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* 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: St000459
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The hook length of the base cell of a partition.
This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
-----------------------------------------------------------------------------
References:
-----------------------------------------------------------------------------
Code:
def statistic(l):
if l:
return l[0]+len(l)-1
return 0
-----------------------------------------------------------------------------
Statistic values:
[] => 0
[1] => 1
[2] => 2
[1,1] => 2
[3] => 3
[2,1] => 3
[1,1,1] => 3
[4] => 4
[3,1] => 4
[2,2] => 3
[2,1,1] => 4
[1,1,1,1] => 4
[5] => 5
[4,1] => 5
[3,2] => 4
[3,1,1] => 5
[2,2,1] => 4
[2,1,1,1] => 5
[1,1,1,1,1] => 5
[6] => 6
[5,1] => 6
[4,2] => 5
[4,1,1] => 6
[3,3] => 4
[3,2,1] => 5
[3,1,1,1] => 6
[2,2,2] => 4
[2,2,1,1] => 5
[2,1,1,1,1] => 6
[1,1,1,1,1,1] => 6
[7] => 7
[6,1] => 7
[5,2] => 6
[5,1,1] => 7
[4,3] => 5
[4,2,1] => 6
[4,1,1,1] => 7
[3,3,1] => 5
[3,2,2] => 5
[3,2,1,1] => 6
[3,1,1,1,1] => 7
[2,2,2,1] => 5
[2,2,1,1,1] => 6
[2,1,1,1,1,1] => 7
[1,1,1,1,1,1,1] => 7
[8] => 8
[7,1] => 8
[6,2] => 7
[6,1,1] => 8
[5,3] => 6
[5,2,1] => 7
[5,1,1,1] => 8
[4,4] => 5
[4,3,1] => 6
[4,2,2] => 6
[4,2,1,1] => 7
[4,1,1,1,1] => 8
[3,3,2] => 5
[3,3,1,1] => 6
[3,2,2,1] => 6
[3,2,1,1,1] => 7
[3,1,1,1,1,1] => 8
[2,2,2,2] => 5
[2,2,2,1,1] => 6
[2,2,1,1,1,1] => 7
[2,1,1,1,1,1,1] => 8
[1,1,1,1,1,1,1,1] => 8
[9] => 9
[8,1] => 9
[7,2] => 8
[7,1,1] => 9
[6,3] => 7
[6,2,1] => 8
[6,1,1,1] => 9
[5,4] => 6
[5,3,1] => 7
[5,2,2] => 7
[5,2,1,1] => 8
[5,1,1,1,1] => 9
[4,4,1] => 6
[4,3,2] => 6
[4,3,1,1] => 7
[4,2,2,1] => 7
[4,2,1,1,1] => 8
[4,1,1,1,1,1] => 9
[3,3,3] => 5
[3,3,2,1] => 6
[3,3,1,1,1] => 7
[3,2,2,2] => 6
[3,2,2,1,1] => 7
[3,2,1,1,1,1] => 8
[3,1,1,1,1,1,1] => 9
[2,2,2,2,1] => 6
[2,2,2,1,1,1] => 7
[2,2,1,1,1,1,1] => 8
[2,1,1,1,1,1,1,1] => 9
[1,1,1,1,1,1,1,1,1] => 9
[10] => 10
[9,1] => 10
[8,2] => 9
[8,1,1] => 10
[7,3] => 8
[7,2,1] => 9
[7,1,1,1] => 10
[6,4] => 7
[6,3,1] => 8
[6,2,2] => 8
[6,2,1,1] => 9
[6,1,1,1,1] => 10
[5,5] => 6
