*****************************************************************************
* www.FindStat.org - The Combinatorial Statistic Finder *
* *
* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
* *
* This information is distributed in the hope that it will be useful, *
* but WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. *
*****************************************************************************
-----------------------------------------------------------------------------
Statistic identifier: St000992
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The alternating sum of the parts of an integer partition.
For a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$, this is $\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$.
-----------------------------------------------------------------------------
References:
-----------------------------------------------------------------------------
Code:
def statistic(L):
return sum( (-1)^k*L[k] for k in range(len(L)) )
-----------------------------------------------------------------------------
Statistic values:
[] => 0
[1] => 1
[2] => 2
[1,1] => 0
[3] => 3
[2,1] => 1
[1,1,1] => 1
[4] => 4
[3,1] => 2
[2,2] => 0
[2,1,1] => 2
[1,1,1,1] => 0
[5] => 5
[4,1] => 3
[3,2] => 1
[3,1,1] => 3
[2,2,1] => 1
[2,1,1,1] => 1
[1,1,1,1,1] => 1
[6] => 6
[5,1] => 4
[4,2] => 2
[4,1,1] => 4
[3,3] => 0
[3,2,1] => 2
[3,1,1,1] => 2
[2,2,2] => 2
[2,2,1,1] => 0
[2,1,1,1,1] => 2
[1,1,1,1,1,1] => 0
[7] => 7
[6,1] => 5
[5,2] => 3
[5,1,1] => 5
[4,3] => 1
[4,2,1] => 3
[4,1,1,1] => 3
[3,3,1] => 1
[3,2,2] => 3
[3,2,1,1] => 1
[3,1,1,1,1] => 3
[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] => 4
[6,1,1] => 6
[5,3] => 2
[5,2,1] => 4
[5,1,1,1] => 4
[4,4] => 0
[4,3,1] => 2
[4,2,2] => 4
[4,2,1,1] => 2
[4,1,1,1,1] => 4
[3,3,2] => 2
[3,3,1,1] => 0
[3,2,2,1] => 2
[3,2,1,1,1] => 2
[3,1,1,1,1,1] => 2
[2,2,2,2] => 0
[2,2,2,1,1] => 2
[2,2,1,1,1,1] => 0
[2,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1] => 0
[9] => 9
[8,1] => 7
[7,2] => 5
[7,1,1] => 7
[6,3] => 3
[6,2,1] => 5
[6,1,1,1] => 5
[5,4] => 1
[5,3,1] => 3
[5,2,2] => 5
[5,2,1,1] => 3
[5,1,1,1,1] => 5
[4,4,1] => 1
[4,3,2] => 3
[4,3,1,1] => 1
[4,2,2,1] => 3
[4,2,1,1,1] => 3
[4,1,1,1,1,1] => 3
[3,3,3] => 3
[3,3,2,1] => 1
[3,3,1,1,1] => 1
[3,2,2,2] => 1
[3,2,2,1,1] => 3
[3,2,1,1,1,1] => 1
[3,1,1,1,1,1,1] => 3
[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] => 6
[8,1,1] => 8
[7,3] => 4
[7,2,1] => 6
[7,1,1,1] => 6
[6,4] => 2
[6,3,1] => 4
[6,2,2] => 6
[6,2,1,1] => 4
[6,1,1,1,1] => 6
[5,5] => 0
[5,4,1] => 2
[5,3,2] => 4
[5,3,1,1] => 2
[5,2,2,1] => 4
[5,2,1,1,1] => 4
[5,1,1,1,1,1] => 4
[4,4,2] => 2
[4,4,1,1] => 0
[4,3,3] => 4
[4,3,2,1] => 2
[4,3,1,1,1] => 2
[4,2,2,2] => 2
[4,2,2,1,1] => 4
[4,2,1,1,1,1] => 2
[4,1,1,1,1,1,1] => 4
[3,3,3,1] => 2
[3,3,2,2] => 0
[3,3,2,1,1] => 2
[3,3,1,1,1,1] => 0
[3,2,2,2,1] => 2
[3,2,2,1,1,1] => 2
[3,2,1,1,1,1,1] => 2
[3,1,1,1,1,1,1,1] => 2
[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] => 0
[2,1,1,1,1,1,1,1,1] => 2
