*****************************************************************************
* 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: St000698
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core.
For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$.
This statistic counts the $2$-rim hooks that are removed in this process to obtain a $2$-core.
-----------------------------------------------------------------------------
References:
-----------------------------------------------------------------------------
Code:
def statistic(L):
k = 2
return (sum(L)-sum(L.core(k)))/k
-----------------------------------------------------------------------------
Statistic values:
[2] => 1
[1,1] => 1
[3] => 1
[2,1] => 0
[1,1,1] => 1
[4] => 2
[3,1] => 2
[2,2] => 2
[2,1,1] => 2
[1,1,1,1] => 2
[5] => 2
[4,1] => 1
[3,2] => 2
[3,1,1] => 2
[2,2,1] => 2
[2,1,1,1] => 1
[1,1,1,1,1] => 2
[6] => 3
[5,1] => 3
[4,2] => 3
[4,1,1] => 3
[3,3] => 3
[3,2,1] => 0
[3,1,1,1] => 3
[2,2,2] => 3
[2,2,1,1] => 3
[2,1,1,1,1] => 3
[1,1,1,1,1,1] => 3
[7] => 3
[6,1] => 2
[5,2] => 3
[5,1,1] => 3
[4,3] => 2
[4,2,1] => 3
[4,1,1,1] => 2
[3,3,1] => 3
[3,2,2] => 3
[3,2,1,1] => 3
[3,1,1,1,1] => 3
[2,2,2,1] => 2
[2,2,1,1,1] => 3
[2,1,1,1,1,1] => 2
[1,1,1,1,1,1,1] => 3
[8] => 4
[7,1] => 4
[6,2] => 4
[6,1,1] => 4
[5,3] => 4
[5,2,1] => 1
[5,1,1,1] => 4
[4,4] => 4
[4,3,1] => 4
[4,2,2] => 4
[4,2,1,1] => 4
[4,1,1,1,1] => 4
[3,3,2] => 4
[3,3,1,1] => 4
[3,2,2,1] => 4
[3,2,1,1,1] => 1
[3,1,1,1,1,1] => 4
[2,2,2,2] => 4
[2,2,2,1,1] => 4
[2,2,1,1,1,1] => 4
[2,1,1,1,1,1,1] => 4
[1,1,1,1,1,1,1,1] => 4
[9] => 4
[8,1] => 3
[7,2] => 4
[7,1,1] => 4
[6,3] => 3
[6,2,1] => 4
[6,1,1,1] => 3
[5,4] => 4
[5,3,1] => 4
[5,2,2] => 4
[5,2,1,1] => 4
[5,1,1,1,1] => 4
[4,4,1] => 4
[4,3,2] => 3
[4,3,1,1] => 3
[4,2,2,1] => 3
[4,2,1,1,1] => 4
[4,1,1,1,1,1] => 3
[3,3,3] => 4
[3,3,2,1] => 3
[3,3,1,1,1] => 4
[3,2,2,2] => 4
[3,2,2,1,1] => 4
[3,2,1,1,1,1] => 4
[3,1,1,1,1,1,1] => 4
[2,2,2,2,1] => 4
[2,2,2,1,1,1] => 3
[2,2,1,1,1,1,1] => 4
[2,1,1,1,1,1,1,1] => 3
[1,1,1,1,1,1,1,1,1] => 4
[10] => 5
[9,1] => 5
[8,2] => 5
[8,1,1] => 5
[7,3] => 5
[7,2,1] => 2
[7,1,1,1] => 5
[6,4] => 5
[6,3,1] => 5
[6,2,2] => 5
[6,2,1,1] => 5
[6,1,1,1,1] => 5
[5,5] => 5
[5,4,1] => 2
[5,3,2] => 5
[5,3,1,1] => 5
[5,2,2,1] => 5
[5,2,1,1,1] => 2
[5,1,1,1,1,1] => 5
[4,4,2] => 5
[4,4,1,1] => 5
[4,3,3] => 5
[4,3,2,1] => 0
[4,3,1,1,1] => 5
[4,2,2,2] => 5
[4,2,2,1,1] => 5
[4,2,1,1,1,1] => 5
[4,1,1,1,1,1,1] => 5
[3,3,3,1] => 5
[3,3,2,2] => 5
[3,3,2,1,1] => 5
[3,3,1,1,1,1] => 5
[3,2,2,2,1] => 2
[3,2,2,1,1,1] => 5
[3,2,1,1,1,1,1] => 2
[3,1,1,1,1,1,1,1] => 5
[2,2,2,2,2] => 5
[2,2,2,2,1,1] => 5
