*****************************************************************************
* 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: St000063
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The number of linear extensions of a certain poset defined for an integer partition.
The poset is constructed in David Speyer's answer to Matt Fayers' question [3].
The value at the partition $\lambda$ also counts cover-inclusive Dyck tilings of $\lambda\setminus\mu$, summed over all $\mu$, as noticed by Philippe Nadeau in a comment.
This statistic arises in the homogeneous Garnir relations for the universal graded Specht modules for cyclotomic quiver Hecke algebras.
-----------------------------------------------------------------------------
References: [1] Fayers, M. Dyck tilings and the homogeneous Garnir relations for graded Specht modules [[arXiv:1309.6467]]
[2] Kenyon, R. W., Wilson, D. B. Double-dimer pairings and skew Young diagrams [[MathSciNet:2811099]]
[3] Fayers, M. A function from partitions to natural numbers - is it familiar? [[MathOverflow:132338]]
-----------------------------------------------------------------------------
Code:
def statistic( P ):
if P.is_empty():
return 1
cells = P.cells()
m = max( i+j for i,j in cells )
found_max = False
while found_max is False:
i,j = cells.pop()
if i+j == m:
found_max = True
P1 = Partition( P[i+1:] )
P2 = Partition( P.conjugate()[j+1:] ).conjugate()
return binomial(i+j+2,i+1)*statistic(P1)*statistic(P2)
-----------------------------------------------------------------------------
Statistic values:
[] => 1
[1] => 2
[2] => 3
[1,1] => 3
[3] => 4
[2,1] => 6
[1,1,1] => 4
[4] => 5
[3,1] => 8
[2,2] => 6
[2,1,1] => 8
[1,1,1,1] => 5
[5] => 6
[4,1] => 10
[3,2] => 12
[3,1,1] => 12
[2,2,1] => 12
[2,1,1,1] => 10
[1,1,1,1,1] => 6
[6] => 7
[5,1] => 12
[4,2] => 15
[4,1,1] => 15
[3,3] => 10
[3,2,1] => 24
[3,1,1,1] => 15
[2,2,2] => 10
[2,2,1,1] => 15
[2,1,1,1,1] => 12
[1,1,1,1,1,1] => 7
[7] => 8
[6,1] => 14
[5,2] => 18
[5,1,1] => 18
[4,3] => 20
[4,2,1] => 30
[4,1,1,1] => 20
[3,3,1] => 20
[3,2,2] => 20
[3,2,1,1] => 30
[3,1,1,1,1] => 18
[2,2,2,1] => 20
[2,2,1,1,1] => 18
[2,1,1,1,1,1] => 14
[1,1,1,1,1,1,1] => 8
[8] => 9
[7,1] => 16
[6,2] => 21
[6,1,1] => 21
[5,3] => 24
[5,2,1] => 36
