Identifier
Identifier
Values
[] generating graphics... => 1
[1] generating graphics... => 2
[2] generating graphics... => 3
[1,1] generating graphics... => 3
[3] generating graphics... => 4
[2,1] generating graphics... => 6
[1,1,1] generating graphics... => 4
[4] generating graphics... => 5
[3,1] generating graphics... => 8
[2,2] generating graphics... => 6
[2,1,1] generating graphics... => 8
[1,1,1,1] generating graphics... => 5
[5] generating graphics... => 6
[4,1] generating graphics... => 10
[3,2] generating graphics... => 12
[3,1,1] generating graphics... => 12
[2,2,1] generating graphics... => 12
[2,1,1,1] generating graphics... => 10
[1,1,1,1,1] generating graphics... => 6
[6] generating graphics... => 7
[5,1] generating graphics... => 12
[4,2] generating graphics... => 15
[4,1,1] generating graphics... => 15
[3,3] generating graphics... => 10
[3,2,1] generating graphics... => 24
[3,1,1,1] generating graphics... => 15
[2,2,2] generating graphics... => 10
[2,2,1,1] generating graphics... => 15
[2,1,1,1,1] generating graphics... => 12
[1,1,1,1,1,1] generating graphics... => 7
[7] generating graphics... => 8
[6,1] generating graphics... => 14
[5,2] generating graphics... => 18
[5,1,1] generating graphics... => 18
[4,3] generating graphics... => 20
[4,2,1] generating graphics... => 30
[4,1,1,1] generating graphics... => 20
[3,3,1] generating graphics... => 20
[3,2,2] generating graphics... => 20
[3,2,1,1] generating graphics... => 30
[3,1,1,1,1] generating graphics... => 18
[2,2,2,1] generating graphics... => 20
[2,2,1,1,1] generating graphics... => 18
[2,1,1,1,1,1] generating graphics... => 14
[1,1,1,1,1,1,1] generating graphics... => 8
[8] generating graphics... => 9
[7,1] generating graphics... => 16
[6,2] generating graphics... => 21
[6,1,1] generating graphics... => 21
[5,3] generating graphics... => 24
[5,2,1] generating graphics... => 36
[5,1,1,1] generating graphics... => 24
[4,4] generating graphics... => 15
[4,3,1] generating graphics... => 40
[4,2,2] generating graphics... => 30
[4,2,1,1] generating graphics... => 40
[4,1,1,1,1] generating graphics... => 24
[3,3,2] generating graphics... => 30
[3,3,1,1] generating graphics... => 30
[3,2,2,1] generating graphics... => 40
[3,2,1,1,1] generating graphics... => 36
[3,1,1,1,1,1] generating graphics... => 21
[2,2,2,2] generating graphics... => 15
[2,2,2,1,1] generating graphics... => 24
[2,2,1,1,1,1] generating graphics... => 21
[2,1,1,1,1,1,1] generating graphics... => 16
[1,1,1,1,1,1,1,1] generating graphics... => 9
[9] generating graphics... => 10
[8,1] generating graphics... => 18
[7,2] generating graphics... => 24
[7,1,1] generating graphics... => 24
[6,3] generating graphics... => 28
[6,2,1] generating graphics... => 42
[6,1,1,1] generating graphics... => 28
[5,4] generating graphics... => 30
[5,3,1] generating graphics... => 48
[5,2,2] generating graphics... => 36
[5,2,1,1] generating graphics... => 48
[5,1,1,1,1] generating graphics... => 30
[4,4,1] generating graphics... => 30
[4,3,2] generating graphics... => 60
[4,3,1,1] generating graphics... => 60
[4,2,2,1] generating graphics... => 60
[4,2,1,1,1] generating graphics... => 48
[4,1,1,1,1,1] generating graphics... => 28
[3,3,3] generating graphics... => 20
[3,3,2,1] generating graphics... => 60
[3,3,1,1,1] generating graphics... => 36
[3,2,2,2] generating graphics... => 30
[3,2,2,1,1] generating graphics... => 48
[3,2,1,1,1,1] generating graphics... => 42
[3,1,1,1,1,1,1] generating graphics... => 24
[2,2,2,2,1] generating graphics... => 30
[2,2,2,1,1,1] generating graphics... => 28
[2,2,1,1,1,1,1] generating graphics... => 24
[2,1,1,1,1,1,1,1] generating graphics... => 18
[1,1,1,1,1,1,1,1,1] generating graphics... => 10
[10] generating graphics... => 11
[9,1] generating graphics... => 20
[8,2] generating graphics... => 27
[8,1,1] generating graphics... => 27
[7,3] generating graphics... => 32