[5,4,1] => 7
[5,3,2] => 7
[5,3,1,1] => 8
[5,2,2,1] => 8
[5,2,1,1,1] => 9
[5,1,1,1,1,1] => 10
[4,4,2] => 6
[4,4,1,1] => 7
[4,3,3] => 6
[4,3,2,1] => 7
[4,3,1,1,1] => 8
[4,2,2,2] => 7
[4,2,2,1,1] => 8
[4,2,1,1,1,1] => 9
[4,1,1,1,1,1,1] => 10
[3,3,3,1] => 6
[3,3,2,2] => 6
[3,3,2,1,1] => 7
[3,3,1,1,1,1] => 8
[3,2,2,2,1] => 7
[3,2,2,1,1,1] => 8
[3,2,1,1,1,1,1] => 9
[3,1,1,1,1,1,1,1] => 10
[2,2,2,2,2] => 6
[2,2,2,2,1,1] => 7
[2,2,2,1,1,1,1] => 8
[2,2,1,1,1,1,1,1] => 9
[2,1,1,1,1,1,1,1,1] => 10
[1,1,1,1,1,1,1,1,1,1] => 10
[11] => 11
[10,1] => 11
[9,2] => 10
[9,1,1] => 11
[8,3] => 9
[8,2,1] => 10
[8,1,1,1] => 11
[7,4] => 8
[7,3,1] => 9
[7,2,2] => 9
[7,2,1,1] => 10
[7,1,1,1,1] => 11
[6,5] => 7
[6,4,1] => 8
[6,3,2] => 8
[6,3,1,1] => 9
[6,2,2,1] => 9
[6,2,1,1,1] => 10
[6,1,1,1,1,1] => 11
[5,5,1] => 7
[5,4,2] => 7
[5,4,1,1] => 8
[5,3,3] => 7
[5,3,2,1] => 8
[5,3,1,1,1] => 9
[5,2,2,2] => 8
[5,2,2,1,1] => 9
[5,2,1,1,1,1] => 10
[5,1,1,1,1,1,1] => 11
[4,4,3] => 6
[4,4,2,1] => 7
[4,4,1,1,1] => 8
[4,3,3,1] => 7
[4,3,2,2] => 7
[4,3,2,1,1] => 8
[4,3,1,1,1,1] => 9
[4,2,2,2,1] => 8
[4,2,2,1,1,1] => 9
[4,2,1,1,1,1,1] => 10
[4,1,1,1,1,1,1,1] => 11
[3,3,3,2] => 6
[3,3,3,1,1] => 7
[3,3,2,2,1] => 7
[3,3,2,1,1,1] => 8
[3,3,1,1,1,1,1] => 9
[3,2,2,2,2] => 7
[3,2,2,2,1,1] => 8
[3,2,2,1,1,1,1] => 9
[3,2,1,1,1,1,1,1] => 10
[3,1,1,1,1,1,1,1,1] => 11
[2,2,2,2,2,1] => 7
[2,2,2,2,1,1,1] => 8
[2,2,2,1,1,1,1,1] => 9
[2,2,1,1,1,1,1,1,1] => 10
[2,1,1,1,1,1,1,1,1,1] => 11
[1,1,1,1,1,1,1,1,1,1,1] => 11
[12] => 12
[11,1] => 12
[10,2] => 11
[10,1,1] => 12
[9,3] => 10
[9,2,1] => 11
[9,1,1,1] => 12
[8,4] => 9
[8,3,1] => 10
[8,2,2] => 10
[8,2,1,1] => 11
[8,1,1,1,1] => 12
[7,5] => 8
[7,4,1] => 9
[7,3,2] => 9
[7,3,1,1] => 10
[7,2,2,1] => 10
[7,2,1,1,1] => 11
[7,1,1,1,1,1] => 12
[6,6] => 7
[6,5,1] => 8
[6,4,2] => 8
[6,4,1,1] => 9
[6,3,3] => 8
[6,3,2,1] => 9
[6,3,1,1,1] => 10
[6,2,2,2] => 9
[6,2,2,1,1] => 10
[6,2,1,1,1,1] => 11
[6,1,1,1,1,1,1] => 12
[5,5,2] => 7
[5,5,1,1] => 8
[5,4,3] => 7
[5,4,2,1] => 8
[5,4,1,1,1] => 9
[5,3,3,1] => 8
[5,3,2,2] => 8
[5,3,2,1,1] => 9
[5,3,1,1,1,1] => 10
[5,2,2,2,1] => 9
[5,2,2,1,1,1] => 10
[5,2,1,1,1,1,1] => 11
