*****************************************************************************
* 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: St000531
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The leading coefficient of the rook polynomial of an integer partition.
Let $m$ be the minimum of the number of parts and the size of the first part of an integer partition $\lambda$. Then this statistic yields the number of ways to place $m$ non-attacking rooks on the Ferrers board of $\lambda$.
-----------------------------------------------------------------------------
References: [1] [[wikipedia:Rook_polynomial]]
-----------------------------------------------------------------------------
Code:
def statistic(la):
return (matrix([[1]*p + [0]*(la[0]-p) for p in la]).rook_vector())[-1]
-----------------------------------------------------------------------------
Statistic values:
[] => 1
[1] => 1
[2] => 2
[1,1] => 2
[3] => 3
[2,1] => 1
[1,1,1] => 3
[4] => 4
[3,1] => 2
[2,2] => 2
[2,1,1] => 2
[1,1,1,1] => 4
[5] => 5
[4,1] => 3
[3,2] => 4
[3,1,1] => 0
[2,2,1] => 4
[2,1,1,1] => 3
[1,1,1,1,1] => 5
[6] => 6
[5,1] => 4
[4,2] => 6
[4,1,1] => 0
[3,3] => 6
[3,2,1] => 1
[3,1,1,1] => 0
[2,2,2] => 6
[2,2,1,1] => 6
[2,1,1,1,1] => 4
[1,1,1,1,1,1] => 6
[7] => 7
[6,1] => 5
[5,2] => 8
[5,1,1] => 0
[4,3] => 9
[4,2,1] => 2
[4,1,1,1] => 0
[3,3,1] => 2
[3,2,2] => 2
[3,2,1,1] => 2
[3,1,1,1,1] => 0
[2,2,2,1] => 9
[2,2,1,1,1] => 8
[2,1,1,1,1,1] => 5
[1,1,1,1,1,1,1] => 7
[8] => 8
[7,1] => 6
[6,2] => 10
[6,1,1] => 0
[5,3] => 12
[5,2,1] => 3
[5,1,1,1] => 0
[4,4] => 12
[4,3,1] => 4
[4,2,2] => 4
[4,2,1,1] => 0
[4,1,1,1,1] => 0
[3,3,2] => 4
[3,3,1,1] => 4
[3,2,2,1] => 4
[3,2,1,1,1] => 3
[3,1,1,1,1,1] => 0
[2,2,2,2] => 12
[2,2,2,1,1] => 12
[2,2,1,1,1,1] => 10
[2,1,1,1,1,1,1] => 6
[1,1,1,1,1,1,1,1] => 8
[9] => 9
[8,1] => 7
[7,2] => 12
[7,1,1] => 0
[6,3] => 15
[6,2,1] => 4
[6,1,1,1] => 0
[5,4] => 16
[5,3,1] => 6
[5,2,2] => 6
[5,2,1,1] => 0
[5,1,1,1,1] => 0
[4,4,1] => 6
[4,3,2] => 8
[4,3,1,1] => 0
[4,2,2,1] => 0
[4,2,1,1,1] => 0
[4,1,1,1,1,1] => 0
[3,3,3] => 6
[3,3,2,1] => 8
[3,3,1,1,1] => 6
[3,2,2,2] => 6
[3,2,2,1,1] => 6
[3,2,1,1,1,1] => 4
[3,1,1,1,1,1,1] => 0
[2,2,2,2,1] => 16
[2,2,2,1,1,1] => 15
[2,2,1,1,1,1,1] => 12
[2,1,1,1,1,1,1,1] => 7
[1,1,1,1,1,1,1,1,1] => 9
[10] => 10
[9,1] => 8
[8,2] => 14
[8,1,1] => 0
[7,3] => 18
[7,2,1] => 5
[7,1,1,1] => 0
[6,4] => 20
[6,3,1] => 8
[6,2,2] => 8
[6,2,1,1] => 0
[6,1,1,1,1] => 0
[5,5] => 20
[5,4,1] => 9
[5,3,2] => 12
[5,3,1,1] => 0
[5,2,2,1] => 0
[5,2,1,1,1] => 0
[5,1,1,1,1,1] => 0
[4,4,2] => 12
[4,4,1,1] => 0
[4,3,3] => 12
[4,3,2,1] => 1
[4,3,1,1,1] => 0
[4,2,2,2] => 0
[4,2,2,1,1] => 0
[4,2,1,1,1,1] => 0
[4,1,1,1,1,1,1] => 0
[3,3,3,1] => 12
[3,3,2,2] => 12
[3,3,2,1,1] => 12
[3,3,1,1,1,1] => 8
[3,2,2,2,1] => 9
[3,2,2,1,1,1] => 8
