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Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St001713
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St001713: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[0]]
=> 1
[[1,0],[1]]
=> 1
[[2,0],[0]]
=> 2
[[2,0],[1]]
=> 2
[[2,0],[2]]
=> 2
[[1,1],[1]]
=> 0
[[1,0,0],[0,0],[0]]
=> 1
[[1,0,0],[1,0],[0]]
=> 1
[[1,0,0],[1,0],[1]]
=> 1
[[3,0],[0]]
=> 3
[[3,0],[1]]
=> 3
[[3,0],[2]]
=> 3
[[3,0],[3]]
=> 3
[[2,1],[1]]
=> 1
[[2,1],[2]]
=> 1
[[2,0,0],[0,0],[0]]
=> 2
[[2,0,0],[1,0],[0]]
=> 2
[[2,0,0],[1,0],[1]]
=> 2
[[2,0,0],[2,0],[0]]
=> 2
[[2,0,0],[2,0],[1]]
=> 2
[[2,0,0],[2,0],[2]]
=> 2
[[1,1,0],[1,0],[0]]
=> 1
[[1,1,0],[1,0],[1]]
=> 1
[[1,1,0],[1,1],[1]]
=> 1
[[1,0,0,0],[0,0,0],[0,0],[0]]
=> 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
=> 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
=> 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> 1
[[4,0],[0]]
=> 4
[[4,0],[1]]
=> 4
[[4,0],[2]]
=> 4
[[4,0],[3]]
=> 4
[[4,0],[4]]
=> 4
[[3,1],[1]]
=> 2
[[3,1],[2]]
=> 2
[[3,1],[3]]
=> 2
[[2,2],[2]]
=> 0
[[3,0,0],[0,0],[0]]
=> 3
[[3,0,0],[1,0],[0]]
=> 3
[[3,0,0],[1,0],[1]]
=> 3
[[3,0,0],[2,0],[0]]
=> 3
[[3,0,0],[2,0],[1]]
=> 3
[[3,0,0],[2,0],[2]]
=> 3
[[3,0,0],[3,0],[0]]
=> 3
[[3,0,0],[3,0],[1]]
=> 3
[[3,0,0],[3,0],[2]]
=> 3
[[3,0,0],[3,0],[3]]
=> 3
[[2,1,0],[1,0],[0]]
=> 2
[[2,1,0],[1,0],[1]]
=> 2
[[2,1,0],[1,1],[1]]
=> 2
Description
The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern.
Matching statistic: St001880
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 50%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 50%
Values
[[1,0],[0]]
=> [[2]]
=> [1] => ([],1)
=> ? = 1
[[1,0],[1]]
=> [[1]]
=> [1] => ([],1)
=> ? = 1
[[2,0],[0]]
=> [[2,2]]
=> [1,2] => ([(0,1)],2)
=> ? = 2
[[2,0],[1]]
=> [[1,2]]
=> [1,2] => ([(0,1)],2)
=> ? = 2
[[2,0],[2]]
=> [[1,1]]
=> [1,2] => ([(0,1)],2)
=> ? = 2
[[1,1],[1]]
=> [[1],[2]]
=> [2,1] => ([],2)
=> ? = 0
[[1,0,0],[0,0],[0]]
=> [[3]]
=> [1] => ([],1)
=> ? = 1
[[1,0,0],[1,0],[0]]
=> [[2]]
=> [1] => ([],1)
=> ? = 1
[[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => ([],1)
=> ? = 1
[[3,0],[0]]
=> [[2,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,0],[1]]
=> [[1,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,0],[2]]
=> [[1,1,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[2,1],[1]]
=> [[1,2],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ? = 1
[[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 1
[[2,0,0],[0,0],[0]]
=> [[3,3]]
=> [1,2] => ([(0,1)],2)
=> ? = 2
[[2,0,0],[1,0],[0]]
=> [[2,3]]
=> [1,2] => ([(0,1)],2)
=> ? = 2
[[2,0,0],[1,0],[1]]
=> [[1,3]]
=> [1,2] => ([(0,1)],2)
=> ? = 2
[[2,0,0],[2,0],[0]]
=> [[2,2]]
=> [1,2] => ([(0,1)],2)
=> ? = 2
[[2,0,0],[2,0],[1]]
=> [[1,2]]
=> [1,2] => ([(0,1)],2)
=> ? = 2
[[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => ([(0,1)],2)
=> ? = 2
[[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> [2,1] => ([],2)
=> ? = 1
[[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> [2,1] => ([],2)
=> ? = 1
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => ([],2)
=> ? = 1
[[1,0,0,0],[0,0,0],[0,0],[0]]
=> [[4]]
=> [1] => ([],1)
=> ? = 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
=> [[3]]
=> [1] => ([],1)
=> ? = 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
=> [[2]]
=> [1] => ([],1)
=> ? = 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => ([],1)
=> ? = 1
[[4,0],[0]]
=> [[2,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[4,0],[1]]
=> [[1,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[4,0],[3]]
=> [[1,1,1,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[3,1],[1]]
=> [[1,2,2],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ? = 2
[[3,1],[2]]
=> [[1,1,2],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 2
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? = 0
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,0,0],[3,0],[0]]
=> [[2,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,0,0],[3,0],[1]]
=> [[1,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,0,0],[3,0],[2]]
=> [[1,1,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ? = 2
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ? = 2
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ? = 2
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 2
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 2
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 2
[[2,1,0],[2,1],[1]]
=> [[1,2],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ? = 2
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 2
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => ([],3)