[7,2,1] generating graphics... => 48
[7,1,1,1] generating graphics... => 32
[6,4] generating graphics... => 35
[6,3,1] generating graphics... => 56
[6,2,2] generating graphics... => 42
[6,2,1,1] generating graphics... => 56
[6,1,1,1,1] generating graphics... => 35
[5,5] generating graphics... => 21
[5,4,1] generating graphics... => 60
[5,3,2] generating graphics... => 72
[5,3,1,1] generating graphics... => 72
[5,2,2,1] generating graphics... => 72
[5,2,1,1,1] generating graphics... => 60
[5,1,1,1,1,1] generating graphics... => 35
[4,4,2] generating graphics... => 45
[4,4,1,1] generating graphics... => 45
[4,3,3] generating graphics... => 40
[4,3,2,1] generating graphics... => 120
[4,3,1,1,1] generating graphics... => 72
[4,2,2,2] generating graphics... => 45
[4,2,2,1,1] generating graphics... => 72
[4,2,1,1,1,1] generating graphics... => 56
[4,1,1,1,1,1,1] generating graphics... => 32
[3,3,3,1] generating graphics... => 40
[3,3,2,2] generating graphics... => 45
[3,3,2,1,1] generating graphics... => 72
[3,3,1,1,1,1] generating graphics... => 42
[3,2,2,2,1] generating graphics... => 60
[3,2,2,1,1,1] generating graphics... => 56
[3,2,1,1,1,1,1] generating graphics... => 48
[3,1,1,1,1,1,1,1] generating graphics... => 27
[2,2,2,2,2] generating graphics... => 21
[2,2,2,2,1,1] generating graphics... => 35
[2,2,2,1,1,1,1] generating graphics... => 32
[2,2,1,1,1,1,1,1] generating graphics... => 27
[2,1,1,1,1,1,1,1,1] generating graphics... => 20
[1,1,1,1,1,1,1,1,1,1] generating graphics... => 11
[11] generating graphics... => 12
[10,1] generating graphics... => 22
[9,2] generating graphics... => 30
[9,1,1] generating graphics... => 30
[8,3] generating graphics... => 36
[8,2,1] generating graphics... => 54
[8,1,1,1] generating graphics... => 36
[7,4] generating graphics... => 40
[7,3,1] generating graphics... => 64
[7,2,2] generating graphics... => 48
[7,2,1,1] generating graphics... => 64
[7,1,1,1,1] generating graphics... => 40
[6,5] generating graphics... => 42
[6,4,1] generating graphics... => 70
[6,3,2] generating graphics... => 84
[6,3,1,1] generating graphics... => 84
[6,2,2,1] generating graphics... => 84
[6,2,1,1,1] generating graphics... => 70
[6,1,1,1,1,1] generating graphics... => 42
[5,5,1] generating graphics... => 42
[5,4,2] generating graphics... => 90
[5,4,1,1] generating graphics... => 90
[5,3,3] generating graphics... => 60
[5,3,2,1] generating graphics... => 144
[5,3,1,1,1] generating graphics... => 90
[5,2,2,2] generating graphics... => 60
[5,2,2,1,1] generating graphics... => 90
[5,2,1,1,1,1] generating graphics... => 70
[5,1,1,1,1,1,1] generating graphics... => 40
[4,4,3] generating graphics... => 60
[4,4,2,1] generating graphics... => 90
[4,4,1,1,1] generating graphics... => 60
[4,3,3,1] generating graphics... => 80
[4,3,2,2] generating graphics... => 90
[4,3,2,1,1] generating graphics... => 144
[4,3,1,1,1,1] generating graphics... => 84
[4,2,2,2,1] generating graphics... => 90
[4,2,2,1,1,1] generating graphics... => 84
[4,2,1,1,1,1,1] generating graphics... => 64
[4,1,1,1,1,1,1,1] generating graphics... => 36
[3,3,3,2] generating graphics... => 60
[3,3,3,1,1] generating graphics... => 60
[3,3,2,2,1] generating graphics... => 90
[3,3,2,1,1,1] generating graphics... => 84
[3,3,1,1,1,1,1] generating graphics... => 48
[3,2,2,2,2] generating graphics... => 42
[3,2,2,2,1,1] generating graphics... => 70
[3,2,2,1,1,1,1] generating graphics... => 64
[3,2,1,1,1,1,1,1] generating graphics... => 54
[3,1,1,1,1,1,1,1,1] generating graphics... => 30
[2,2,2,2,2,1] generating graphics... => 42
[2,2,2,2,1,1,1] generating graphics... => 40
[2,2,2,1,1,1,1,1] generating graphics... => 36
[2,2,1,1,1,1,1,1,1] generating graphics... => 30
[2,1,1,1,1,1,1,1,1,1] generating graphics... => 22
[1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 12
[12] generating graphics... => 13
[11,1] generating graphics... => 24
