- FindStat

Identifier

Mp00040: Integer compositions —to partition⟶ Integer partitions
St001918: Integer partitions ⟶ ℤ

Values

[1] => [1] => 0

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

[2] => [2] => 1

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

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

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

[3] => [3] => 2

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

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

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

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

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

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

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

[4] => [4] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[5] => [5] => 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[6] => [6] => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 2280 entries. <<<

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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.

Map

to partition

Description

Sends a composition to the partition obtained by sorting the entries.