=> ? = 0
[[2,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4]]
=> [1,2] => ([(0,1)],2)
=> ? = 2
[[2,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4]]
=> [1,2] => ([(0,1)],2)
=> ? = 2
[[2,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4]]
=> [1,2] => ([(0,1)],2)
=> ? = 2
[[2,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4]]
=> [1,2] => ([(0,1)],2)
=> ? = 2
[[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> [1,2] => ([(0,1)],2)
=> ? = 2
[[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> [1,2] => ([(0,1)],2)
=> ? = 2
[[2,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3]]
=> [1,2] => ([(0,1)],2)
=> ? = 2
[[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> [1,2] => ([(0,1)],2)
=> ? = 2
[[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> [1,2] => ([(0,1)],2)
=> ? = 2
[[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => ([(0,1)],2)
=> ? = 2
[[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> [2,1] => ([],2)
=> ? = 1
[[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> [2,1] => ([],2)
=> ? = 1
[[1,1,0,0],[1,0,0],[1,0],[1]]
=> [[1],[4]]
=> [2,1] => ([],2)
=> ? = 1
[[5,0],[0]]
=> [[2,2,2,2,2]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[5,0],[1]]
=> [[1,2,2,2,2]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[5,0],[2]]
=> [[1,1,2,2,2]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[5,0],[3]]
=> [[1,1,1,2,2]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[5,0],[4]]
=> [[1,1,1,1,2]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[4,0,0],[0,0],[0]]
=> [[3,3,3,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[4,0,0],[1,0],[0]]
=> [[2,3,3,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[4,0,0],[1,0],[1]]
=> [[1,3,3,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[4,0,0],[2,0],[0]]
=> [[2,2,3,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[4,0,0],[2,0],[1]]
=> [[1,2,3,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[4,0,0],[3,0],[0]]
=> [[2,2,2,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[4,0,0],[3,0],[1]]
=> [[1,2,2,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[4,0,0],[3,0],[3]]
=> [[1,1,1,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[4,0,0],[4,0],[0]]
=> [[2,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[4,0,0],[4,0],[1]]
=> [[1,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[4,0,0],[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[4,0,0],[4,0],[3]]
=> [[1,1,1,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 4
[[3,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
[[3,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 3
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001879
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 50%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 50%
Values
[[1,0],[0]]
=> [[2]]
=> [1] => ([],1)
=> ? = 1 - 1
[[1,0],[1]]
=> [[1]]
=> [1] => ([],1)
=> ? = 1 - 1
[[2,0],[0]]
=> [[2,2]]
=> [1,2] => ([(0,1)],2)
=> ? = 2 - 1
[[2,0],[1]]
=> [[1,2]]
=> [1,2] => ([(0,1)],2)
=> ? = 2 - 1
[[2,0],[2]]
=> [[1,1]]
=> [1,2] => ([(0,1)],2)
=> ? = 2 - 1
[[1,1],[1]]
=> [[1],[2]]
=> [2,1] => ([],2)
=> ? = 0 - 1
[[1,0,0],[0,0],[0]]
=> [[3]]
=> [1] => ([],1)
=> ? = 1 - 1
[[1,0,0],[1,0],[0]]
=> [[2]]
=> [1] => ([],1)
=> ? = 1 - 1
[[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => ([],1)
=> ? = 1 - 1
[[3,0],[0]]
=> [[2,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,0],[1]]
=> [[1,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,0],[2]]
=> [[1,1,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[2,1],[1]]
=> [[1,2],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ? = 1 - 1
[[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 1 - 1
[[2,0,0],[0,0],[0]]
=> [[3,3]]
=> [1,2] => ([(0,1)],2)
=> ? = 2 - 1
[[2,0,0],[1,0],[0]]
=> [[2,3]]
=> [1,2] => ([(0,1)],2)
=> ? = 2 - 1
[[2,0,0],[1,0],[1]]
=> [[1,3]]
=> [1,2] => ([(0,1)],2)
=> ? = 2 - 1
[[2,0,0],[2,0],[0]]
=> [[2,2]]
=> [1,2] => ([(0,1)],2)
=> ? = 2 - 1
[[2,0,0],[2,0],[1]]
=> [[1,2]]
=> [1,2] => ([(0,1)],2)
=> ? = 2 - 1
[[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => ([(0,1)],2)
=> ? = 2 - 1
[[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> [2,1] => ([],2)
=> ? = 1 - 1
[[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> [2,1] => ([],2)
=> ? = 1 - 1
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => ([],2)
=> ? = 1 - 1
[[1,0,0,0],[0,0,0],[0,0],[0]]
=> [[4]]
=> [1] => ([],1)
=> ? = 1 - 1
[[1,0,0,0],[1,0,0],[0,0],[0]]
=> [[3]]
=> [1] => ([],1)
=> ? = 1 - 1
[[1,0,0,0],[1,0,0],[1,0],[0]]
=> [[2]]
=> [1] => ([],1)
=> ? = 1 - 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [1] => ([],1)
