- FindStat

Identifier

Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000496: 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}} => 2

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

[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}} => 2

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

[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}} => 2

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

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

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

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

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

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

[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}} => 2

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

[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}} => 3

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

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

[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}} => 2

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

[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}} => 2

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

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

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

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

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

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

[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}} => 2

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

[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}} => 3

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 8 + 4\ q + 2\ q^{2}$

$F_{5} = 16 + 12\ q + 9\ q^{2} + 4\ q^{3} + q^{4}$

$F_{6} = 32 + 32\ q + 30\ q^{2} + 20\ q^{3} + 12\ q^{4} + 4\ q^{5} + 2\ q^{6}$

$F_{7} = 64 + 80\ q + 88\ q^{2} + 73\ q^{3} + 56\ q^{4} + 34\ q^{5} + 20\ q^{6} + 9\ q^{7} + 4\ q^{8} + q^{9}$

Description

The rcs statistic of a set partition.
Let

$S = B_1,\ldots,B_k$ be a set partition with ordered blocks

$B_i$ and with

$\operatorname{min} B_a < \operatorname{min} B_b$ for

$a < b$ .
According to [1, Definition 3], a rcs (right-closer-smaller) of

$S$ is given by a pair

$i > j$ such that

$j = \operatorname{max} B_b$ and

$i \in B_a$ for

$a < b$ .

Map

to noncrossing partition

Description

Biane's map to noncrossing set partitions.