- FindStat

Identifier

St001571: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [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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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.

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()

Created

Jul 16, 2020 at 21:08 by Rene Marczinzik

Updated

Oct 02, 2020 at 18:22 by Martin Rubey