[5,1,1,1] => 24
[4,4] => 15
[4,3,1] => 40
[4,2,2] => 30
[4,2,1,1] => 40
[4,1,1,1,1] => 24
[3,3,2] => 30
[3,3,1,1] => 30
[3,2,2,1] => 40
[3,2,1,1,1] => 36
[3,1,1,1,1,1] => 21
[2,2,2,2] => 15
[2,2,2,1,1] => 24
[2,2,1,1,1,1] => 21
[2,1,1,1,1,1,1] => 16
[1,1,1,1,1,1,1,1] => 9
[9] => 10
[8,1] => 18
[7,2] => 24
[7,1,1] => 24
[6,3] => 28
[6,2,1] => 42
[6,1,1,1] => 28
[5,4] => 30
[5,3,1] => 48
[5,2,2] => 36
[5,2,1,1] => 48
[5,1,1,1,1] => 30
[4,4,1] => 30
[4,3,2] => 60
[4,3,1,1] => 60
[4,2,2,1] => 60
[4,2,1,1,1] => 48
[4,1,1,1,1,1] => 28
[3,3,3] => 20
[3,3,2,1] => 60
[3,3,1,1,1] => 36
[3,2,2,2] => 30
[3,2,2,1,1] => 48
[3,2,1,1,1,1] => 42
[3,1,1,1,1,1,1] => 24
[2,2,2,2,1] => 30
[2,2,2,1,1,1] => 28
[2,2,1,1,1,1,1] => 24
[2,1,1,1,1,1,1,1] => 18
[1,1,1,1,1,1,1,1,1] => 10
[10] => 11
[9,1] => 20
[8,2] => 27
[8,1,1] => 27
[7,3] => 32
[7,2,1] => 48
[7,1,1,1] => 32
[6,4] => 35
[6,3,1] => 56
[6,2,2] => 42
[6,2,1,1] => 56
[6,1,1,1,1] => 35
[5,5] => 21
[5,4,1] => 60
[5,3,2] => 72
[5,3,1,1] => 72
[5,2,2,1] => 72
[5,2,1,1,1] => 60
[5,1,1,1,1,1] => 35
[4,4,2] => 45
[4,4,1,1] => 45
[4,3,3] => 40
[4,3,2,1] => 120
[4,3,1,1,1] => 72
[4,2,2,2] => 45
[4,2,2,1,1] => 72
[4,2,1,1,1,1] => 56
[4,1,1,1,1,1,1] => 32
[3,3,3,1] => 40
[3,3,2,2] => 45
[3,3,2,1,1] => 72
[3,3,1,1,1,1] => 42
[3,2,2,2,1] => 60
[3,2,2,1,1,1] => 56
[3,2,1,1,1,1,1] => 48
[3,1,1,1,1,1,1,1] => 27
[2,2,2,2,2] => 21
[2,2,2,2,1,1] => 35
[2,2,2,1,1,1,1] => 32
[2,2,1,1,1,1,1,1] => 27
[2,1,1,1,1,1,1,1,1] => 20
[1,1,1,1,1,1,1,1,1,1] => 11
[11] => 12
[10,1] => 22
[9,2] => 30
[9,1,1] => 30
[8,3] => 36
[8,2,1] => 54
[8,1,1,1] => 36
[7,4] => 40
[7,3,1] => 64
[7,2,2] => 48
[7,2,1,1] => 64
[7,1,1,1,1] => 40
[6,5] => 42
[6,4,1] => 70
[6,3,2] => 84
[6,3,1,1] => 84
[6,2,2,1] => 84
[6,2,1,1,1] => 70
[6,1,1,1,1,1] => 42
[5,5,1] => 42
[5,4,2] => 90
[5,4,1,1] => 90
[5,3,3] => 60
[5,3,2,1] => 144
[5,3,1,1,1] => 90
[5,2,2,2] => 60
[5,2,2,1,1] => 90
[5,2,1,1,1,1] => 70
[5,1,1,1,1,1,1] => 40
[4,4,3] => 60
[4,4,2,1] => 90
[4,4,1,1,1] => 60
[4,3,3,1] => 80
[4,3,2,2] => 90
[4,3,2,1,1] => 144
[4,3,1,1,1,1] => 84
[4,2,2,2,1] => 90
[4,2,2,1,1,1] => 84
[4,2,1,1,1,1,1] => 64
[4,1,1,1,1,1,1,1] => 36
[3,3,3,2] => 60
[3,3,3,1,1] => 60
[3,3,2,2,1] => 90
[3,3,2,1,1,1] => 84
[3,3,1,1,1,1,1] => 48
