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-----------------------------------------------------------------------------
Statistic identifier: St000928
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The sum of the coefficients of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1].
-----------------------------------------------------------------------------
References: [1] Garsia, A. M., Goupil, A. Character polynomials, their $q$-analogs and the Kronecker product [[MathSciNet:2576382]]
-----------------------------------------------------------------------------
Code:
def statistic(L):
return L.character_polynomial()(*[1]*sum(L))
-----------------------------------------------------------------------------
Statistic values:
[2] => 0
[1,1] => -1
[3] => 1
[2,1] => 0
[1,1,1] => 1
[4] => 0
[3,1] => -1
[2,2] => -1
[2,1,1] => 1
[1,1,1,1] => -1
[5] => 1
[4,1] => -1
[3,2] => 1
[3,1,1] => 1
[2,2,1] => 0
[2,1,1,1] => -1
[1,1,1,1,1] => 0
[6] => 1
[5,1] => -1
[4,2] => 1
[4,1,1] => 0
[3,3] => -1
[3,2,1] => 0
[3,1,1,1] => -1
[2,2,2] => -1
[2,2,1,1] => 0
[2,1,1,1,1] => 2
[1,1,1,1,1,1] => 0
[7] => 1
[6,1] => -2
[5,2] => 0
[5,1,1] => 2
[4,3] => 0
[4,2,1] => 0
[4,1,1,1] => -1
[3,3,1] => -1
[3,2,2] => 0
[3,2,1,1] => 1
[3,1,1,1,1] => 0
[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] => 1
[7,1] => -1
[6,2] => 0
[6,1,1] => 2
[5,3] => 1
[5,2,1] => -1
[5,1,1,1] => -1
[4,4] => -1
[4,3,1] => 1
[4,2,2] => -1
[4,2,1,1] => -1
[4,1,1,1,1] => 2
[3,3,2] => 0
[3,3,1,1] => 1
[3,2,2,1] => 1
[3,2,1,1,1] => -1
[3,1,1,1,1,1] => 0
[2,2,2,2] => -1
[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] => 2
[8,1] => -3
[7,2] => 1
[7,1,1] => 2
[6,3] => 1
[6,2,1] => -1
[6,1,1,1] => -2
[5,4] => -1
[5,3,1] => -1
[5,2,2] => 1
[5,2,1,1] => 1
[5,1,1,1,1] => 0
[4,4,1] => 0
[4,3,2] => 1
[4,3,1,1] => 0
[4,2,2,1] => 0
[4,2,1,1,1] => 1
[4,1,1,1,1,1] => -1
[3,3,3] => -1
[3,3,2,1] => -1
[3,3,1,1,1] => -1
[3,2,2,2] => 0
[3,2,2,1,1] => 0
[3,2,1,1,1,1] => 0
[3,1,1,1,1,1,1] => -1
[2,2,2,2,1] => 0
[2,2,2,1,1,1] => 2
[2,2,1,1,1,1,1] => -1
[2,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1] => -1
[10] => 2
[9,1] => -2
[8,2] => 1
[8,1,1] => 2
[7,3] => 0
[7,2,1] => -1
[7,1,1,1] => -1
[6,4] => 0
[6,3,1] => -1
[6,2,2] => 0
[6,2,1,1] => 1
[6,1,1,1,1] => 1
[5,5] => 1
[5,4,1] => 1
[5,3,2] => 0
[5,3,1,1] => 0
[5,2,2,1] => 0
[5,2,1,1,1] => 1
[5,1,1,1,1,1] => -2
[4,4,2] => 0
[4,4,1,1] => 0
[4,3,3] => -1
[4,3,2,1] => 0
[4,3,1,1,1] => -1
[4,2,2,2] => 0
[4,2,2,1,1] => -2
[4,2,1,1,1,1] => 0
[4,1,1,1,1,1,1] => 2
[3,3,3,1] => 2
[3,3,2,2] => -1
[3,3,2,1,1] => 1
[3,3,1,1,1,1] => 2
[3,2,2,2,1] => 2
[3,2,2,1,1,1] => -1
[3,2,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1] => 1
