- FindStat

Identifier

Mp00218: Set partitions —inverse Wachs-White-rho⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St000609: Set partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1,2,3,4,6},{5}} => {{1,2,3,4,6},{5}} => {{1,2,3,4},{5,6}} => 1

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

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

{{1,2,3,5,6},{4}} => {{1,2,3,5,6},{4}} => {{1,2,3,6},{4,5}} => 1

{{1,2,3,5},{4,6}} => {{1,2,3,6},{4,5}} => {{1,2,3,5},{4,6}} => 1

{{1,2,3,5},{4},{6}} => {{1,2,3,5},{4},{6}} => {{1,2,3},{4,5},{6}} => 1

{{1,2,3,6},{4,5}} => {{1,2,3,5},{4,6}} => {{1,2,3},{4,5,6}} => 2

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

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

{{1,2,3,6},{4},{5}} => {{1,2,3,6},{4},{5}} => {{1,2,3},{4},{5,6}} => 2

{{1,2,3},{4,6},{5}} => {{1,2,3},{4,6},{5}} => {{1,2,3},{4,6},{5}} => 1

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

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

{{1,2,4,5,6},{3}} => {{1,2,4,5,6},{3}} => {{1,2,5,6},{3,4}} => 1

{{1,2,4,5},{3,6}} => {{1,2,6},{3,4,5}} => {{1,2,4,5},{3,6}} => 1

{{1,2,4,5},{3},{6}} => {{1,2,4,5},{3},{6}} => {{1,2,5},{3,4},{6}} => 1

{{1,2,4,6},{3,5}} => {{1,2,5},{3,4,6}} => {{1,2,4},{3,5,6}} => 2

{{1,2,4},{3,5,6}} => {{1,2,5,6},{3,4}} => {{1,2,4,6},{3,5}} => 1

{{1,2,4},{3,5},{6}} => {{1,2,5},{3,4},{6}} => {{1,2,4},{3,5},{6}} => 1

{{1,2,4,6},{3},{5}} => {{1,2,4,6},{3},{5}} => {{1,2},{3,4,6},{5}} => 2

{{1,2,4},{3,6},{5}} => {{1,2,6},{3,4},{5}} => {{1,2,4},{3},{5,6}} => 2

{{1,2,4},{3},{5,6}} => {{1,2,4},{3},{5,6}} => {{1,2,6},{3,4},{5}} => 1

{{1,2,4},{3},{5},{6}} => {{1,2,4},{3},{5},{6}} => {{1,2},{3,4},{5},{6}} => 1

{{1,2,5,6},{3,4}} => {{1,2,4},{3,5,6}} => {{1,2,6},{3,4,5}} => 2

{{1,2,5},{3,4,6}} => {{1,2,4,5},{3,6}} => {{1,2,5},{3,4,6}} => 2

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$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 8 + 5\ q + 2\ q^{2}$

$F_{5} = 16 + 17\ q + 13\ q^{2} + 5\ q^{3} + q^{4}$

$F_{6} = 32 + 49\ q + 54\ q^{2} + 39\ q^{3} + 20\ q^{4} + 7\ q^{5} + 2\ q^{6}$

$F_{7} = 64 + 129\ q + 183\ q^{2} + 187\ q^{3} + 148\ q^{4} + 92\ q^{5} + 48\ q^{6} + 19\ q^{7} + 6\ q^{8} + q^{9}$

Description

The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.

Map

intertwining number to dual major index

Description

A bijection sending the intertwining number of a set partition to its dual major index.
More precisely, St000490The intertwining number of a set partition.

$(P) =$ St000493The los statistic of a set partition.

$(\phi(P))$ for all set partitions

$P$ .

Map

inverse Wachs-White-rho

Description

The inverse of a transformation of set partitions due to Wachs and White.
Return the set partition of

$\{1,...,n\}$ corresponding to the set of arcs, interpreted as a rook placement, applying Wachs and White's bijection

$\rho^{-1}$ .
Note that our index convention differs from the convention in [1]: regarding the rook board as a lower-right triangular grid, we refer with

$(i,j)$ to the cell in the

$i$ -th column from the right and the

$j$ -th row from the top.