[10,2] generating graphics... => 33
[10,1,1] generating graphics... => 33
[9,3] generating graphics... => 40
[9,2,1] generating graphics... => 60
[9,1,1,1] generating graphics... => 40
[8,4] generating graphics... => 45
[8,3,1] generating graphics... => 72
[8,2,2] generating graphics... => 54
[8,2,1,1] generating graphics... => 72
[8,1,1,1,1] generating graphics... => 45
[7,5] generating graphics... => 48
[7,4,1] generating graphics... => 80
[7,3,2] generating graphics... => 96
[7,3,1,1] generating graphics... => 96
[7,2,2,1] generating graphics... => 96
[7,2,1,1,1] generating graphics... => 80
[7,1,1,1,1,1] generating graphics... => 48
[6,6] generating graphics... => 28
[6,5,1] generating graphics... => 84
[6,4,2] generating graphics... => 105
[6,4,1,1] generating graphics... => 105
[6,3,3] generating graphics... => 70
[6,3,2,1] generating graphics... => 168
[6,3,1,1,1] generating graphics... => 105
[6,2,2,2] generating graphics... => 70
[6,2,2,1,1] generating graphics... => 105
[6,2,1,1,1,1] generating graphics... => 84
[6,1,1,1,1,1,1] generating graphics... => 48
[5,5,2] generating graphics... => 63
[5,5,1,1] generating graphics... => 63
[5,4,3] generating graphics... => 120
[5,4,2,1] generating graphics... => 180
[5,4,1,1,1] generating graphics... => 120
[5,3,3,1] generating graphics... => 120
[5,3,2,2] generating graphics... => 120
[5,3,2,1,1] generating graphics... => 180
[5,3,1,1,1,1] generating graphics... => 105
[5,2,2,2,1] generating graphics... => 120
[5,2,2,1,1,1] generating graphics... => 105
[5,2,1,1,1,1,1] generating graphics... => 80
[5,1,1,1,1,1,1,1] generating graphics... => 45
[4,4,4] generating graphics... => 35
[4,4,3,1] generating graphics... => 120
[4,4,2,2] generating graphics... => 90
[4,4,2,1,1] generating graphics... => 120
[4,4,1,1,1,1] generating graphics... => 70
[4,3,3,2] generating graphics... => 120
[4,3,3,1,1] generating graphics... => 120
[4,3,2,2,1] generating graphics... => 180
[4,3,2,1,1,1] generating graphics... => 168
[4,3,1,1,1,1,1] generating graphics... => 96
[4,2,2,2,2] generating graphics... => 63
[4,2,2,2,1,1] generating graphics... => 105
[4,2,2,1,1,1,1] generating graphics... => 96
[4,2,1,1,1,1,1,1] generating graphics... => 72
[4,1,1,1,1,1,1,1,1] generating graphics... => 40
[3,3,3,3] generating graphics... => 35
[3,3,3,2,1] generating graphics... => 120
[3,3,3,1,1,1] generating graphics... => 70
[3,3,2,2,2] generating graphics... => 63
[3,3,2,2,1,1] generating graphics... => 105
[3,3,2,1,1,1,1] generating graphics... => 96
[3,3,1,1,1,1,1,1] generating graphics... => 54
[3,2,2,2,2,1] generating graphics... => 84
[3,2,2,2,1,1,1] generating graphics... => 80
[3,2,2,1,1,1,1,1] generating graphics... => 72
[3,2,1,1,1,1,1,1,1] generating graphics... => 60
[3,1,1,1,1,1,1,1,1,1] generating graphics... => 33
[2,2,2,2,2,2] generating graphics... => 28
[2,2,2,2,2,1,1] generating graphics... => 48
[2,2,2,2,1,1,1,1] generating graphics... => 45
[2,2,2,1,1,1,1,1,1] generating graphics... => 40
[2,2,1,1,1,1,1,1,1,1] generating graphics... => 33
[2,1,1,1,1,1,1,1,1,1,1] generating graphics... => 24
[1,1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 13
[5,4,3,1] generating graphics... => 240
[5,4,2,2] generating graphics... => 180
[5,4,2,1,1] generating graphics... => 240
[5,3,3,2] generating graphics... => 180
[5,3,3,1,1] generating graphics... => 180
[5,3,2,2,1] generating graphics... => 240
[4,4,3,2] generating graphics... => 180
[4,4,3,1,1] generating graphics... => 180
[4,4,2,2,1] generating graphics... => 180
[4,3,3,2,1] generating graphics... => 240
[5,4,3,2] generating graphics... => 360
[5,4,3,1,1] generating graphics... => 360
[5,4,2,2,1] generating graphics... => 360
[5,3,3,2,1] generating graphics... => 360
[4,4,3,2,1] generating graphics... => 360
[5,4,3,2,1] generating graphics... => 720
click to show generating function       
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