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Matching statistic: St001937
St001937: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 2
[1,2] => 2
[2,1] => 1
[1,1,1] => 3
[1,1,2] => 3
[1,2,1] => 3
[2,1,1] => 2
[1,1,3] => 3
[1,3,1] => 2
[3,1,1] => 2
[1,2,2] => 3
[2,1,2] => 2
[2,2,1] => 1
[1,2,3] => 3
[1,3,2] => 2
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 2
[3,2,1] => 1
[1,1,1,1] => 4
[1,1,1,2] => 4
[1,1,2,1] => 4
[1,2,1,1] => 4
[2,1,1,1] => 3
[1,1,1,3] => 4
[1,1,3,1] => 4
[1,3,1,1] => 3
[3,1,1,1] => 3
[1,1,1,4] => 4
[1,1,4,1] => 3
[1,4,1,1] => 3
[4,1,1,1] => 3
[1,1,2,2] => 4
[1,2,1,2] => 4
[1,2,2,1] => 4
[2,1,1,2] => 3
[2,1,2,1] => 3
[2,2,1,1] => 2
[1,1,2,3] => 4
[1,1,3,2] => 4
[1,2,1,3] => 4
[1,2,3,1] => 4
[1,3,1,2] => 3
[1,3,2,1] => 3
[2,1,1,3] => 3
[2,1,3,1] => 2
[2,3,1,1] => 2
[3,1,1,2] => 3
[3,1,2,1] => 3
Description
The size of the center of a parking function.
The center of a parking function $p_1,\dots,p_n$ is the longest subsequence $a_1,\dots,a_k$ such that $a_i\leq i$.
Matching statistic: St001880
Mp00302: Parking functions —insertion tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 60%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 60%
Values
[1] => [[1]]
=> [1] => ([],1)
=> ? = 1
[1,1] => [[1,1]]
=> [1,2] => ([(0,1)],2)
=> ? = 2
[1,2] => [[1,2]]
=> [1,2] => ([(0,1)],2)
=> ? = 2
[2,1] => [[1],[2]]
=> [2,1] => ([],2)
=> ? = 1
[1,1,1] => [[1,1,1]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[1,1,2] => [[1,1,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[1,2,1] => [[1,1],[2]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 3
[2,1,1] => [[1,1],[2]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 2
[1,1,3] => [[1,1,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[1,3,1] => [[1,1],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 2
[3,1,1] => [[1,1],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 2
[1,2,2] => [[1,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[2,1,2] => [[1,2],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ? = 2
[2,2,1] => [[1,2],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ? = 1
[1,2,3] => [[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[1,3,2] => [[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 2
[2,1,3] => [[1,3],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ? = 1
[2,3,1] => [[1,3],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ? = 1
[3,1,2] => [[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 2
[3,2,1] => [[1],[2],[3]]
=> [3,2,1] => ([],3)
=> ? = 1
[1,1,1,1] => [[1,1,1,1]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,1,2] => [[1,1,1,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,2,1] => [[1,1,1],[2]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 4
[1,2,1,1] => [[1,1,1],[2]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 4
[2,1,1,1] => [[1,1,1],[2]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 3
[1,1,1,3] => [[1,1,1,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,3,1] => [[1,1,1],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 4
[1,3,1,1] => [[1,1,1],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 3
[3,1,1,1] => [[1,1,1],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 3
[1,1,1,4] => [[1,1,1,4]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,4,1] => [[1,1,1],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 3
[1,4,1,1] => [[1,1,1],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 3
[4,1,1,1] => [[1,1,1],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 3
[1,1,2,2] => [[1,1,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,1,2] => [[1,1,2],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? = 4
[1,2,2,1] => [[1,1,2],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? = 4
[2,1,1,2] => [[1,1,2],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[2,1,2,1] => [[1,1],[2,2]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? = 3
[2,2,1,1] => [[1,1],[2,2]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? = 2
[1,1,2,3] => [[1,1,2,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,3,2] => [[1,1,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 4
[1,2,1,3] => [[1,1,3],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? = 4
[1,2,3,1] => [[1,1,3],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? = 4
[1,3,1,2] => [[1,1,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 3
[1,3,2,1] => [[1,1],[2],[3]]
=> [4,3,1,2] => ([(2,3)],4)
=> ? = 3
[2,1,1,3] => [[1,1,3],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[2,1,3,1] => [[1,1],[2,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? = 2
[2,3,1,1] => [[1,1],[2,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? = 2
[3,1,1,2] => [[1,1,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 3
[3,1,2,1] => [[1,1],[2],[3]]
=> [4,3,1,2] => ([(2,3)],4)
=> ? = 3
[3,2,1,1] => [[1,1],[2],[3]]
=> [4,3,1,2] => ([(2,3)],4)
=> ? = 2
[1,1,2,4] => [[1,1,2,4]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,4,2] => [[1,1,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 3
[1,2,1,4] => [[1,1,4],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? = 4
[1,2,4,1] => [[1,1,4],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[1,4,1,2] => [[1,1,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 3
[1,4,2,1] => [[1,1],[2],[4]]
=> [4,3,1,2] => ([(2,3)],4)
=> ? = 3
