- FindStat

Identifier

Mp00151: Permutations —to cycle type⟶ Set partitions
St000839: Set partitions ⟶ ℤ

Values

[1] => {{1}} => 1

[1,2] => {{1},{2}} => 2

[2,1] => {{1,2}} => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,3,4,5] => {{1},{2},{3},{4},{5}} => 5

[1,2,3,5,4] => {{1},{2},{3},{4,5}} => 4

[1,2,4,3,5] => {{1},{2},{3,4},{5}} => 5

[1,2,4,5,3] => {{1},{2},{3,4,5}} => 3

[1,2,5,3,4] => {{1},{2},{3,4,5}} => 3

[1,2,5,4,3] => {{1},{2},{3,5},{4}} => 4

[1,3,2,4,5] => {{1},{2,3},{4},{5}} => 5

[1,3,2,5,4] => {{1},{2,3},{4,5}} => 4

[1,3,4,2,5] => {{1},{2,3,4},{5}} => 5

[1,3,4,5,2] => {{1},{2,3,4,5}} => 2

[1,3,5,2,4] => {{1},{2,3,4,5}} => 2

[1,3,5,4,2] => {{1},{2,3,5},{4}} => 4

[1,4,2,3,5] => {{1},{2,3,4},{5}} => 5

[1,4,2,5,3] => {{1},{2,3,4,5}} => 2

[1,4,3,2,5] => {{1},{2,4},{3},{5}} => 5

[1,4,3,5,2] => {{1},{2,4,5},{3}} => 3

[1,4,5,2,3] => {{1},{2,4},{3,5}} => 3

[1,4,5,3,2] => {{1},{2,3,4,5}} => 2

[1,5,2,3,4] => {{1},{2,3,4,5}} => 2

[1,5,2,4,3] => {{1},{2,3,5},{4}} => 4

[1,5,3,2,4] => {{1},{2,4,5},{3}} => 3

[1,5,3,4,2] => {{1},{2,5},{3},{4}} => 4

[1,5,4,2,3] => {{1},{2,3,4,5}} => 2

[1,5,4,3,2] => {{1},{2,5},{3,4}} => 3

[2,1,3,4,5] => {{1,2},{3},{4},{5}} => 5

[2,1,3,5,4] => {{1,2},{3},{4,5}} => 4

[2,1,4,3,5] => {{1,2},{3,4},{5}} => 5

[2,1,4,5,3] => {{1,2},{3,4,5}} => 3

[2,1,5,3,4] => {{1,2},{3,4,5}} => 3

[2,1,5,4,3] => {{1,2},{3,5},{4}} => 4

[2,3,1,4,5] => {{1,2,3},{4},{5}} => 5

[2,3,1,5,4] => {{1,2,3},{4,5}} => 4

[2,3,4,1,5] => {{1,2,3,4},{5}} => 5

[2,3,4,5,1] => {{1,2,3,4,5}} => 1

[2,3,5,1,4] => {{1,2,3,4,5}} => 1

[2,3,5,4,1] => {{1,2,3,5},{4}} => 4

[2,4,1,3,5] => {{1,2,3,4},{5}} => 5

[2,4,1,5,3] => {{1,2,3,4,5}} => 1

[2,4,3,1,5] => {{1,2,4},{3},{5}} => 5

[2,4,3,5,1] => {{1,2,4,5},{3}} => 3

[2,4,5,1,3] => {{1,2,4},{3,5}} => 3

[2,4,5,3,1] => {{1,2,3,4,5}} => 1

[2,5,1,3,4] => {{1,2,3,4,5}} => 1

[2,5,1,4,3] => {{1,2,3,5},{4}} => 4

[2,5,3,1,4] => {{1,2,4,5},{3}} => 3

[2,5,3,4,1] => {{1,2,5},{3},{4}} => 4

[2,5,4,1,3] => {{1,2,3,4,5}} => 1

[2,5,4,3,1] => {{1,2,5},{3,4}} => 3

[3,1,2,4,5] => {{1,2,3},{4},{5}} => 5

[3,1,2,5,4] => {{1,2,3},{4,5}} => 4

[3,1,4,2,5] => {{1,2,3,4},{5}} => 5

[3,1,4,5,2] => {{1,2,3,4,5}} => 1

[3,1,5,2,4] => {{1,2,3,4,5}} => 1

[3,1,5,4,2] => {{1,2,3,5},{4}} => 4

[3,2,1,4,5] => {{1,3},{2},{4},{5}} => 5

[3,2,1,5,4] => {{1,3},{2},{4,5}} => 4

[3,2,4,1,5] => {{1,3,4},{2},{5}} => 5

[3,2,4,5,1] => {{1,3,4,5},{2}} => 2

[3,2,5,1,4] => {{1,3,4,5},{2}} => 2

[3,2,5,4,1] => {{1,3,5},{2},{4}} => 4

[3,4,1,2,5] => {{1,3},{2,4},{5}} => 5

[3,4,1,5,2] => {{1,3},{2,4,5}} => 2

[3,4,2,1,5] => {{1,2,3,4},{5}} => 5

[3,4,2,5,1] => {{1,2,3,4,5}} => 1

[3,4,5,1,2] => {{1,2,3,4,5}} => 1

[3,4,5,2,1] => {{1,3,5},{2,4}} => 2

[3,5,1,2,4] => {{1,3},{2,4,5}} => 2

[3,5,1,4,2] => {{1,3},{2,5},{4}} => 4

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$F_{1} = q$

$F_{2} = q + q^{2}$

$F_{3} = 2\ q + 2\ q^{2} + 2\ q^{3}$

$F_{4} = 6\ q + 6\ q^{2} + 6\ q^{3} + 6\ q^{4}$

$F_{5} = 24\ q + 24\ q^{2} + 24\ q^{3} + 24\ q^{4} + 24\ q^{5}$

$F_{6} = 120\ q + 120\ q^{2} + 120\ q^{3} + 120\ q^{4} + 120\ q^{5} + 120\ q^{6}$

$F_{7} = 720\ q + 720\ q^{2} + 720\ q^{3} + 720\ q^{4} + 720\ q^{5} + 720\ q^{6} + 720\ q^{7}$

Description

The largest opener of a set partition.
An opener (or left hand endpoint) of a set partition is a number that is minimal in its block. For this statistic, singletons are considered as openers.

Map

to cycle type

Description

Let

$\pi=c_1\dots c_r$ a permutation of size

$n$ decomposed in its cyclic parts. The associated set partition of

$[n]$ then is

$S=S_1\cup\dots\cup S_r$ such that

$S_i$ is the set of integers in the cycle

$c_i$ .
A permutation is cyclic [1] if and only if its cycle type is a hook partition [2].