,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2
[1,2,1,3] => [4] => [[4],[]]
=> ([],1)
=> ? = 2
[1,2,3,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[1,3,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2
[1,3,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[2,1,1,3] => [4] => [[4],[]]
=> ([],1)
=> ? = 2
[2,1,3,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[2,3,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[3,1,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2
[3,1,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3
[1,1,3,3,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,3,5,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,5,3,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,1,3,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,1,5,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,3,1,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,3,5,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,5,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,5,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,5,1,3,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,5,3,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,5,3,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,1,3,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,1,5,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,3,1,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,3,5,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,5,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,5,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,1,1,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,1,5,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,5,1,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,5,1,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,5,1,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,5,3,1,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,1,1,3,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,1,3,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,1,3,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,3,1,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,3,1,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,3,3,1,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,4,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,4,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,4,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,4,2,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,4,2,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,4,4,2,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,2,4,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,4,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,4,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1,4,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,4,1,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,4,4,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,4,1,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,4,1,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,4,2,1,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,4,2,4,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,4,4,1,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,4,4,2,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[4,1,2,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[4,1,2,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001876
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 20%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => [[1],[]]
=> ([],1)
=> ? = 0 - 1
[1,1] => [2] => [[2],[]]
=> ([],1)
=> ? = 1 - 1
[1,2] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? = 0 - 1
[2,1] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? = 0 - 1
[1,1,1] => [3] => [[3],[]]
=> ([],1)
=> ? = 2 - 1
[1,1,2] => [3] => [[3],[]]
=> ([],1)
=> ? = 1 - 1
[1,2,1] => [3] => [[3],[]]
=> ([],1)
=> ? = 2 - 1
[2,1,1] => [3] => [[3],[]]
=> ([],1)
=> ? = 2 - 1
[1,1,3] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? = 1 - 1
[1,3,1] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? = 1 - 1
[3,1,1] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? = 1 - 1
[1,2,2] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? = 1 - 1
[2,1,2] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? = 1 - 1
[2,2,1] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? = 1 - 1
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0 - 1
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0 - 1
[2,1,3] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0 - 1
[2,3,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0 - 1
[3,1,2] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0 - 1
[3,2,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0 - 1
[1,1,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[1,1,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 - 1
[1,1,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[1,2,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[2,1,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[1,1,1,3] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 - 1
[1,1,3,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[1,3,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[3,1,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2 - 1
[1,1,4,1] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2 - 1
[1,4,1,1] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2 - 1
[4,1,1,1] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2 - 1
[1,1,2,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 - 1
[1,2,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 - 1
[1,2,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[2,1,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 - 1
[2,1,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[2,2,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[1,1,2,3] => [4] => [[4],[]]
=> ([],1)
=> ? = 1 - 1
[1,1,3,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 - 1
[1,2,1,3] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 - 1
[1,2,3,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[1,3,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 - 1
[1,3,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[2,1,1,3] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 - 1
[2,1,3,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[2,3,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[3,1,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 - 1
[3,1,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[1,1,3,3,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,3,5,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,5,3,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,3,1,3,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,3,1,5,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,3,3,1,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,3,3,5,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,3,5,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,3,5,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,5,1,3,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,5,3,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,5,3,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,1,1,3,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,1,1,5,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,1,3,1,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,1,3,5,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,1,5,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,1,5,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,3,1,1,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,3,1,5,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,3,5,1,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,5,1,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,5,1,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,5,3,1,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[5,1,1,3,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[5,1,3,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[5,1,3,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[5,3,1,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[5,3,1,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[5,3,3,1,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,2,2,4,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,2,4,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,2,4,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,4,2,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,4,2,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,4,4,2,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,1,2,4,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,1,4,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,1,4,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,2,1,4,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,2,4,1,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,2,4,4,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,4,1,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,4,1,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,4,2,1,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,4,2,4,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,4,4,1,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,4,4,2,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[4,1,2,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[4,1,2,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St001877
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 20%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => [[1],[]]
=> ([],1)
=> ? = 0 - 1
[1,1] => [2] => [[2],[]]
=> ([],1)
=> ? = 1 - 1
[1,2] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? = 0 - 1
[2,1] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? = 0 - 1
[1,1,1] => [3] => [[3],[]]
=> ([],1)
=> ? = 2 - 1
[1,1,2] => [3] => [[3],[]]
=> ([],1)
=> ? = 1 - 1
[1,2,1] => [3] => [[3],[]]
=> ([],1)
=> ? = 2 - 1
[2,1,1] => [3] => [[3],[]]
=> ([],1)
=> ? = 2 - 1
[1,1,3] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? = 1 - 1
[1,3,1] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? = 1 - 1
[3,1,1] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? = 1 - 1
[1,2,2] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? = 1 - 1
[2,1,2] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? = 1 - 1
[2,2,1] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? = 1 - 1
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0 - 1
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0 - 1
[2,1,3] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0 - 1
[2,3,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0 - 1
[3,1,2] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0 - 1
[3,2,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0 - 1
[1,1,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[1,1,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 - 1