[3,2,1,1,1,1,1] => 5
[3,1,1,1,1,1,1,1] => 0
[2,2,2,2,2] => 20
[2,2,2,2,1,1] => 20
[2,2,2,1,1,1,1] => 18
[2,2,1,1,1,1,1,1] => 14
[2,1,1,1,1,1,1,1,1] => 8
[1,1,1,1,1,1,1,1,1,1] => 10
[11] => 11
[10,1] => 9
[9,2] => 16
[9,1,1] => 0
[8,3] => 21
[8,2,1] => 6
[8,1,1,1] => 0
[7,4] => 24
[7,3,1] => 10
[7,2,2] => 10
[7,2,1,1] => 0
[7,1,1,1,1] => 0
[6,5] => 25
[6,4,1] => 12
[6,3,2] => 16
[6,3,1,1] => 0
[6,2,2,1] => 0
[6,2,1,1,1] => 0
[6,1,1,1,1,1] => 0
[5,5,1] => 12
[5,4,2] => 18
[5,4,1,1] => 0
[5,3,3] => 18
[5,3,2,1] => 2
[5,3,1,1,1] => 0
[5,2,2,2] => 0
[5,2,2,1,1] => 0
[5,2,1,1,1,1] => 0
[5,1,1,1,1,1,1] => 0
[4,4,3] => 18
[4,4,2,1] => 2
[4,4,1,1,1] => 0
[4,3,3,1] => 2
[4,3,2,2] => 2
[4,3,2,1,1] => 2
[4,3,1,1,1,1] => 0
[4,2,2,2,1] => 0
[4,2,2,1,1,1] => 0
[4,2,1,1,1,1,1] => 0
[4,1,1,1,1,1,1,1] => 0
[3,3,3,2] => 18
[3,3,3,1,1] => 18
[3,3,2,2,1] => 18
[3,3,2,1,1,1] => 16
[3,3,1,1,1,1,1] => 10
[3,2,2,2,2] => 12
[3,2,2,2,1,1] => 12
[3,2,2,1,1,1,1] => 10
[3,2,1,1,1,1,1,1] => 6
[3,1,1,1,1,1,1,1,1] => 0
[2,2,2,2,2,1] => 25
[2,2,2,2,1,1,1] => 24
[2,2,2,1,1,1,1,1] => 21
[2,2,1,1,1,1,1,1,1] => 16
[2,1,1,1,1,1,1,1,1,1] => 9
[1,1,1,1,1,1,1,1,1,1,1] => 11
[12] => 12
[11,1] => 10
[10,2] => 18
[10,1,1] => 0
[9,3] => 24
[9,2,1] => 7
[9,1,1,1] => 0
[8,4] => 28
[8,3,1] => 12
[8,2,2] => 12
[8,2,1,1] => 0
[8,1,1,1,1] => 0
[7,5] => 30
[7,4,1] => 15
[7,3,2] => 20
[7,3,1,1] => 0
[7,2,2,1] => 0
[7,2,1,1,1] => 0
[7,1,1,1,1,1] => 0
[6,6] => 30
[6,5,1] => 16
[6,4,2] => 24
[6,4,1,1] => 0
[6,3,3] => 24
[6,3,2,1] => 3
[6,3,1,1,1] => 0
[6,2,2,2] => 0
[6,2,2,1,1] => 0
[6,2,1,1,1,1] => 0
[6,1,1,1,1,1,1] => 0
[5,5,2] => 24
[5,5,1,1] => 0
[5,4,3] => 27
[5,4,2,1] => 4
[5,4,1,1,1] => 0
[5,3,3,1] => 4
[5,3,2,2] => 4
[5,3,2,1,1] => 0
[5,3,1,1,1,1] => 0
[5,2,2,2,1] => 0
[5,2,2,1,1,1] => 0
[5,2,1,1,1,1,1] => 0
[5,1,1,1,1,1,1,1] => 0
[4,4,4] => 24
[4,4,3,1] => 4
[4,4,2,2] => 4
[4,4,2,1,1] => 4
[4,4,1,1,1,1] => 0
[4,3,3,2] => 4
[4,3,3,1,1] => 4
[4,3,2,2,1] => 4
[4,3,2,1,1,1] => 3
[4,3,1,1,1,1,1] => 0
[4,2,2,2,2] => 0
[4,2,2,2,1,1] => 0
[4,2,2,1,1,1,1] => 0
[4,2,1,1,1,1,1,1] => 0
[4,1,1,1,1,1,1,1,1] => 0
[3,3,3,3] => 24
[3,3,3,2,1] => 27
[3,3,3,1,1,1] => 24
[3,3,2,2,2] => 24
[3,3,2,2,1,1] => 24
[3,3,2,1,1,1,1] => 20
[3,3,1,1,1,1,1,1] => 12
[3,2,2,2,2,1] => 16
[3,2,2,2,1,1,1] => 15
[3,2,2,1,1,1,1,1] => 12
[3,2,1,1,1,1,1,1,1] => 7
[3,1,1,1,1,1,1,1,1,1] => 0
[2,2,2,2,2,2] => 30
[2,2,2,2,2,1,1] => 30
[2,2,2,2,1,1,1,1] => 28
[2,2,2,1,1,1,1,1,1] => 24
[2,2,1,1,1,1,1,1,1,1] => 18
[2,1,1,1,1,1,1,1,1,1,1] => 10
[1,1,1,1,1,1,1,1,1,1,1,1] => 12
-----------------------------------------------------------------------------
Created: Jun 10, 2016 at 23:50 by Martin Rubey
-----------------------------------------------------------------------------
Last Updated: Dec 22, 2020 at 13:32 by Martin Rubey