[5,1,1,1,1,1,1,1] => 12
[4,4,4] => 6
[4,4,3,1] => 7
[4,4,2,2] => 7
[4,4,2,1,1] => 8
[4,4,1,1,1,1] => 9
[4,3,3,2] => 7
[4,3,3,1,1] => 8
[4,3,2,2,1] => 8
[4,3,2,1,1,1] => 9
[4,3,1,1,1,1,1] => 10
[4,2,2,2,2] => 8
[4,2,2,2,1,1] => 9
[4,2,2,1,1,1,1] => 10
[4,2,1,1,1,1,1,1] => 11
[4,1,1,1,1,1,1,1,1] => 12
[3,3,3,3] => 6
[3,3,3,2,1] => 7
[3,3,3,1,1,1] => 8
[3,3,2,2,2] => 7
[3,3,2,2,1,1] => 8
[3,3,2,1,1,1,1] => 9
[3,3,1,1,1,1,1,1] => 10
[3,2,2,2,2,1] => 8
[3,2,2,2,1,1,1] => 9
[3,2,2,1,1,1,1,1] => 10
[3,2,1,1,1,1,1,1,1] => 11
[3,1,1,1,1,1,1,1,1,1] => 12
[2,2,2,2,2,2] => 7
[2,2,2,2,2,1,1] => 8
[2,2,2,2,1,1,1,1] => 9
[2,2,2,1,1,1,1,1,1] => 10
[2,2,1,1,1,1,1,1,1,1] => 11
[2,1,1,1,1,1,1,1,1,1,1] => 12
[1,1,1,1,1,1,1,1,1,1,1,1] => 12
[13] => 13
[12,1] => 13
[10,3] => 11
[9,2,2] => 11
[8,5] => 9
[8,4,1] => 10
[8,3,2] => 10
[8,3,1,1] => 11
[7,6] => 8
[7,5,1] => 9
[7,4,2] => 9
[7,3,3] => 9
[6,6,1] => 8
[6,5,2] => 8
[6,5,1,1] => 9
[6,4,3] => 8
[6,4,2,1] => 9
[6,3,2,2] => 9
[6,3,1,1,1,1] => 11
[6,2,2,1,1,1] => 11
[6,1,1,1,1,1,1,1] => 13
[5,5,3] => 7
[5,4,4] => 7
[5,4,3,1] => 8
[5,4,2,2] => 8
[5,4,2,1,1] => 9
[5,4,1,1,1,1] => 10
[5,3,3,2] => 8
[5,3,3,1,1] => 9
[5,3,2,2,1] => 9
[5,3,2,1,1,1] => 10
[5,3,1,1,1,1,1] => 11
[5,2,2,2,1,1] => 10
[5,2,2,1,1,1,1] => 11
[5,2,1,1,1,1,1,1] => 12
[4,4,4,1] => 7
[4,4,3,2] => 7
[4,4,3,1,1] => 8
[4,4,2,2,1] => 8
[4,3,3,3] => 7
[4,3,3,2,1] => 8
[3,3,3,3,1] => 7
[3,3,3,2,2] => 7
[3,3,2,2,2,1] => 8
[3,3,2,1,1,1,1,1] => 10
[3,2,2,2,2,2] => 8
[3,2,2,2,2,1,1] => 9
[3,1,1,1,1,1,1,1,1,1,1] => 13
[2,2,2,2,2,2,1] => 8
[2,2,2,2,1,1,1,1,1] => 10
[1,1,1,1,1,1,1,1,1,1,1,1,1] => 13
[14] => 14
[13,1] => 14
[12,2] => 13
[12,1,1] => 14
[9,5] => 10
[8,6] => 9
[8,5,1] => 10
[8,4,2] => 10
[8,3,1,1,1] => 12
[7,7] => 8
[7,5,2] => 9
[7,4,3] => 9
[6,6,2] => 8
[6,6,1,1] => 9
[6,5,3] => 8
[6,5,1,1,1] => 10
[6,4,4] => 8
[6,4,2,2] => 9
[6,2,2,2,2] => 10
[6,1,1,1,1,1,1,1,1] => 14
[5,5,4] => 7
[5,5,1,1,1,1] => 10
[5,4,3,2] => 8
[5,4,3,1,1] => 9
[5,4,2,2,1] => 9
[5,4,2,1,1,1] => 10
[5,4,1,1,1,1,1] => 11
[5,3,3,3] => 8
[5,3,3,2,1] => 9
[5,3,2,2,2] => 9
[5,3,2,2,1,1] => 10
[5,3,2,1,1,1,1] => 11
[5,3,1,1,1,1,1,1] => 12
[5,2,2,2,2,1] => 10