[2,2,2,2,2] => -2
[2,2,2,2,1,1] => 0
[2,2,2,1,1,1,1] => -2
[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] => 2
[10,1] => -4
[9,2] => 1
[9,1,1] => 3
[8,3] => 0
[8,2,1] => 0
[8,1,1,1] => -3
[7,4] => -1
[7,3,1] => 0
[7,2,2] => -1
[7,2,1,1] => 0
[7,1,1,1,1] => 2
[6,5] => 0
[6,4,1] => 0
[6,3,2] => 2
[6,3,1,1] => 1
[6,2,2,1] => 0
[6,2,1,1,1] => -1
[6,1,1,1,1,1] => 0
[5,5,1] => -1
[5,4,2] => 0
[5,4,1,1] => 0
[5,3,3] => -1
[5,3,2,1] => -1
[5,3,1,1,1] => -1
[5,2,2,2] => 0
[5,2,2,1,1] => -1
[5,2,1,1,1,1] => 1
[5,1,1,1,1,1,1] => 0
[4,4,3] => -1
[4,4,2,1] => 0
[4,4,1,1,1] => 0
[4,3,3,1] => 2
[4,3,2,2] => -1
[4,3,2,1,1] => 2
[4,3,1,1,1,1] => 0
[4,2,2,2,1] => 2
[4,2,2,1,1,1] => 0
[4,2,1,1,1,1,1] => 0
[4,1,1,1,1,1,1,1] => -1
[3,3,3,2] => 0
[3,3,3,1,1] => -3
[3,3,2,2,1] => -1
[3,3,2,1,1,1] => 0
[3,3,1,1,1,1,1] => -2
[3,2,2,2,2] => -1
[3,2,2,2,1,1] => -1
[3,2,2,1,1,1,1] => 1
[3,2,1,1,1,1,1,1] => 0
[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] => 0
[2,1,1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,1,1,1] => -1
[12] => 3
[11,1] => -3
[10,2] => 1
[10,1,1] => 4
[9,3] => 0
[9,2,1] => -1
[9,1,1,1] => -3
[8,4] => 1
[8,3,1] => -2
[8,2,2] => -1
[8,2,1,1] => 2
[8,1,1,1,1] => 1
[7,5] => 0
[7,4,1] => 1
[7,3,2] => 1
[7,3,1,1] => 1
[7,2,2,1] => 1
[7,2,1,1,1] => -1
[7,1,1,1,1,1] => 0
[6,6] => 0
[6,5,1] => 1
[6,4,2] => -1
[6,4,1,1] => -2
[6,3,3] => -2
[6,3,2,1] => -1
[6,3,1,1,1] => 0
[6,2,2,2] => 0
[6,2,2,1,1] => 0
[6,2,1,1,1,1] => -1
[6,1,1,1,1,1,1] => 0
[5,5,2] => -2
[5,5,1,1] => 1
[5,4,3] => 3
[5,4,2,1] => 1
[5,4,1,1,1] => 1
[5,3,3,1] => 0
[5,3,2,2] => 1
[5,3,2,1,1] => 0
[5,3,1,1,1,1] => 1
[5,2,2,2,1] => 0
[5,2,2,1,1,1] => 2
[5,2,1,1,1,1,1] => -1
[5,1,1,1,1,1,1,1] => 1
[4,4,4] => -1
[4,4,3,1] => 0
[4,4,2,2] => -2
[4,4,2,1,1] => 0
[4,4,1,1,1,1] => -1
[4,3,3,2] => 0
[4,3,3,1,1] => -2
[4,3,2,2,1] => 1
[4,3,2,1,1,1] => -1
[4,3,1,1,1,1,1] => 0
[4,2,2,2,2] => -1
[4,2,2,2,1,1] => -3
[4,2,2,1,1,1,1] => 1
[4,2,1,1,1,1,1,1] => -1
[4,1,1,1,1,1,1,1,1] => 1
[3,3,3,3] => 2
[3,3,3,2,1] => 0
[3,3,3,1,1,1] => 2
[3,3,2,2,2] => 1
[3,3,2,2,1,1] => 1
[3,3,2,1,1,1,1] => 0
[3,3,1,1,1,1,1,1] => 1
[3,2,2,2,2,1] => 1
[3,2,2,2,1,1,1] => 0
[3,2,2,1,1,1,1,1] => 0
[3,2,1,1,1,1,1,1,1] => 1
[3,1,1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2,2] => 1
[2,2,2,2,2,1,1] => -1
[2,2,2,2,1,1,1,1] => -1
[2,2,2,1,1,1,1,1,1] => -2
[2,2,1,1,1,1,1,1,1,1] => 0
[2,1,1,1,1,1,1,1,1,1,1] => -2
[1,1,1,1,1,1,1,1,1,1,1,1] => 0
-----------------------------------------------------------------------------
Created: Aug 07, 2017 at 14:00 by Christian Stump
-----------------------------------------------------------------------------
Last Updated: Aug 07, 2017 at 14:00 by Christian Stump