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Identifier
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