- FindStat

Identifier

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000247: Set partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => {{1,2,3,4,5,6}} => 0

[1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6] => {{1,2,3,4,5},{6}} => 1

[1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,1,6,5] => {{1,2,3,4},{5,6}} => 0

[1,0,1,0,1,0,1,1,0,1,0,0] => [2,3,4,6,1,5] => {{1,2,3,4,5,6}} => 0

[1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,6] => {{1,2,3,4},{5},{6}} => 2

[1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,1,5,6,4] => {{1,2,3},{4,5,6}} => 0

[1,0,1,0,1,1,0,0,1,1,0,0] => [2,3,1,5,4,6] => {{1,2,3},{4,5},{6}} => 1

[1,0,1,0,1,1,0,1,0,0,1,0] => [2,3,5,1,6,4] => {{1,2,3,4,5,6}} => 0

[1,0,1,0,1,1,0,1,0,1,0,0] => [2,3,5,6,1,4] => {{1,2,3,5},{4,6}} => 0

[1,0,1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => {{1,2,3,4,5},{6}} => 1

[1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,1,4,6,5] => {{1,2,3},{4},{5,6}} => 1

[1,0,1,0,1,1,1,0,0,1,0,0] => [2,3,1,6,4,5] => {{1,2,3},{4,5,6}} => 0

[1,0,1,0,1,1,1,0,1,0,0,0] => [2,3,6,1,4,5] => {{1,2,3,4,5,6}} => 0

[1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6] => {{1,2,3},{4},{5},{6}} => 3

[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => {{1,2},{3,4,5,6}} => 0

[1,0,1,1,0,0,1,0,1,1,0,0] => [2,1,4,5,3,6] => {{1,2},{3,4,5},{6}} => 1

[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => {{1,2},{3,4},{5,6}} => 0

[1,0,1,1,0,0,1,1,0,1,0,0] => [2,1,4,6,3,5] => {{1,2},{3,4,5,6}} => 0

[1,0,1,1,0,0,1,1,1,0,0,0] => [2,1,4,3,5,6] => {{1,2},{3,4},{5},{6}} => 2

[1,0,1,1,0,1,0,0,1,0,1,0] => [2,4,1,5,6,3] => {{1,2,3,4,5,6}} => 0

[1,0,1,1,0,1,0,0,1,1,0,0] => [2,4,1,5,3,6] => {{1,2,3,4,5},{6}} => 1

[1,0,1,1,0,1,0,1,0,0,1,0] => [2,4,5,1,6,3] => {{1,2,4},{3,5,6}} => 0

[1,0,1,1,0,1,0,1,0,1,0,0] => [2,4,5,6,1,3] => {{1,2,3,4,5,6}} => 0

[1,0,1,1,0,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => {{1,2,4},{3,5},{6}} => 1

[1,0,1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,6,5] => {{1,2,3,4},{5,6}} => 0

[1,0,1,1,0,1,1,0,0,1,0,0] => [2,4,1,6,3,5] => {{1,2,3,4,5,6}} => 0

[1,0,1,1,0,1,1,0,1,0,0,0] => [2,4,6,1,3,5] => {{1,2,4},{3,5,6}} => 0

[1,0,1,1,0,1,1,1,0,0,0,0] => [2,4,1,3,5,6] => {{1,2,3,4},{5},{6}} => 2

[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,3,5,6,4] => {{1,2},{3},{4,5,6}} => 1

[1,0,1,1,1,0,0,0,1,1,0,0] => [2,1,3,5,4,6] => {{1,2},{3},{4,5},{6}} => 2

[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,5,3,6,4] => {{1,2},{3,4,5,6}} => 0

[1,0,1,1,1,0,0,1,0,1,0,0] => [2,1,5,6,3,4] => {{1,2},{3,5},{4,6}} => 0

[1,0,1,1,1,0,0,1,1,0,0,0] => [2,1,5,3,4,6] => {{1,2},{3,4,5},{6}} => 1

[1,0,1,1,1,0,1,0,0,0,1,0] => [2,5,1,3,6,4] => {{1,2,3,4,5,6}} => 0

[1,0,1,1,1,0,1,0,0,1,0,0] => [2,5,1,6,3,4] => {{1,2,3,5},{4,6}} => 0

[1,0,1,1,1,0,1,0,1,0,0,0] => [2,5,6,1,3,4] => {{1,2,3,4,5,6}} => 0

[1,0,1,1,1,0,1,1,0,0,0,0] => [2,5,1,3,4,6] => {{1,2,3,4,5},{6}} => 1

[1,0,1,1,1,1,0,0,0,0,1,0] => [2,1,3,4,6,5] => {{1,2},{3},{4},{5,6}} => 2

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Description

The number of singleton blocks of a set partition.

Map

to 321-avoiding permutation (Billey-Jockusch-Stanley)

Description

The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.

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].