Identifier
-
Mp00138:
Dyck paths
—to noncrossing partition⟶
Set partitions
St000229: Set partitions ⟶ ℤ
Values
[1,0] => {{1}} => 1
[1,0,1,0] => {{1},{2}} => 2
[1,1,0,0] => {{1,2}} => 2
[1,0,1,0,1,0] => {{1},{2},{3}} => 3
[1,0,1,1,0,0] => {{1},{2,3}} => 3
[1,1,0,0,1,0] => {{1,2},{3}} => 3
[1,1,0,1,0,0] => {{1,3},{2}} => 4
[1,1,1,0,0,0] => {{1,2,3}} => 3
[1,0,1,0,1,0,1,0] => {{1},{2},{3},{4}} => 4
[1,0,1,0,1,1,0,0] => {{1},{2},{3,4}} => 4
[1,0,1,1,0,0,1,0] => {{1},{2,3},{4}} => 4
[1,0,1,1,0,1,0,0] => {{1},{2,4},{3}} => 5
[1,0,1,1,1,0,0,0] => {{1},{2,3,4}} => 4
[1,1,0,0,1,0,1,0] => {{1,2},{3},{4}} => 4
[1,1,0,0,1,1,0,0] => {{1,2},{3,4}} => 4
[1,1,0,1,0,0,1,0] => {{1,3},{2},{4}} => 5
[1,1,0,1,0,1,0,0] => {{1,4},{2},{3}} => 6
[1,1,0,1,1,0,0,0] => {{1,3,4},{2}} => 5
[1,1,1,0,0,0,1,0] => {{1,2,3},{4}} => 4
[1,1,1,0,0,1,0,0] => {{1,4},{2,3}} => 6
[1,1,1,0,1,0,0,0] => {{1,2,4},{3}} => 5
[1,1,1,1,0,0,0,0] => {{1,2,3,4}} => 4
[1,0,1,0,1,0,1,0,1,0] => {{1},{2},{3},{4},{5}} => 5
[1,0,1,0,1,0,1,1,0,0] => {{1},{2},{3},{4,5}} => 5
[1,0,1,0,1,1,0,0,1,0] => {{1},{2},{3,4},{5}} => 5
[1,0,1,0,1,1,0,1,0,0] => {{1},{2},{3,5},{4}} => 6
[1,0,1,0,1,1,1,0,0,0] => {{1},{2},{3,4,5}} => 5
[1,0,1,1,0,0,1,0,1,0] => {{1},{2,3},{4},{5}} => 5
[1,0,1,1,0,0,1,1,0,0] => {{1},{2,3},{4,5}} => 5
[1,0,1,1,0,1,0,0,1,0] => {{1},{2,4},{3},{5}} => 6
[1,0,1,1,0,1,0,1,0,0] => {{1},{2,5},{3},{4}} => 7
[1,0,1,1,0,1,1,0,0,0] => {{1},{2,4,5},{3}} => 6
[1,0,1,1,1,0,0,0,1,0] => {{1},{2,3,4},{5}} => 5
[1,0,1,1,1,0,0,1,0,0] => {{1},{2,5},{3,4}} => 7
[1,0,1,1,1,0,1,0,0,0] => {{1},{2,3,5},{4}} => 6
[1,0,1,1,1,1,0,0,0,0] => {{1},{2,3,4,5}} => 5
[1,1,0,0,1,0,1,0,1,0] => {{1,2},{3},{4},{5}} => 5
[1,1,0,0,1,0,1,1,0,0] => {{1,2},{3},{4,5}} => 5
[1,1,0,0,1,1,0,0,1,0] => {{1,2},{3,4},{5}} => 5
[1,1,0,0,1,1,0,1,0,0] => {{1,2},{3,5},{4}} => 6
[1,1,0,0,1,1,1,0,0,0] => {{1,2},{3,4,5}} => 5
[1,1,0,1,0,0,1,0,1,0] => {{1,3},{2},{4},{5}} => 6
[1,1,0,1,0,0,1,1,0,0] => {{1,3},{2},{4,5}} => 6
[1,1,0,1,0,1,0,0,1,0] => {{1,4},{2},{3},{5}} => 7
[1,1,0,1,0,1,0,1,0,0] => {{1,5},{2},{3},{4}} => 8
[1,1,0,1,0,1,1,0,0,0] => {{1,4,5},{2},{3}} => 7
[1,1,0,1,1,0,0,0,1,0] => {{1,3,4},{2},{5}} => 6
[1,1,0,1,1,0,0,1,0,0] => {{1,5},{2},{3,4}} => 8
[1,1,0,1,1,0,1,0,0,0] => {{1,3,5},{2},{4}} => 7
[1,1,0,1,1,1,0,0,0,0] => {{1,3,4,5},{2}} => 6
[1,1,1,0,0,0,1,0,1,0] => {{1,2,3},{4},{5}} => 5
[1,1,1,0,0,0,1,1,0,0] => {{1,2,3},{4,5}} => 5
[1,1,1,0,0,1,0,0,1,0] => {{1,4},{2,3},{5}} => 7
[1,1,1,0,0,1,0,1,0,0] => {{1,5},{2,3},{4}} => 8
[1,1,1,0,0,1,1,0,0,0] => {{1,4,5},{2,3}} => 7
[1,1,1,0,1,0,0,0,1,0] => {{1,2,4},{3},{5}} => 6
[1,1,1,0,1,0,0,1,0,0] => {{1,5},{2,4},{3}} => 9
[1,1,1,0,1,0,1,0,0,0] => {{1,2,5},{3},{4}} => 7
[1,1,1,0,1,1,0,0,0,0] => {{1,2,4,5},{3}} => 6
[1,1,1,1,0,0,0,0,1,0] => {{1,2,3,4},{5}} => 5
[1,1,1,1,0,0,0,1,0,0] => {{1,5},{2,3,4}} => 8
[1,1,1,1,0,0,1,0,0,0] => {{1,2,5},{3,4}} => 7
