Identifier
Values
([],1) => ([],1) => [2] => [1,1] => 1
([(0,1)],2) => ([(0,1)],2) => [3] => [1,1,1] => 1
([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => [4] => [1,1,1,1] => 1
([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => [4,2] => [2,2,1,1] => 1
([(0,3),(2,1),(3,2)],4) => ([(0,3),(2,1),(3,2)],4) => [5] => [1,1,1,1,1] => 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => [4,2,2,2] => [4,4,1,1] => 3
([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => [8] => [1,1,1,1,1,1,1,1] => 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => [5,2] => [2,2,1,1,1] => 1
([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,4),(2,3),(3,1),(4,2)],5) => [6] => [1,1,1,1,1,1] => 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => [5,2] => [2,2,1,1,1] => 1
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6) => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6) => [5,2,2,2] => [4,4,1,1,1] => 3
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => [9] => [1,1,1,1,1,1,1,1,1] => 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => [6,2] => [2,2,1,1,1,1] => 1
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => [6,4] => [2,2,2,2,1,1] => 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => [6,2] => [2,2,1,1,1,1] => 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => [6,2] => [2,2,1,1,1,1] => 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6) => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6) => [5,2,2,2] => [4,4,1,1,1] => 3
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => [5,3,3] => [3,3,3,1,1] => 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [5,5] => [2,2,2,2,2] => 2
([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [7] => [1,1,1,1,1,1,1] => 1
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => [9] => [1,1,1,1,1,1,1,1,1] => 1
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => [7,2] => [2,2,1,1,1,1,1] => 1
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => [6,5] => [2,2,2,2,2,1] => 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => [6,2,2] => [3,3,1,1,1,1] => 2
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7) => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7) => [7,2] => [2,2,1,1,1,1,1] => 1
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => [6,5] => [2,2,2,2,2,1] => 1
([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => [7,4] => [2,2,2,2,1,1,1] => 1
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => [7,2] => [2,2,1,1,1,1,1] => 1
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => [10] => [1,1,1,1,1,1,1,1,1,1] => 1
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => [8] => [1,1,1,1,1,1,1,1] => 1
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => [7,4] => [2,2,2,2,1,1,1] => 1
([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => [7,2] => [2,2,1,1,1,1,1] => 1
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Description
The Cartan determinant of the integer partition.
Let $p=[p_1,...,p_r]$ be a given integer partition with highest part t. Let $A=K[x]/(x^t)$ be the finite dimensional algebra over the field $K$ and $M$ the direct sum of the indecomposable $A$-modules of vector space dimension $p_i$ for each $i$. Then the Cartan determinant of $p$ is the Cartan determinant of the endomorphism algebra of $M$ over $A$.
Explicitly, this is the determinant of the matrix $\left(\min(\bar p_i, \bar p_j)\right)_{i,j}$, where $\bar p$ is the set of distinct parts of the partition.
Map
conjugate
Description
Return the conjugate partition of the partition.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
Map
rowmotion cycle type
Description
The cycle type of rowmotion on the order ideals of a poset.
Map
to poset
Description
Return the poset corresponding to the lattice.