- FindStat

Identifier

Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
St001099: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[-1,-2,-3,-4] => [1,1,1,1] => 15

[1,2,4,-3] => [2] => 0

[1,2,-4,3] => [2] => 0

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

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

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

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

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

[-1,-2,4,-3] => [2,1,1] => 12

[-1,-2,-4,3] => [2,1,1] => 12

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

[1,3,-2,4] => [2] => 0

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

[1,-3,2,4] => [2] => 0

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

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

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

[-1,3,-2,-4] => [2,1,1] => 12

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

[-1,-3,2,-4] => [2,1,1] => 12

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

[1,3,4,-2] => [3] => 0

[1,3,-4,2] => [3] => 0

[1,-3,4,2] => [3] => 0

[1,-3,-4,-2] => [3] => 0

[-1,3,4,-2] => [3,1] => 6

[-1,3,-4,2] => [3,1] => 6

[-1,-3,4,2] => [3,1] => 6

[-1,-3,-4,-2] => [3,1] => 6

[1,4,2,-3] => [3] => 0

[1,4,-2,3] => [3] => 0

[1,-4,2,3] => [3] => 0

[1,-4,-2,-3] => [3] => 0

[-1,4,2,-3] => [3,1] => 6

[-1,4,-2,3] => [3,1] => 6

[-1,-4,2,3] => [3,1] => 6

[-1,-4,-2,-3] => [3,1] => 6

[1,4,3,-2] => [2] => 0

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

[1,-4,3,2] => [2] => 0

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

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

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

[-1,4,-3,-2] => [2,1,1] => 12

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

[-1,-4,-3,2] => [2,1,1] => 12

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

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

[2,-1,3,4] => [2] => 0

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

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

[2,-1,-3,-4] => [2,1,1] => 12

[-2,1,3,4] => [2] => 0

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

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

[-2,1,-3,-4] => [2,1,1] => 12

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

[2,1,4,-3] => [2] => 0

[2,1,-4,3] => [2] => 0

[2,-1,4,3] => [2] => 0

[2,-1,4,-3] => [2,2] => 10

[2,-1,-4,3] => [2,2] => 10

[2,-1,-4,-3] => [2] => 0

[-2,1,4,3] => [2] => 0

>>> Load all 4127 entries. <<<

search for individual values

searching the database for the individual values of this statistic

Description

The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees.
For a generating function

$f$ the associated formal group law is the symmetric function

$f(f^{(-1)}(x_1) + f^{(-1)}(x_2), \dots)$ , see [1].
This statistic records the coefficient of the monomial symmetric function

$m_\lambda$ times the product of the factorials of the parts of

$\lambda$ in the formal group law for leaf labelled binary trees, with generating function

$f(x) = 1-\sqrt{1-2x}$ , see [1, sec. 3.2]
Fix a set of distinguishable vertices and a coloring of the vertices so that

$\lambda_i$ are colored

$i$ . This statistic gives the number of rooted binary trees with leaves labeled with this set of vertices and internal vertices unlabeled so that no pair of 'twin' leaves have the same color.

Map

odd cycle type

Description

The partition corresponding to the odd cycles.
A cycle of length

$\ell$ of a signed permutation

$\pi$ can be written in two line notation as

$\begin{array}{cccc} a_1 & a_2 & \dots & a_\ell \\ \pi(a_1) & \pi(a_2) & \dots & \pi(a_\ell) \end{array}$
where

$a_i > 0$ for all

$i$ ,

$a_{i+1} = |\pi(a_i)|$ for

$i < \ell$ and

$a_1 = |\pi(a_\ell)|$ .
The cycle is odd, if the number of negative elements in the second row is odd.
This map records the integer partition given by the lengths of the odd cycles.
The integer partition of even cycles together with the integer partition of the odd cycles determines the conjugacy class of the signed permutation.