[1,1,1,1,1,1,1,1,1,1] => 0
[11] => 11
[10,1] => 9
[9,2] => 7
[9,1,1] => 9
[8,3] => 5
[8,2,1] => 7
[8,1,1,1] => 7
[7,4] => 3
[7,3,1] => 5
[7,2,2] => 7
[7,2,1,1] => 5
[7,1,1,1,1] => 7
[6,5] => 1
[6,4,1] => 3
[6,3,2] => 5
[6,3,1,1] => 3
[6,2,2,1] => 5
[6,2,1,1,1] => 5
[6,1,1,1,1,1] => 5
[5,5,1] => 1
[5,4,2] => 3
[5,4,1,1] => 1
[5,3,3] => 5
[5,3,2,1] => 3
[5,3,1,1,1] => 3
[5,2,2,2] => 3
[5,2,2,1,1] => 5
[5,2,1,1,1,1] => 3
[5,1,1,1,1,1,1] => 5
[4,4,3] => 3
[4,4,2,1] => 1
[4,4,1,1,1] => 1
[4,3,3,1] => 3
[4,3,2,2] => 1
[4,3,2,1,1] => 3
[4,3,1,1,1,1] => 1
[4,2,2,2,1] => 3
[4,2,2,1,1,1] => 3
[4,2,1,1,1,1,1] => 3
[4,1,1,1,1,1,1,1] => 3
[3,3,3,2] => 1
[3,3,3,1,1] => 3
[3,3,2,2,1] => 1
[3,3,2,1,1,1] => 1
[3,3,1,1,1,1,1] => 1
[3,2,2,2,2] => 3
[3,2,2,2,1,1] => 1
[3,2,2,1,1,1,1] => 3
[3,2,1,1,1,1,1,1] => 1
[3,1,1,1,1,1,1,1,1] => 3
[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
[12] => 12
[11,1] => 10
[10,2] => 8
[10,1,1] => 10
[9,3] => 6
[9,2,1] => 8
[9,1,1,1] => 8
[8,4] => 4
[8,3,1] => 6
[8,2,2] => 8
[8,2,1,1] => 6
[8,1,1,1,1] => 8
[7,5] => 2
[7,4,1] => 4
[7,3,2] => 6
[7,3,1,1] => 4
[7,2,2,1] => 6
[7,2,1,1,1] => 6
[7,1,1,1,1,1] => 6
[6,6] => 0
[6,5,1] => 2
[6,4,2] => 4
[6,4,1,1] => 2
[6,3,3] => 6
[6,3,2,1] => 4
[6,3,1,1,1] => 4
[6,2,2,2] => 4
[6,2,2,1,1] => 6
[6,2,1,1,1,1] => 4
[6,1,1,1,1,1,1] => 6
[5,5,2] => 2
[5,5,1,1] => 0
[5,4,3] => 4
[5,4,2,1] => 2
[5,4,1,1,1] => 2
[5,3,3,1] => 4
[5,3,2,2] => 2
[5,3,2,1,1] => 4
[5,3,1,1,1,1] => 2
[5,2,2,2,1] => 4
[5,2,2,1,1,1] => 4
[5,2,1,1,1,1,1] => 4
[5,1,1,1,1,1,1,1] => 4
[4,4,4] => 4
[4,4,3,1] => 2
[4,4,2,2] => 0
[4,4,2,1,1] => 2
[4,4,1,1,1,1] => 0
[4,3,3,2] => 2
[4,3,3,1,1] => 4
[4,3,2,2,1] => 2
[4,3,2,1,1,1] => 2
[4,3,1,1,1,1,1] => 2
[4,2,2,2,2] => 4
[4,2,2,2,1,1] => 2
[4,2,2,1,1,1,1] => 4
[4,2,1,1,1,1,1,1] => 2
[4,1,1,1,1,1,1,1,1] => 4
[3,3,3,3] => 0
[3,3,3,2,1] => 2
[3,3,3,1,1,1] => 2
[3,3,2,2,2] => 2
[3,3,2,2,1,1] => 0
[3,3,2,1,1,1,1] => 2
[3,3,1,1,1,1,1,1] => 0
[3,2,2,2,2,1] => 2
[3,2,2,2,1,1,1] => 2
[3,2,2,1,1,1,1,1] => 2
[3,2,1,1,1,1,1,1,1] => 2
[3,1,1,1,1,1,1,1,1,1] => 2
[2,2,2,2,2,2] => 0
[2,2,2,2,2,1,1] => 2
[2,2,2,2,1,1,1,1] => 0
[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
[5,4,3,1] => 3
[5,4,2,2] => 1
[5,4,2,1,1] => 3
[5,3,3,2] => 3
[5,3,3,1,1] => 5
[5,3,2,2,1] => 3
[4,4,3,2] => 1
[4,4,3,1,1] => 3
[4,4,2,2,1] => 1
[4,3,3,2,1] => 3
[5,4,3,2] => 2
[5,4,3,1,1] => 4
[5,4,2,2,1] => 2
[5,3,3,2,1] => 4
[4,4,3,2,1] => 2
[5,4,3,2,1] => 3
[7,5,3,1] => 4
[7,5,4,3,1] => 4
[6,5,4,3,2,1] => 3
[11,7,5,1] => 8
[9,7,5,3,1] => 5
[7,6,5,4,3,2,1] => 4
[9,7,5,5,3,1] => 4
[11,9,7,5,3,1] => 6
[11,8,7,5,4,1] => 8
[8,7,6,5,4,3,2,1] => 4
[11,9,7,6,5,3,1] => 6
[13,11,9,7,5,3,1] => 7
[13,11,9,7,7,5,3,1] => 8
[17,13,11,9,7,5,1] => 9
[15,13,11,9,7,5,3,1] => 8
[29,23,19,17,13,11,7,1] => 16
-----------------------------------------------------------------------------
Created: Oct 17, 2017 at 11:36 by Christian Stump
-----------------------------------------------------------------------------
Last Updated: Sep 07, 2020 at 19:49 by Martin Rubey