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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