[1,1,1,1,0,1,0,0,0,0] => {{1,2,3,5},{4}} => 6
[1,1,1,1,1,0,0,0,0,0] => {{1,2,3,4,5}} => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => {{1},{2},{3},{4},{5},{6}} => 6
[1,0,1,0,1,0,1,0,1,1,0,0] => {{1},{2},{3},{4},{5,6}} => 6
[1,0,1,0,1,0,1,1,0,0,1,0] => {{1},{2},{3},{4,5},{6}} => 6
[1,0,1,0,1,0,1,1,0,1,0,0] => {{1},{2},{3},{4,6},{5}} => 7
[1,0,1,0,1,0,1,1,1,0,0,0] => {{1},{2},{3},{4,5,6}} => 6
[1,0,1,0,1,1,0,0,1,0,1,0] => {{1},{2},{3,4},{5},{6}} => 6
[1,0,1,0,1,1,0,0,1,1,0,0] => {{1},{2},{3,4},{5,6}} => 6
[1,0,1,0,1,1,0,1,0,0,1,0] => {{1},{2},{3,5},{4},{6}} => 7
[1,0,1,0,1,1,0,1,0,1,0,0] => {{1},{2},{3,6},{4},{5}} => 8
[1,0,1,0,1,1,0,1,1,0,0,0] => {{1},{2},{3,5,6},{4}} => 7
[1,0,1,0,1,1,1,0,0,0,1,0] => {{1},{2},{3,4,5},{6}} => 6
[1,0,1,0,1,1,1,0,0,1,0,0] => {{1},{2},{3,6},{4,5}} => 8
[1,0,1,0,1,1,1,0,1,0,0,0] => {{1},{2},{3,4,6},{5}} => 7
[1,0,1,0,1,1,1,1,0,0,0,0] => {{1},{2},{3,4,5,6}} => 6
[1,0,1,1,0,0,1,0,1,0,1,0] => {{1},{2,3},{4},{5},{6}} => 6
[1,0,1,1,0,0,1,0,1,1,0,0] => {{1},{2,3},{4},{5,6}} => 6
[1,0,1,1,0,0,1,1,0,0,1,0] => {{1},{2,3},{4,5},{6}} => 6
[1,0,1,1,0,0,1,1,0,1,0,0] => {{1},{2,3},{4,6},{5}} => 7
[1,0,1,1,0,0,1,1,1,0,0,0] => {{1},{2,3},{4,5,6}} => 6
[1,0,1,1,0,1,0,0,1,0,1,0] => {{1},{2,4},{3},{5},{6}} => 7
[1,0,1,1,0,1,0,0,1,1,0,0] => {{1},{2,4},{3},{5,6}} => 7
[1,0,1,1,0,1,0,1,0,0,1,0] => {{1},{2,5},{3},{4},{6}} => 8
[1,0,1,1,0,1,0,1,0,1,0,0] => {{1},{2,6},{3},{4},{5}} => 9
[1,0,1,1,0,1,0,1,1,0,0,0] => {{1},{2,5,6},{3},{4}} => 8
[1,0,1,1,0,1,1,0,0,0,1,0] => {{1},{2,4,5},{3},{6}} => 7
[1,0,1,1,0,1,1,0,0,1,0,0] => {{1},{2,6},{3},{4,5}} => 9
[1,0,1,1,0,1,1,0,1,0,0,0] => {{1},{2,4,6},{3},{5}} => 8
[1,0,1,1,0,1,1,1,0,0,0,0] => {{1},{2,4,5,6},{3}} => 7
[1,0,1,1,1,0,0,0,1,0,1,0] => {{1},{2,3,4},{5},{6}} => 6
[1,0,1,1,1,0,0,0,1,1,0,0] => {{1},{2,3,4},{5,6}} => 6
[1,0,1,1,1,0,0,1,0,0,1,0] => {{1},{2,5},{3,4},{6}} => 8
[1,0,1,1,1,0,0,1,0,1,0,0] => {{1},{2,6},{3,4},{5}} => 9
[1,0,1,1,1,0,0,1,1,0,0,0] => {{1},{2,5,6},{3,4}} => 8
[1,0,1,1,1,0,1,0,0,0,1,0] => {{1},{2,3,5},{4},{6}} => 7
[1,0,1,1,1,0,1,0,0,1,0,0] => {{1},{2,6},{3,5},{4}} => 10
[1,0,1,1,1,0,1,0,1,0,0,0] => {{1},{2,3,6},{4},{5}} => 8
[1,0,1,1,1,0,1,1,0,0,0,0] => {{1},{2,3,5,6},{4}} => 7
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Description
Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition.
This is, for a set partition $P = \{B_1,\ldots,B_k\}$ of $\{1,\ldots,n\}$, the statistic is
$$d(P) = \sum_i \big(\operatorname{max}(B_i)-\operatorname{min}(B_i)+1\big).$$
This statistic is called dimension index in [2]
This is, for a set partition $P = \{B_1,\ldots,B_k\}$ of $\{1,\ldots,n\}$, the statistic is
$$d(P) = \sum_i \big(\operatorname{max}(B_i)-\operatorname{min}(B_i)+1\big).$$
This statistic is called dimension index in [2]
Map
to noncrossing partition
Description
Biane's map to noncrossing set partitions.
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