[5,2,2,1,1,1,1,1] => 12
[4,4,4,2] => 7
[4,4,4,1,1] => 8
[4,4,3,3] => 7
[4,4,3,2,1] => 8
[4,3,2,2,2,1] => 9
[3,3,3,3,2] => 7
[3,3,3,3,1,1] => 8
[3,3,3,2,2,1] => 8
[3,3,2,2,2,2] => 8
[3,3,1,1,1,1,1,1,1,1] => 12
[3,2,2,2,2,1,1,1] => 10
[2,2,2,2,2,2,2] => 8
[1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 14
[15] => 15
[14,1] => 15
[12,3] => 13
[11,2,2] => 13
[9,6] => 10
[9,5,1] => 11
[8,7] => 9
[8,5,2] => 10
[7,5,3] => 9
[6,6,3] => 8
[6,5,4] => 8
[6,5,1,1,1,1] => 11
[6,4,3,1,1] => 10
[6,3,3,3] => 9
[6,3,1,1,1,1,1,1] => 13
[6,2,2,2,2,1] => 11
[5,5,5] => 7
[5,4,3,3] => 8
[5,4,3,2,1] => 9
[5,4,3,1,1,1] => 10
[5,3,2,2,2,1] => 10
[5,3,2,2,1,1,1] => 11
[5,3,1,1,1,1,1,1,1] => 13
[4,4,4,3] => 7
[4,4,4,1,1,1] => 9
[4,3,3,3,2] => 8
[3,3,3,3,3] => 7
[3,3,3,3,2,1] => 8
[3,3,3,2,2,2] => 8
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 15
[16] => 16
[15,1] => 16
[12,4] => 13
[12,1,1,1,1] => 16
[10,6] => 11
[8,8] => 9
[8,6,2] => 10
[8,5,3] => 10
[7,6,3] => 9
[7,5,3,1] => 10
[6,6,4] => 8
[6,6,2,2] => 9
[6,5,5] => 8
[6,4,3,3] => 9
[5,5,3,3] => 8
[5,5,2,2,2] => 9
[5,4,4,3] => 8
[5,4,3,2,1,1] => 10
[5,4,2,2,2,1] => 10
[4,4,4,4] => 7
[4,4,4,2,2] => 8
[4,4,3,3,2] => 8
[4,3,3,3,3] => 8
[4,3,3,3,2,1] => 9
[3,3,3,3,2,2] => 8
[3,3,3,3,1,1,1,1] => 10
[3,3,2,2,2,2,2] => 9
[2,2,2,2,2,2,2,2] => 9
[2,2,2,2,2,2,1,1,1,1] => 11
[2,2,2,2,1,1,1,1,1,1,1,1] => 13
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 16
[17] => 17
[9,8] => 10
[8,8,1] => 10
[8,6,3] => 10
[7,5,3,2] => 10
[6,5,3,3] => 9
[6,5,2,2,2] => 10
[6,4,4,3] => 9
[6,4,4,1,1,1] => 11
[6,3,3,3,2] => 10
[6,3,3,3,1,1] => 11
[5,5,4,3] => 8
[5,5,4,1,1,1] => 10
[5,5,2,2,2,1] => 10
[5,4,4,4] => 8
[5,4,3,2,2,1] => 10
[5,3,3,3,2,1] => 10
[4,4,4,3,2] => 8
[4,4,4,3,1,1] => 9
[4,4,4,2,2,1] => 9
[4,4,3,3,3] => 8
[3,3,3,3,3,2] => 8
[3,2,2,2,2,2,2,2] => 10
[3,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 17
-----------------------------------------------------------------------------
Created: Apr 06, 2016 at 11:26 by Christian Stump
-----------------------------------------------------------------------------
Last Updated: Dec 29, 2023 at 18:11 by Martin Rubey