[1,1,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[1,2,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[2,1,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[1,1,1,3] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 - 1
[1,1,3,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[1,3,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[3,1,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2 - 1
[1,1,4,1] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2 - 1
[1,4,1,1] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2 - 1
[4,1,1,1] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2 - 1
[1,1,2,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 - 1
[1,2,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 - 1
[1,2,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[2,1,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 - 1
[2,1,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[2,2,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[1,1,2,3] => [4] => [[4],[]]
=> ([],1)
=> ? = 1 - 1
[1,1,3,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 - 1
[1,2,1,3] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 - 1
[1,2,3,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[1,3,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 - 1
[1,3,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[2,1,1,3] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 - 1
[2,1,3,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[2,3,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[3,1,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 - 1
[3,1,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 - 1
[1,1,3,3,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,3,5,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,1,5,3,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,3,1,3,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,3,1,5,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,3,3,1,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,3,3,5,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,3,5,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,3,5,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,5,1,3,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,5,3,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,5,3,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,1,1,3,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,1,1,5,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,1,3,1,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,1,3,5,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,1,5,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,1,5,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,3,1,1,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,3,1,5,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,3,5,1,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,5,1,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,5,1,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,5,3,1,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[5,1,1,3,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[5,1,3,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[5,1,3,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[5,3,1,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[5,3,1,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[5,3,3,1,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,2,2,4,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,2,4,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,2,4,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,4,2,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,4,2,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,4,4,2,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,1,2,4,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,1,4,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,1,4,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,2,1,4,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,2,4,1,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,2,4,4,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,4,1,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,4,1,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,4,2,1,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,4,2,4,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,4,4,1,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,4,4,2,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[4,1,2,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[4,1,2,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St001875
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 20%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => [[1],[]]
=> ([],1)
=> ? = 0 + 2
[1,1] => [2] => [[2],[]]
=> ([],1)
=> ? = 1 + 2
[1,2] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? = 0 + 2
[2,1] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? = 0 + 2
[1,1,1] => [3] => [[3],[]]
=> ([],1)
=> ? = 2 + 2
[1,1,2] => [3] => [[3],[]]
=> ([],1)
=> ? = 1 + 2
[1,2,1] => [3] => [[3],[]]
=> ([],1)
=> ? = 2 + 2
[2,1,1] => [3] => [[3],[]]
=> ([],1)
=> ? = 2 + 2
[1,1,3] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? = 1 + 2
[1,3,1] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? = 1 + 2
[3,1,1] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? = 1 + 2
[1,2,2] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? = 1 + 2
[2,1,2] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? = 1 + 2
[2,2,1] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? = 1 + 2
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0 + 2
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0 + 2
[2,1,3] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0 + 2
[2,3,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0 + 2
[3,1,2] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0 + 2
[3,2,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? = 0 + 2
[1,1,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 + 2
[1,1,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 + 2
[1,1,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 + 2
[1,2,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 + 2
[2,1,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 + 2
[1,1,1,3] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 + 2
[1,1,3,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 + 2
[1,3,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 + 2
[3,1,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 + 2
[1,1,1,4] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2 + 2
[1,1,4,1] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2 + 2
[1,4,1,1] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2 + 2
[4,1,1,1] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? = 2 + 2
[1,1,2,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 + 2
[1,2,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 + 2
[1,2,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 + 2
[2,1,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 + 2
[2,1,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 + 2
[2,2,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 + 2
[1,1,2,3] => [4] => [[4],[]]
=> ([],1)
=> ? = 1 + 2
[1,1,3,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 + 2
[1,2,1,3] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 + 2
[1,2,3,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 + 2
[1,3,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 + 2
[1,3,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 + 2
[2,1,1,3] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 + 2
[2,1,3,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 + 2
[2,3,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 + 2
[3,1,1,2] => [4] => [[4],[]]
=> ([],1)
=> ? = 2 + 2
[3,1,2,1] => [4] => [[4],[]]
=> ([],1)
=> ? = 3 + 2
[1,1,3,3,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[1,1,3,5,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[1,1,5,3,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[1,3,1,3,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[1,3,1,5,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[1,3,3,1,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[1,3,3,5,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[1,3,5,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[1,3,5,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[1,5,1,3,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[1,5,3,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[1,5,3,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[3,1,1,3,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[3,1,1,5,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[3,1,3,1,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[3,1,3,5,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[3,1,5,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[3,1,5,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[3,3,1,1,5] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[3,3,1,5,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[3,3,5,1,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[3,5,1,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[3,5,1,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[3,5,3,1,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[5,1,1,3,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[5,1,3,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[5,1,3,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[5,3,1,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[5,3,1,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[5,3,3,1,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[1,2,2,4,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[1,2,4,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[1,2,4,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[1,4,2,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[1,4,2,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[1,4,4,2,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[2,1,2,4,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[2,1,4,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[2,1,4,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[2,2,1,4,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[2,2,4,1,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[2,2,4,4,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[2,4,1,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[2,4,1,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[2,4,2,1,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[2,4,2,4,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[2,4,4,1,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[2,4,4,2,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[4,1,2,2,4] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
[4,1,2,4,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 3 = 1 + 2
Description
The number of simple modules with projective dimension at most 1.
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