[2,1,1,4] => [[1,1,4],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[2,1,4,1] => [[1,1],[2,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? = 2
[2,4,1,1] => [[1,1],[2,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? = 2
[4,1,1,2] => [[1,1,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 3
[4,1,2,1] => [[1,1],[2],[4]]
=> [4,3,1,2] => ([(2,3)],4)
=> ? = 3
[1,1,3,3] => [[1,1,3,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,3,4] => [[1,1,3,4]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,2,2] => [[1,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,2,3] => [[1,2,2,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,2,4] => [[1,2,2,4]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,3,3] => [[1,2,3,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,1,1,1] => [[1,1,1,1,1]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,1,1,2] => [[1,1,1,1,2]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,1,1,3] => [[1,1,1,1,3]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,1,1,4] => [[1,1,1,1,4]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,1,1,5] => [[1,1,1,1,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,1,2,2] => [[1,1,1,2,2]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,1,2,3] => [[1,1,1,2,3]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,1,2,4] => [[1,1,1,2,4]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,1,2,5] => [[1,1,1,2,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,1,3,3] => [[1,1,1,3,3]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,1,3,4] => [[1,1,1,3,4]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,1,3,5] => [[1,1,1,3,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,1,4,4] => [[1,1,1,4,4]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,1,4,5] => [[1,1,1,4,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,2,2,2] => [[1,1,2,2,2]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,2,2,3] => [[1,1,2,2,3]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,2,2,4] => [[1,1,2,2,4]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,2,2,5] => [[1,1,2,2,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,2,3,3] => [[1,1,2,3,3]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,2,3,4] => [[1,1,2,3,4]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,2,3,5] => [[1,1,2,3,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,2,4,4] => [[1,1,2,4,4]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,2,4,5] => [[1,1,2,4,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,3,3,3] => [[1,1,3,3,3]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,3,3,4] => [[1,1,3,3,4]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,3,3,5] => [[1,1,3,3,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,3,4,4] => [[1,1,3,4,4]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,3,4,5] => [[1,1,3,4,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,2,2,2] => [[1,2,2,2,2]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,2,2,3] => [[1,2,2,2,3]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,2,2,4] => [[1,2,2,2,4]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001879
Mp00302: Parking functions —insertion tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 60%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 60%
Values
[1] => [[1]]
=> [1] => ([],1)
=> ? = 1 - 1
[1,1] => [[1,1]]
=> [1,2] => ([(0,1)],2)
=> ? = 2 - 1
[1,2] => [[1,2]]
=> [1,2] => ([(0,1)],2)
=> ? = 2 - 1
[2,1] => [[1],[2]]
=> [2,1] => ([],2)
=> ? = 1 - 1
[1,1,1] => [[1,1,1]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,1,2] => [[1,1,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2,1] => [[1,1],[2]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 3 - 1
[2,1,1] => [[1,1],[2]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 2 - 1
[1,1,3] => [[1,1,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,1] => [[1,1],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 2 - 1
[3,1,1] => [[1,1],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 2 - 1
[1,2,2] => [[1,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,1,2] => [[1,2],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ? = 2 - 1
[2,2,1] => [[1,2],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ? = 1 - 1
[1,2,3] => [[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,3,2] => [[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 2 - 1
[2,1,3] => [[1,3],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ? = 1 - 1
[2,3,1] => [[1,3],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ? = 1 - 1
[3,1,2] => [[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 2 - 1
[3,2,1] => [[1],[2],[3]]
=> [3,2,1] => ([],3)
=> ? = 1 - 1
[1,1,1,1] => [[1,1,1,1]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,1,1,2] => [[1,1,1,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,1,2,1] => [[1,1,1],[2]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 4 - 1
[1,2,1,1] => [[1,1,1],[2]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 4 - 1
[2,1,1,1] => [[1,1,1],[2]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 3 - 1
[1,1,1,3] => [[1,1,1,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,1,3,1] => [[1,1,1],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 4 - 1
[1,3,1,1] => [[1,1,1],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 3 - 1
[3,1,1,1] => [[1,1,1],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 3 - 1
