- FindStat

Identifier

St000063: Integer partitions ⟶ ℤ

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

>>> Load all 288 entries. <<<

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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