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Matching statistic: St000474
St000474: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> -1
[2]
=> 2
[1,1]
=> -2
[3]
=> 3
[2,1]
=> 0
[1,1,1]
=> -3
[4]
=> 4
[3,1]
=> 0
[2,2]
=> 2
[2,1,1]
=> -2
[1,1,1,1]
=> -4
[5]
=> 5
[4,1]
=> 0
[3,2]
=> 3
[3,1,1]
=> -1
[2,2,1]
=> 1
[2,1,1,1]
=> -3
[1,1,1,1,1]
=> -5
[6]
=> 6
[5,1]
=> 0
[4,2]
=> 4
[4,1,1]
=> -1
[3,3]
=> 3
[3,2,1]
=> 1
[3,1,1,1]
=> -3
[2,2,2]
=> 2
[2,2,1,1]
=> -2
[2,1,1,1,1]
=> -4
[1,1,1,1,1,1]
=> -6
[7]
=> 7
[6,1]
=> 0
[5,2]
=> 5
[5,1,1]
=> -1
[4,3]
=> 4
[4,2,1]
=> 1
[4,1,1,1]
=> -2
[3,3,1]
=> 1
[3,2,2]
=> 3
[3,2,1,1]
=> -1
[3,1,1,1,1]
=> -4
[2,2,2,1]
=> 2
[2,2,1,1,1]
=> -3
[2,1,1,1,1,1]
=> -5
[1,1,1,1,1,1,1]
=> -7
[8]
=> 8
[7,1]
=> 0
[6,2]
=> 6
[6,1,1]
=> -1
[5,3]
=> 5
[5,2,1]
=> 1
Description
Dyson's crank of a partition.
Let $\lambda$ be a partition and let $o(\lambda)$ be the number of parts that are equal to 1 ([[St000475]]), and let $\mu(\lambda)$ be the number of parts that are strictly larger than $o(\lambda)$ ([[St000473]]). Dyson's crank is then defined as
$$crank(\lambda) = \begin{cases} \text{ largest part of }\lambda & o(\lambda) = 0\\ \mu(\lambda) - o(\lambda) & o(\lambda) > 0. \end{cases}$$
Matching statistic: St001880
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 24%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 24%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> ? = -1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> ? ∊ {-2,2}
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(1,2)],3)
=> ? ∊ {-2,2}
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> ? ∊ {-3,0}
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> ? ∊ {-3,0}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> ? ∊ {-4,-2,0,2}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> ? ∊ {-4,-2,0,2}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {-4,-2,0,2}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> ? ∊ {-4,-2,0,2}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> ? ∊ {-5,-3,-1,0,3,5}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> ? ∊ {-5,-3,-1,0,3,5}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {-5,-3,-1,0,3,5}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {-5,-3,-1,0,3,5}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {-5,-3,-1,0,3,5}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(4,5)],6)
=> ? ∊ {-5,-3,-1,0,3,5}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6)],7)
=> ? ∊ {-6,-4,-3,-2,-1,0,2,3,6}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {-6,-4,-3,-2,-1,0,2,3,6}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ? ∊ {-6,-4,-3,-2,-1,0,2,3,6}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4)],5)
=> ? ∊ {-6,-4,-3,-2,-1,0,2,3,6}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {-6,-4,-3,-2,-1,0,2,3,6}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(2,3),(2,4)],5)
=> ? ∊ {-6,-4,-3,-2,-1,0,2,3,6}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(2,4)],5)
=> ? ∊ {-6,-4,-3,-2,-1,0,2,3,6}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {-6,-4,-3,-2,-1,0,2,3,6}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ([(5,6)],7)
=> ? ∊ {-6,-4,-3,-2,-1,0,2,3,6}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7)],8)
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,7}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,7}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,7}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,7}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,7}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,7}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,7}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,7}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,7}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,7}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,5),(2,5),(3,4),(3,5)],6)
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,7}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,7}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ([(6,7)],8)
=> ? ∊ {-7,-5,-4,-3,-2,-1,-1,0,1,2,3,4,7}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8)],9)
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-1,-1,0,0,1,1,2,3,4,6,8}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-1,-1,0,0,1,1,2,3,4,6,8}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-1,-1,0,0,1,1,2,3,4,6,8}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(4,6),(5,6)],7)
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-1,-1,0,0,1,1,2,3,4,6,8}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-1,-1,0,0,1,1,2,3,4,6,8}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-1,-1,0,0,1,1,2,3,4,6,8}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4),(1,5)],6)
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-1,-1,0,0,1,1,2,3,4,6,8}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-1,-1,0,0,1,1,2,3,4,6,8}
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 4
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-1,-1,0,0,1,1,2,3,4,6,8}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-1,-1,0,0,1,1,2,3,4,6,8}
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(1,4)],5)
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-1,-1,0,0,1,1,2,3,4,6,8}
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-1,-1,0,0,1,1,2,3,4,6,8}
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> ? ∊ {-8,-6,-5,-4,-4,-3,-2,-2,-1,-1,0,0,1,1,2,3,4,6,8}
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
=> 2
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
=> 2
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> 2
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 1
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> 2
[6,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
=> 2
[6,3,2,1]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> 3
[6,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
=> 1
[5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
[5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 5
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
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