[1,1,1,4] => [[1,1,1,4]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,1,4,1] => [[1,1,1],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 3 - 1
[1,4,1,1] => [[1,1,1],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 3 - 1
[4,1,1,1] => [[1,1,1],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 3 - 1
[1,1,2,2] => [[1,1,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,1,2] => [[1,1,2],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 1
[1,2,2,1] => [[1,1,2],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 1
[2,1,1,2] => [[1,1,2],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[2,1,2,1] => [[1,1],[2,2]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? = 3 - 1
[2,2,1,1] => [[1,1],[2,2]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? = 2 - 1
[1,1,2,3] => [[1,1,2,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,1,3,2] => [[1,1,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 4 - 1
[1,2,1,3] => [[1,1,3],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 1
[1,2,3,1] => [[1,1,3],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 1
[1,3,1,2] => [[1,1,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 3 - 1
[1,3,2,1] => [[1,1],[2],[3]]
=> [4,3,1,2] => ([(2,3)],4)
=> ? = 3 - 1
[2,1,1,3] => [[1,1,3],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[2,1,3,1] => [[1,1],[2,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? = 2 - 1
[2,3,1,1] => [[1,1],[2,3]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? = 2 - 1
[3,1,1,2] => [[1,1,2],[3]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 3 - 1
[3,1,2,1] => [[1,1],[2],[3]]
=> [4,3,1,2] => ([(2,3)],4)
=> ? = 3 - 1
[3,2,1,1] => [[1,1],[2],[3]]
=> [4,3,1,2] => ([(2,3)],4)
=> ? = 2 - 1
[1,1,2,4] => [[1,1,2,4]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,1,4,2] => [[1,1,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 3 - 1
[1,2,1,4] => [[1,1,4],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 1
[1,2,4,1] => [[1,1,4],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? = 3 - 1
[1,4,1,2] => [[1,1,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 3 - 1
[1,4,2,1] => [[1,1],[2],[4]]
=> [4,3,1,2] => ([(2,3)],4)
=> ? = 3 - 1
[2,1,1,4] => [[1,1,4],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? = 2 - 1
[2,1,4,1] => [[1,1],[2,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? = 2 - 1
[2,4,1,1] => [[1,1],[2,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? = 2 - 1
[4,1,1,2] => [[1,1,2],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 3 - 1
[4,1,2,1] => [[1,1],[2],[4]]
=> [4,3,1,2] => ([(2,3)],4)
=> ? = 3 - 1
[1,1,3,3] => [[1,1,3,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,1,3,4] => [[1,1,3,4]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,2,2] => [[1,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,2,3] => [[1,2,2,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,2,4] => [[1,2,2,4]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,3,3] => [[1,2,3,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,1,1,1,1] => [[1,1,1,1,1]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,1,1,2] => [[1,1,1,1,2]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,1,1,3] => [[1,1,1,1,3]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,1,1,4] => [[1,1,1,1,4]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,1,1,5] => [[1,1,1,1,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,1,2,2] => [[1,1,1,2,2]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,1,2,3] => [[1,1,1,2,3]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,1,2,4] => [[1,1,1,2,4]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,1,2,5] => [[1,1,1,2,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,1,3,3] => [[1,1,1,3,3]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,1,3,4] => [[1,1,1,3,4]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,1,3,5] => [[1,1,1,3,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,1,4,4] => [[1,1,1,4,4]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,1,4,5] => [[1,1,1,4,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,2,2,2] => [[1,1,2,2,2]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,2,2,3] => [[1,1,2,2,3]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,2,2,4] => [[1,1,2,2,4]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,2,2,5] => [[1,1,2,2,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,2,3,3] => [[1,1,2,3,3]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,2,3,4] => [[1,1,2,3,4]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,2,3,5] => [[1,1,2,3,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,2,4,4] => [[1,1,2,4,4]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,2,4,5] => [[1,1,2,4,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,3,3,3] => [[1,1,3,3,3]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,3,3,4] => [[1,1,3,3,4]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,3,3,5] => [[1,1,3,3,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,3,4,4] => [[1,1,3,4,4]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,3,4,5] => [[1,1,3,4,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,2,2,2] => [[1,2,2,2,2]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,2,2,3] => [[1,2,2,2,3]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,2,2,4] => [[1,2,2,2,4]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
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