- FindStat

Identifier

St000063: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        []=>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
                    

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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)

Created

May 31, 2013 at 11:49 by Christian Stump

Updated

Mar 19, 2019 at 23:37 by Martin Rubey