=> ? = 1 - 1
[[4,0],[0]]
=> [[2,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[4,0],[1]]
=> [[1,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[4,0],[3]]
=> [[1,1,1,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[3,1],[1]]
=> [[1,2,2],[2]]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ? = 2 - 1
[[3,1],[2]]
=> [[1,1,2],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? = 2 - 1
[[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? = 2 - 1
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? = 0 - 1
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,0,0],[3,0],[0]]
=> [[2,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,0,0],[3,0],[1]]
=> [[1,2,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,0,0],[3,0],[2]]
=> [[1,1,2]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,0,0],[3,0],[3]]
=> [[1,1,1]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ? = 2 - 1
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ? = 2 - 1
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ? = 2 - 1
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 2 - 1
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 2 - 1
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 2 - 1
[[2,1,0],[2,1],[1]]
=> [[1,2],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ? = 2 - 1
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => ([(1,2)],3)
=> ? = 2 - 1
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => ([],3)
=> ? = 0 - 1
[[2,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4]]
=> [1,2] => ([(0,1)],2)
=> ? = 2 - 1
[[2,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4]]
=> [1,2] => ([(0,1)],2)
=> ? = 2 - 1
[[2,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4]]
=> [1,2] => ([(0,1)],2)
=> ? = 2 - 1
[[2,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4]]
=> [1,2] => ([(0,1)],2)
=> ? = 2 - 1
[[2,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3]]
=> [1,2] => ([(0,1)],2)
=> ? = 2 - 1
[[2,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3]]
=> [1,2] => ([(0,1)],2)
=> ? = 2 - 1
[[2,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3]]
=> [1,2] => ([(0,1)],2)
=> ? = 2 - 1
[[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> [1,2] => ([(0,1)],2)
=> ? = 2 - 1
[[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> [1,2] => ([(0,1)],2)
=> ? = 2 - 1
[[2,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1]]
=> [1,2] => ([(0,1)],2)
=> ? = 2 - 1
[[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> [2,1] => ([],2)
=> ? = 1 - 1
[[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> [2,1] => ([],2)
=> ? = 1 - 1
[[1,1,0,0],[1,0,0],[1,0],[1]]
=> [[1],[4]]
=> [2,1] => ([],2)
=> ? = 1 - 1
[[5,0],[0]]
=> [[2,2,2,2,2]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[5,0],[1]]
=> [[1,2,2,2,2]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[5,0],[2]]
=> [[1,1,2,2,2]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[5,0],[3]]
=> [[1,1,1,2,2]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[5,0],[4]]
=> [[1,1,1,1,2]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[5,0],[5]]
=> [[1,1,1,1,1]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[4,0,0],[0,0],[0]]
=> [[3,3,3,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[4,0,0],[1,0],[0]]
=> [[2,3,3,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[4,0,0],[1,0],[1]]
=> [[1,3,3,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[4,0,0],[2,0],[0]]
=> [[2,2,3,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[4,0,0],[2,0],[1]]
=> [[1,2,3,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[4,0,0],[2,0],[2]]
=> [[1,1,3,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[4,0,0],[3,0],[0]]
=> [[2,2,2,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[4,0,0],[3,0],[1]]
=> [[1,2,2,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[4,0,0],[3,0],[2]]
=> [[1,1,2,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[4,0,0],[3,0],[3]]
=> [[1,1,1,3]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[4,0,0],[4,0],[0]]
=> [[2,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[4,0,0],[4,0],[1]]
=> [[1,2,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[4,0,0],[4,0],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[4,0,0],[4,0],[3]]
=> [[1,1,1,2]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[4,0,0],[4,0],[4]]
=> [[1,1,1,1]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[3,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[3,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 3 - 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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