- FindStat

Identifier

Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001840: Set partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of descents of a set partition.
The Mahonian representation of a set partition $\{B_1,\dots,B_k\}$ of $\{1,\dots,n\}$ is the restricted growth word $w_1\dots w_n\}$ obtained by sorting the blocks of the set partition according to their maximal element, and setting $w_i$ to the index of the block containing $i$.
The word $w$ has a descent at position $i$ if $w_i > w_{i+1}$.

Map

to noncrossing partition

Description

Biane's map to noncrossing set partitions.