[2,2,2,1,1,1,1] => 5
[2,2,1,1,1,1,1,1] => 5
[2,1,1,1,1,1,1,1,1] => 5
[1,1,1,1,1,1,1,1,1,1] => 5
[11] => 5
[10,1] => 4
[9,2] => 5
[9,1,1] => 5
[8,3] => 4
[8,2,1] => 5
[8,1,1,1] => 4
[7,4] => 5
[7,3,1] => 5
[7,2,2] => 5
[7,2,1,1] => 5
[7,1,1,1,1] => 5
[6,5] => 4
[6,4,1] => 5
[6,3,2] => 4
[6,3,1,1] => 4
[6,2,2,1] => 4
[6,2,1,1,1] => 5
[6,1,1,1,1,1] => 4
[5,5,1] => 5
[5,4,2] => 5
[5,4,1,1] => 5
[5,3,3] => 5
[5,3,2,1] => 4
[5,3,1,1,1] => 5
[5,2,2,2] => 5
[5,2,2,1,1] => 5
[5,2,1,1,1,1] => 5
[5,1,1,1,1,1,1] => 5
[4,4,3] => 5
[4,4,2,1] => 4
[4,4,1,1,1] => 5
[4,3,3,1] => 4
[4,3,2,2] => 4
[4,3,2,1,1] => 4
[4,3,1,1,1,1] => 4
[4,2,2,2,1] => 5
[4,2,2,1,1,1] => 4
[4,2,1,1,1,1,1] => 5
[4,1,1,1,1,1,1,1] => 4
[3,3,3,2] => 5
[3,3,3,1,1] => 5
[3,3,2,2,1] => 5
[3,3,2,1,1,1] => 4
[3,3,1,1,1,1,1] => 5
[3,2,2,2,2] => 5
[3,2,2,2,1,1] => 5
[3,2,2,1,1,1,1] => 5
[3,2,1,1,1,1,1,1] => 5
[3,1,1,1,1,1,1,1,1] => 5
[2,2,2,2,2,1] => 4
[2,2,2,2,1,1,1] => 5
[2,2,2,1,1,1,1,1] => 4
[2,2,1,1,1,1,1,1,1] => 5
[2,1,1,1,1,1,1,1,1,1] => 4
[1,1,1,1,1,1,1,1,1,1,1] => 5
[12] => 6
[11,1] => 6
[10,2] => 6
[10,1,1] => 6
[9,3] => 6
[9,2,1] => 3
[9,1,1,1] => 6
[8,4] => 6
[8,3,1] => 6
[8,2,2] => 6
[8,2,1,1] => 6
[8,1,1,1,1] => 6
[7,5] => 6
[7,4,1] => 3
[7,3,2] => 6
[7,3,1,1] => 6
[7,2,2,1] => 6
[7,2,1,1,1] => 3
[7,1,1,1,1,1] => 6
[6,6] => 6
[6,5,1] => 6
[6,4,2] => 6
[6,4,1,1] => 6
[6,3,3] => 6
[6,3,2,1] => 1
[6,3,1,1,1] => 6
[6,2,2,2] => 6
[6,2,2,1,1] => 6
[6,2,1,1,1,1] => 6
[6,1,1,1,1,1,1] => 6
[5,5,2] => 6
[5,5,1,1] => 6
[5,4,3] => 3
[5,4,2,1] => 6
[5,4,1,1,1] => 3
[5,3,3,1] => 6
[5,3,2,2] => 6
[5,3,2,1,1] => 6
[5,3,1,1,1,1] => 6
[5,2,2,2,1] => 3
[5,2,2,1,1,1] => 6
[5,2,1,1,1,1,1] => 3
[5,1,1,1,1,1,1,1] => 6
[4,4,4] => 6
[4,4,3,1] => 6
[4,4,2,2] => 6
[4,4,2,1,1] => 6
[4,4,1,1,1,1] => 6
[4,3,3,2] => 6
[4,3,3,1,1] => 6
[4,3,2,2,1] => 6
[4,3,2,1,1,1] => 1
[4,3,1,1,1,1,1] => 6
[4,2,2,2,2] => 6
[4,2,2,2,1,1] => 6
[4,2,2,1,1,1,1] => 6
[4,2,1,1,1,1,1,1] => 6
[4,1,1,1,1,1,1,1,1] => 6
[3,3,3,3] => 6
[3,3,3,2,1] => 3
[3,3,3,1,1,1] => 6
[3,3,2,2,2] => 6
[3,3,2,2,1,1] => 6
[3,3,2,1,1,1,1] => 6
[3,3,1,1,1,1,1,1] => 6
[3,2,2,2,2,1] => 6
[3,2,2,2,1,1,1] => 3
[3,2,2,1,1,1,1,1] => 6
[3,2,1,1,1,1,1,1,1] => 3
[3,1,1,1,1,1,1,1,1,1] => 6
[2,2,2,2,2,2] => 6
[2,2,2,2,2,1,1] => 6
[2,2,2,2,1,1,1,1] => 6
[2,2,2,1,1,1,1,1,1] => 6
[2,2,1,1,1,1,1,1,1,1] => 6
[2,1,1,1,1,1,1,1,1,1,1] => 6
[1,1,1,1,1,1,1,1,1,1,1,1] => 6
-----------------------------------------------------------------------------
Created: Feb 20, 2017 at 10:33 by Christian Stump
-----------------------------------------------------------------------------
Last Updated: Feb 20, 2017 at 11:17 by Christian Stump