[3,2,2,2,2] => 42
[3,2,2,2,1,1] => 70
[3,2,2,1,1,1,1] => 64
[3,2,1,1,1,1,1,1] => 54
[3,1,1,1,1,1,1,1,1] => 30
[2,2,2,2,2,1] => 42
[2,2,2,2,1,1,1] => 40
[2,2,2,1,1,1,1,1] => 36
[2,2,1,1,1,1,1,1,1] => 30
[2,1,1,1,1,1,1,1,1,1] => 22
[1,1,1,1,1,1,1,1,1,1,1] => 12
[12] => 13
[11,1] => 24
[10,2] => 33
[10,1,1] => 33
[9,3] => 40
[9,2,1] => 60
[9,1,1,1] => 40
[8,4] => 45
[8,3,1] => 72
[8,2,2] => 54
[8,2,1,1] => 72
[8,1,1,1,1] => 45
[7,5] => 48
[7,4,1] => 80
[7,3,2] => 96
[7,3,1,1] => 96
[7,2,2,1] => 96
[7,2,1,1,1] => 80
[7,1,1,1,1,1] => 48
[6,6] => 28
[6,5,1] => 84
[6,4,2] => 105
[6,4,1,1] => 105
[6,3,3] => 70
[6,3,2,1] => 168
[6,3,1,1,1] => 105
[6,2,2,2] => 70
[6,2,2,1,1] => 105
[6,2,1,1,1,1] => 84
[6,1,1,1,1,1,1] => 48
[5,5,2] => 63
[5,5,1,1] => 63
[5,4,3] => 120
[5,4,2,1] => 180
[5,4,1,1,1] => 120
[5,3,3,1] => 120
[5,3,2,2] => 120
[5,3,2,1,1] => 180
[5,3,1,1,1,1] => 105
[5,2,2,2,1] => 120
[5,2,2,1,1,1] => 105
[5,2,1,1,1,1,1] => 80
[5,1,1,1,1,1,1,1] => 45
[4,4,4] => 35
[4,4,3,1] => 120
[4,4,2,2] => 90
[4,4,2,1,1] => 120
[4,4,1,1,1,1] => 70
[4,3,3,2] => 120
[4,3,3,1,1] => 120
[4,3,2,2,1] => 180
[4,3,2,1,1,1] => 168
[4,3,1,1,1,1,1] => 96
[4,2,2,2,2] => 63
[4,2,2,2,1,1] => 105
[4,2,2,1,1,1,1] => 96
[4,2,1,1,1,1,1,1] => 72
[4,1,1,1,1,1,1,1,1] => 40
[3,3,3,3] => 35
[3,3,3,2,1] => 120
[3,3,3,1,1,1] => 70
[3,3,2,2,2] => 63
[3,3,2,2,1,1] => 105
[3,3,2,1,1,1,1] => 96
[3,3,1,1,1,1,1,1] => 54
[3,2,2,2,2,1] => 84
[3,2,2,2,1,1,1] => 80
[3,2,2,1,1,1,1,1] => 72
[3,2,1,1,1,1,1,1,1] => 60
[3,1,1,1,1,1,1,1,1,1] => 33
[2,2,2,2,2,2] => 28
[2,2,2,2,2,1,1] => 48
[2,2,2,2,1,1,1,1] => 45
[2,2,2,1,1,1,1,1,1] => 40
[2,2,1,1,1,1,1,1,1,1] => 33
[2,1,1,1,1,1,1,1,1,1,1] => 24
[1,1,1,1,1,1,1,1,1,1,1,1] => 13
[5,4,3,1] => 240
[5,4,2,2] => 180
[5,4,2,1,1] => 240
[5,3,3,2] => 180
[5,3,3,1,1] => 180
[5,3,2,2,1] => 240
[4,4,3,2] => 180
[4,4,3,1,1] => 180
[4,4,2,2,1] => 180
[4,3,3,2,1] => 240
[5,4,3,2] => 360
[5,4,3,1,1] => 360
[5,4,2,2,1] => 360
[5,3,3,2,1] => 360
[4,4,3,2,1] => 360
[5,4,3,2,1] => 720
-----------------------------------------------------------------------------
Created: May 31, 2013 at 11:49 by Christian Stump
-----------------------------------------------------------------------------
Last Updated: Mar 19, 2019 at 23:37 by Martin Rubey