- FindStat

Identifier

Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
St001094: Set partitions ⟶ ℤ

Values

{{1}} => {{1}} => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1,2,3,4,5}} => {{1,2,3,4,5}} => 10

{{1,2,3,4},{5}} => {{1,2,3,4},{5}} => 9

{{1,2,3,5},{4}} => {{1,2,3},{4,5}} => 9

{{1,2,3},{4,5}} => {{1,2,3,5},{4}} => 8

{{1,2,3},{4},{5}} => {{1,2,3},{4},{5}} => 7

{{1,2,4,5},{3}} => {{1,2},{3,4,5}} => 9

{{1,2,4},{3,5}} => {{1,2,5},{3,4}} => 8

{{1,2,4},{3},{5}} => {{1,2},{3,4},{5}} => 7

{{1,2,5},{3,4}} => {{1,2,4,5},{3}} => 8

{{1,2},{3,4,5}} => {{1,2,4},{3,5}} => 7

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

{{1,2,5},{3},{4}} => {{1,2},{3},{4,5}} => 7

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

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

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

{{1,3,4,5},{2}} => {{1},{2,3,4,5}} => 9

{{1,3,4},{2,5}} => {{1,5},{2,3,4}} => 8

{{1,3,4},{2},{5}} => {{1},{2,3,4},{5}} => 7

{{1,3,5},{2,4}} => {{1,4,5},{2,3}} => 8

{{1,3},{2,4,5}} => {{1,4},{2,3,5}} => 7

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

{{1,3,5},{2},{4}} => {{1},{2,3},{4,5}} => 7

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

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

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

{{1,4,5},{2,3}} => {{1,3,4,5},{2}} => 8

{{1,4},{2,3,5}} => {{1,3,4},{2,5}} => 7

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

{{1,5},{2,3,4}} => {{1,3},{2,4,5}} => 7

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

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

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

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

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

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

{{1,4,5},{2},{3}} => {{1},{2},{3,4,5}} => 7

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => 15

{{1,2,3,4,5},{6}} => {{1,2,3,4,5},{6}} => 14

{{1,2,3,4,6},{5}} => {{1,2,3,4},{5,6}} => 14

{{1,2,3,4},{5,6}} => {{1,2,3,4,6},{5}} => 13

{{1,2,3,4},{5},{6}} => {{1,2,3,4},{5},{6}} => 12

{{1,2,3,5,6},{4}} => {{1,2,3},{4,5,6}} => 14

{{1,2,3,5},{4,6}} => {{1,2,3,6},{4,5}} => 13

{{1,2,3,5},{4},{6}} => {{1,2,3},{4,5},{6}} => 12

{{1,2,3,6},{4,5}} => {{1,2,3,5,6},{4}} => 13

{{1,2,3},{4,5,6}} => {{1,2,3,5},{4,6}} => 12

{{1,2,3},{4,5},{6}} => {{1,2,3,5},{4},{6}} => 11

{{1,2,3,6},{4},{5}} => {{1,2,3},{4},{5,6}} => 12

{{1,2,3},{4,6},{5}} => {{1,2,3},{4,6},{5}} => 11

{{1,2,3},{4},{5,6}} => {{1,2,3,6},{4},{5}} => 10

{{1,2,3},{4},{5},{6}} => {{1,2,3},{4},{5},{6}} => 9

{{1,2,4,5,6},{3}} => {{1,2},{3,4,5,6}} => 14

{{1,2,4,5},{3,6}} => {{1,2,6},{3,4,5}} => 13

{{1,2,4,5},{3},{6}} => {{1,2},{3,4,5},{6}} => 12

{{1,2,4,6},{3,5}} => {{1,2,5,6},{3,4}} => 13

{{1,2,4},{3,5,6}} => {{1,2,5},{3,4,6}} => 12

{{1,2,4},{3,5},{6}} => {{1,2,5},{3,4},{6}} => 11

{{1,2,4,6},{3},{5}} => {{1,2},{3,4},{5,6}} => 12

{{1,2,4},{3,6},{5}} => {{1,2},{3,4,6},{5}} => 11

{{1,2,4},{3},{5,6}} => {{1,2,6},{3,4},{5}} => 10

{{1,2,4},{3},{5},{6}} => {{1,2},{3,4},{5},{6}} => 9

{{1,2,5,6},{3,4}} => {{1,2,4,5,6},{3}} => 13

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Description

The depth index of a set partition.
For a set partition $\Pi$ of $\{1,\dots,n\}$ with arcs $\mathcal A$, this is $$\sum_{i=1}^{|\mathcal A|} (n-i) - \sum_{j=1}^n depth(j) + \sum_{\alpha\in\mathcal A} depth(\alpha),$$
where the depth of an element $i$ is the number of arcs $(k,\ell)$ with $k < i < \ell$, and the depth of an arc $(i,j)$ is the number of arcs $(k,\ell)$ with $k < i$ and $j < \ell$.

Map

dual major index to intertwining number

Description

A bijection sending the dual major index of a set partition to its intertwining number.
More precisely, St000493The los statistic of a set partition.$(P) = $St000490The intertwining number of a set partition.$(\phi(P))$ for all set partitions $P$.
This is the inverse of Mp00171intertwining number to dual major index.