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Statistic identifier: St001571
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Collection: Integer partitions
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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.
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References:
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Code:
def statistic(p):
p = list(set(p))
return matrix([[min(p[i], p[j]) for i in range(len(p))] for j in range(len(p))]).det()
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Statistic values:
[1] => 1
[2] => 2
[1,1] => 1
[3] => 3
[2,1] => 1
[1,1,1] => 1
[4] => 4
[3,1] => 2
[2,2] => 2
[2,1,1] => 1
[1,1,1,1] => 1
[5] => 5
[4,1] => 3
[3,2] => 2
[3,1,1] => 2
[2,2,1] => 1
[2,1,1,1] => 1
[1,1,1,1,1] => 1
[6] => 6
[5,1] => 4
[4,2] => 4
[4,1,1] => 3
[3,3] => 3
[3,2,1] => 1
[3,1,1,1] => 2
[2,2,2] => 2
[2,2,1,1] => 1
[2,1,1,1,1] => 1
[1,1,1,1,1,1] => 1
[7] => 7
[6,1] => 5
[5,2] => 6
[5,1,1] => 4
[4,3] => 3
[4,2,1] => 2
[4,1,1,1] => 3
[3,3,1] => 2
[3,2,2] => 2
[3,2,1,1] => 1
[3,1,1,1,1] => 2
[2,2,2,1] => 1
[2,2,1,1,1] => 1
[2,1,1,1,1,1] => 1
[1,1,1,1,1,1,1] => 1
[8] => 8
[7,1] => 6
[6,2] => 8
[6,1,1] => 5
[5,3] => 6
[5,2,1] => 3
[5,1,1,1] => 4
[4,4] => 4
[4,3,1] => 2
[4,2,2] => 4
[4,2,1,1] => 2
[4,1,1,1,1] => 3
[3,3,2] => 2
[3,3,1,1] => 2
[3,2,2,1] => 1
[3,2,1,1,1] => 1
[3,1,1,1,1,1] => 2
[2,2,2,2] => 2
[2,2,2,1,1] => 1
[2,2,1,1,1,1] => 1
[2,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1] => 1
[9] => 9
[8,1] => 7
[7,2] => 10
[7,1,1] => 6
[6,3] => 9
[6,2,1] => 4
[6,1,1,1] => 5
[5,4] => 4
[5,3,1] => 4
[5,2,2] => 6
[5,2,1,1] => 3
[5,1,1,1,1] => 4
[4,4,1] => 3
[4,3,2] => 2
[4,3,1,1] => 2
[4,2,2,1] => 2
[4,2,1,1,1] => 2
[4,1,1,1,1,1] => 3
[3,3,3] => 3
[3,3,2,1] => 1
[3,3,1,1,1] => 2
[3,2,2,2] => 2
[3,2,2,1,1] => 1
[3,2,1,1,1,1] => 1
[3,1,1,1,1,1,1] => 2
[2,2,2,2,1] => 1
[2,2,2,1,1,1] => 1
[2,2,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1] => 1
[10] => 10
[9,1] => 8
[8,2] => 12
[8,1,1] => 7
[7,3] => 12
[7,2,1] => 5
[7,1,1,1] => 6
[6,4] => 8
[6,3,1] => 6
[6,2,2] => 8
[6,2,1,1] => 4
[6,1,1,1,1] => 5
[5,5] => 5
[5,4,1] => 3
[5,3,2] => 4
[5,3,1,1] => 4
[5,2,2,1] => 3
[5,2,1,1,1] => 3
[5,1,1,1,1,1] => 4
[4,4,2] => 4
[4,4,1,1] => 3
[4,3,3] => 3
[4,3,2,1] => 1
[4,3,1,1,1] => 2
[4,2,2,2] => 4
[4,2,2,1,1] => 2
[4,2,1,1,1,1] => 2
[4,1,1,1,1,1,1] => 3
[3,3,3,1] => 2
[3,3,2,2] => 2
[3,3,2,1,1] => 1
[3,3,1,1,1,1] => 2
[3,2,2,2,1] => 1
[3,2,2,1,1,1] => 1
[3,2,1,1,1,1,1] => 1
[3,1,1,1,1,1,1,1] => 2
[2,2,2,2,2] => 2
[2,2,2,2,1,1] => 1
[2,2,2,1,1,1,1] => 1
[2,2,1,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1] => 1
[11] => 11
[10,1] => 9
[9,2] => 14
[9,1,1] => 8
[8,3] => 15
[8,2,1] => 6
[8,1,1,1] => 7
[7,4] => 12
[7,3,1] => 8
[7,2,2] => 10
[7,2,1,1] => 5
[7,1,1,1,1] => 6
[6,5] => 5
[6,4,1] => 6
[6,3,2] => 6
[6,3,1,1] => 6
[6,2,2,1] => 4
[6,2,1,1,1] => 4
[6,1,1,1,1,1] => 5
[5,5,1] => 4
[5,4,2] => 4
[5,4,1,1] => 3
[5,3,3] => 6
[5,3,2,1] => 2
[5,3,1,1,1] => 4
[5,2,2,2] => 6
[5,2,2,1,1] => 3
[5,2,1,1,1,1] => 3
[5,1,1,1,1,1,1] => 4
[4,4,3] => 3
[4,4,2,1] => 2
[4,4,1,1,1] => 3
[4,3,3,1] => 2
[4,3,2,2] => 2
[4,3,2,1,1] => 1
[4,3,1,1,1,1] => 2
[4,2,2,2,1] => 2
[4,2,2,1,1,1] => 2
[4,2,1,1,1,1,1] => 2
[4,1,1,1,1,1,1,1] => 3
[3,3,3,2] => 2
[3,3,3,1,1] => 2
[3,3,2,2,1] => 1
[3,3,2,1,1,1] => 1
[3,3,1,1,1,1,1] => 2
[3,2,2,2,2] => 2
[3,2,2,2,1,1] => 1
[3,2,2,1,1,1,1] => 1
[3,2,1,1,1,1,1,1] => 1
[3,1,1,1,1,1,1,1,1] => 2
[2,2,2,2,2,1] => 1
[2,2,2,2,1,1,1] => 1
[2,2,2,1,1,1,1,1] => 1
[2,2,1,1,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1,1,1,1] => 1
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Created: Jul 16, 2020 at 21:08 by Rene Marczinzik
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Last Updated: Oct 02, 2020 at 18:22 by Martin Rubey