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Identifier

St001918: Integer partitions ⟶ ℤ

Values

[1] => 0

[2] => 1

[1,1] => 0

[3] => 2

[2,1] => 1

[1,1,1] => 0

[4] => 3

[3,1] => 2

[2,2] => 1

[2,1,1] => 1

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

[5] => 4

[4,1] => 3

[3,2] => 4

[3,1,1] => 2

[2,2,1] => 1

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

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

[6] => 5

[5,1] => 4

[4,2] => 3

[4,1,1] => 3

[3,3] => 2

[3,2,1] => 4

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

[2,2,2] => 1

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

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

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

[7] => 6

[6,1] => 5

[5,2] => 8

[5,1,1] => 4

[4,3] => 9

[4,2,1] => 3

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

[3,3,1] => 2

[3,2,2] => 4

[3,2,1,1] => 4

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

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

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

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

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

[8] => 7

[7,1] => 6

[6,2] => 5

[6,1,1] => 5

[5,3] => 12

[5,2,1] => 8

[5,1,1,1] => 4

[4,4] => 3

[4,3,1] => 9

[4,2,2] => 3

[4,2,1,1] => 3

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

[3,3,2] => 4

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

[3,2,2,1] => 4

[3,2,1,1,1] => 4

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

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

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

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

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

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

[9] => 8

[8,1] => 7

[7,2] => 12

[7,1,1] => 6

[6,3] => 5

[6,2,1] => 5

[6,1,1,1] => 5

[5,4] => 16

[5,3,1] => 12

[5,2,2] => 8

[5,2,1,1] => 8

[5,1,1,1,1] => 4

[4,4,1] => 3

[4,3,2] => 9

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

[4,2,2,1] => 3

[4,2,1,1,1] => 3

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

[3,3,3] => 2

[3,3,2,1] => 4

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

[3,2,2,2] => 4

[3,2,2,1,1] => 4

[3,2,1,1,1,1] => 4

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

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

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

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

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

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

[10] => 9

[9,1] => 8

[8,2] => 7

[8,1,1] => 7

[7,3] => 18

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Description

The degree of the cyclic sieving polynomial corresponding to an integer partition.
Let $\lambda$ be an integer partition of $n$ and let $N$ be the least common multiple of the parts of $\lambda$. Fix an arbitrary permutation $\pi$ of cycle type $\lambda$. Then $\pi$ induces a cyclic action of order $N$ on $\{1,\dots,n\}$.
The corresponding character can be identified with the cyclic sieving polynomial $C_\lambda(q)$ of this action, modulo $q^N-1$. Explicitly, it is
$$ \sum_{p\in\lambda} [p]_{q^{N/p}}, $$
where $[p]_q = 1+\dots+q^{p-1}$ is the $q$-integer.
This statistic records the degree of $C_\lambda(q)$. Equivalently, it equals
$$ \left(1 - \frac{1}{\lambda_1}\right) N, $$
where $\lambda_1$ is the largest part of $\lambda$.
The statistic is undefined for the empty partition.

Code

def statistic(la):
    return (la[0] - 1) * lcm(la) / la[0]

def statistic_alternative(la):
    return CyclicSievingPolynomial(la).degree()

Created

Aug 21, 2023 at 14:19 by Martin Rubey

Updated

Aug 21, 2023 at 14:19 by Martin Rubey