edit this statistic or download as text // json
Identifier
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
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
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