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Identifier

St001593: Integer partitions ⟶ ℤ

Values

[1] => 1

[2] => 1

[1,1] => 0

[3] => 1

[2,1] => 1

[1,1,1] => 0

[4] => 1

[3,1] => 2

[2,2] => 0

[2,1,1] => 0

[1,1,1,1] => 0

[5] => 1

[4,1] => 3

[3,2] => 2

[3,1,1] => 0

[2,2,1] => 0

[2,1,1,1] => 0

[1,1,1,1,1] => 0

[6] => 1

[5,1] => 4

[4,2] => 5

[4,1,1] => 0

[3,3] => 0

[3,2,1] => 2

[3,1,1,1] => 0

[2,2,2] => 0

[2,2,1,1] => 0

[2,1,1,1,1] => 0

[1,1,1,1,1,1] => 0

[7] => 1

[6,1] => 5

[5,2] => 9

[5,1,1] => 0

[4,3] => 5

[4,2,1] => 7

[4,1,1,1] => 0

[3,3,1] => 0

[3,2,2] => 0

[3,2,1,1] => 0

[3,1,1,1,1] => 0

[2,2,2,1] => 0

[2,2,1,1,1] => 0

[2,1,1,1,1,1] => 0

[1,1,1,1,1,1,1] => 0

[8] => 1

[7,1] => 6

[6,2] => 14

[6,1,1] => 0

[5,3] => 14

[5,2,1] => 16

[5,1,1,1] => 0

[4,4] => 0

[4,3,1] => 12

[4,2,2] => 0

[4,2,1,1] => 0

[4,1,1,1,1] => 0

[3,3,2] => 0

[3,3,1,1] => 0

[3,2,2,1] => 0

[3,2,1,1,1] => 0

[3,1,1,1,1,1] => 0

[2,2,2,2] => 0

[2,2,2,1,1] => 0

[2,2,1,1,1,1] => 0

[2,1,1,1,1,1,1] => 0

[1,1,1,1,1,1,1,1] => 0

[9] => 1

[8,1] => 7

[7,2] => 20

[7,1,1] => 0

[6,3] => 28

[6,2,1] => 30

[6,1,1,1] => 0

[5,4] => 14

[5,3,1] => 42

[5,2,2] => 0

[5,2,1,1] => 0

[5,1,1,1,1] => 0

[4,4,1] => 0

[4,3,2] => 12

[4,3,1,1] => 0

[4,2,2,1] => 0

[4,2,1,1,1] => 0

[4,1,1,1,1,1] => 0

[3,3,3] => 0

[3,3,2,1] => 0

[3,3,1,1,1] => 0

[3,2,2,2] => 0

[3,2,2,1,1] => 0

[3,2,1,1,1,1] => 0

[3,1,1,1,1,1,1] => 0

[2,2,2,2,1] => 0

[2,2,2,1,1,1] => 0

[2,2,1,1,1,1,1] => 0

[2,1,1,1,1,1,1,1] => 0

[1,1,1,1,1,1,1,1,1] => 0

[10] => 1

[9,1] => 8

[8,2] => 27

[8,1,1] => 0

[7,3] => 48

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Description

This is the number of standard Young tableaux of the given shifted shape.
For an integer partition $\lambda = (\lambda_1,\dots,\lambda_k)$, the shifted diagram is obtained by moving the $i$-th row in the diagram $i-1$ boxes to the right, i.e.,
$$\lambda^∗ = \{(i, j) | 1 \leq i \leq k, i \leq j \leq \lambda_i + i − 1 \}.$$
In particular, this statistic is zero if and only if $\lambda_{i+1} = \lambda_i$ for some $i$.

References

[1] Cho, Y., Kim, J. S., Lee, E. Enumeration of Gelfand-Cetlin type reduced words arXiv:2009.06906

Code

def statistic(L):
    return multinomial(list(L)) * prod( ZZ(L[i] - L[j]) / ZZ(L[i] + L[j]) for j in range(len(L)) for i in range(j))

Created

Sep 18, 2020 at 10:32 by Christian Stump

Updated

Sep 18, 2020 at 10:32 by Christian Stump