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Identifier

St001389: Integer partitions ⟶ ℤ

Values

[1] => 1

[2] => 2

[1,1] => 1

[3] => 3

[2,1] => 2

[1,1,1] => 1

[4] => 4

[3,1] => 3

[2,2] => 3

[2,1,1] => 2

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

[5] => 5

[4,1] => 4

[3,2] => 5

[3,1,1] => 3

[2,2,1] => 3

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

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

[6] => 6

[5,1] => 5

[4,2] => 7

[4,1,1] => 4

[3,3] => 6

[3,2,1] => 5

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

[2,2,2] => 4

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

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

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

[7] => 7

[6,1] => 6

[5,2] => 9

[5,1,1] => 5

[4,3] => 9

[4,2,1] => 7

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

[3,3,1] => 6

[3,2,2] => 7

[3,2,1,1] => 5

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

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

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

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

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

[8] => 8

[7,1] => 7

[6,2] => 11

[6,1,1] => 6

[5,3] => 12

[5,2,1] => 9

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

[4,4] => 10

[4,3,1] => 9

[4,2,2] => 10

[4,2,1,1] => 7

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

[3,3,2] => 9

[3,3,1,1] => 6

[3,2,2,1] => 7

[3,2,1,1,1] => 5

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

[2,2,2,2] => 5

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

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

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

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

[9] => 9

[8,1] => 8

[7,2] => 13

[7,1,1] => 7

[6,3] => 15

[6,2,1] => 11

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

[5,4] => 14

[5,3,1] => 12

[5,2,2] => 13

[5,2,1,1] => 9

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

[4,4,1] => 10

[4,3,2] => 14

[4,3,1,1] => 9

[4,2,2,1] => 10

[4,2,1,1,1] => 7

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

[3,3,3] => 10

[3,3,2,1] => 9

[3,3,1,1,1] => 6

[3,2,2,2] => 9

[3,2,2,1,1] => 7

[3,2,1,1,1,1] => 5

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

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

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

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

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

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

[10] => 10

[9,1] => 9

[8,2] => 15

[8,1,1] => 8

[7,3] => 18

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Description

The number of partitions of the same length below the given integer partition.
For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is
$$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$

Code

def statistic(la):
    k = len(la)
    return det(matrix([[binomial(la[k-i], j-i+1) for j in range(1, k+1)] for i in range(1, k+1)]))

Created

May 11, 2019 at 18:25 by Martin Rubey

Updated

May 11, 2019 at 18:25 by Martin Rubey