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Identifier

St001387: Integer partitions ⟶ ℤ

Values

[] => 1

[1] => 1

[2] => 1

[1,1] => 2

[3] => 1

[2,1] => 6

[1,1,1] => 3

[4] => 1

[3,1] => 12

[2,2] => 6

[2,1,1] => 12

[1,1,1,1] => 4

[5] => 1

[4,1] => 20

[3,2] => 30

[3,1,1] => 30

[2,2,1] => 20

[2,1,1,1] => 20

[1,1,1,1,1] => 5

[6] => 1

[5,1] => 30

[4,2] => 90

[4,1,1] => 60

[3,3] => 20

[3,2,1] => 120

[3,1,1,1] => 60

[2,2,2] => 15

[2,2,1,1] => 30

[2,1,1,1,1] => 30

[1,1,1,1,1,1] => 6

[7] => 1

[6,1] => 42

[5,2] => 210

[5,1,1] => 105

[4,3] => 140

[4,2,1] => 420

[4,1,1,1] => 140

[3,3,1] => 105

[3,2,2] => 105

[3,2,1,1] => 210

[3,1,1,1,1] => 105

[2,2,2,1] => 42

[2,2,1,1,1] => 42

[2,1,1,1,1,1] => 42

[1,1,1,1,1,1,1] => 7

[8] => 1

[7,1] => 56

[6,2] => 420

[6,1,1] => 168

[5,3] => 560

[5,2,1] => 1120

[5,1,1,1] => 280

[4,4] => 70

[4,3,1] => 840

[4,2,2] => 420

[4,2,1,1] => 840

[4,1,1,1,1] => 280

[3,3,2] => 168

[3,3,1,1] => 168

[3,2,2,1] => 336

[3,2,1,1,1] => 336

[3,1,1,1,1,1] => 168

[2,2,2,2] => 28

[2,2,2,1,1] => 56

[2,2,1,1,1,1] => 56

[2,1,1,1,1,1,1] => 56

[1,1,1,1,1,1,1,1] => 8

[9] => 1

[8,1] => 72

[7,2] => 756

[7,1,1] => 252

[6,3] => 1680

[6,2,1] => 2520

[6,1,1,1] => 504

[5,4] => 630

[5,3,1] => 3780

[5,2,2] => 1260

[5,2,1,1] => 2520

[5,1,1,1,1] => 630

[4,4,1] => 504

[4,3,2] => 1512

[4,3,1,1] => 1512

[4,2,2,1] => 1512

[4,2,1,1,1] => 1512

[4,1,1,1,1,1] => 504

[3,3,3] => 84

[3,3,2,1] => 504

[3,3,1,1,1] => 252

[3,2,2,2] => 252

[3,2,2,1,1] => 504

[3,2,1,1,1,1] => 504

[3,1,1,1,1,1,1] => 252

[2,2,2,2,1] => 72

[2,2,2,1,1,1] => 72

[2,2,1,1,1,1,1] => 72

[2,1,1,1,1,1,1,1] => 72

[1,1,1,1,1,1,1,1,1] => 9

[10] => 1

[9,1] => 90

[8,2] => 1260

[8,1,1] => 360

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Description

Number of standard Young tableaux of the skew shape tracing the border of the given partition.
Let $\lambda \vdash n$ be a diagram with the given partition as shape.
Add $n$ additional boxes, one in each column $1,\dotsc,n$, and let this be $\mu$.
The statistic is the number of standard Young tableaux of skew shape $\mu/\lambda$,
which is equal to $\frac{n!}{\prod_{i} (\mu_i - \lambda_i)!}$.
For example, $\lambda=[2,1,1]$ gives $\mu = [4,2,1,1]$.
The first row in the skew shape $\mu/\lambda$ has two boxes, so the number of SYT of
shape $\mu/\lambda$ is then $4!/2 = 12$.
This statistic shows up in the study of skew specialized Macdonald polynomials,
where a type of charge statistic give rise to a $q$-analogue of the above formula.

Code

def statistic(la):
    la = Partition(la)
    n = la.size()
    la_t = la.conjugate()
    mu = Partition([c + 1 for c in la_t] + [1]*(n-len(la_t)))
    return StandardTableaux(SkewPartition([mu.conjugate(), la])).cardinality()

Created

Apr 30, 2019 at 11:28 by Per Alexandersson

Updated

Apr 23, 2022 at 10:56 by Martin Rubey