- FindStat

Identifier

Mp00151: Permutations —to cycle type⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000703: Permutations ⟶ ℤ

Values

[1] => {{1}} => [1] => 0

[1,2] => {{1},{2}} => [1,2] => 0

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

[1,2,3] => {{1},{2},{3}} => [1,2,3] => 0

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

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

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

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

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

[1,2,3,4] => {{1},{2},{3},{4}} => [1,2,3,4] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,3,4,5] => {{1},{2},{3},{4},{5}} => [1,2,3,4,5] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 5\ q$

$F_{4} = 1 + 20\ q + 3\ q^{2}$

$F_{5} = 1 + 84\ q + 35\ q^{2}$

$F_{6} = 1 + 409\ q + 295\ q^{2} + 15\ q^{3}$

Description

The number of deficiencies of a permutation.
This is defined as

$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$
The number of exceedances is St000155The number of exceedances (also excedences) of a permutation..

Map

to permutation

Description

Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.

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].