Identifier
-
Mp00129:
Dyck paths
—to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶
Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000229: Set partitions ⟶ ℤ
Values
[1,0] => [1] => {{1}} => 1
[1,0,1,0] => [2,1] => {{1,2}} => 2
[1,1,0,0] => [1,2] => {{1},{2}} => 2
[1,0,1,0,1,0] => [2,3,1] => {{1,2,3}} => 3
[1,0,1,1,0,0] => [2,1,3] => {{1,2},{3}} => 3
[1,1,0,0,1,0] => [1,3,2] => {{1},{2,3}} => 3
[1,1,0,1,0,0] => [3,1,2] => {{1,3},{2}} => 4
[1,1,1,0,0,0] => [1,2,3] => {{1},{2},{3}} => 3
[1,0,1,0,1,0,1,0] => [2,3,4,1] => {{1,2,3,4}} => 4
[1,0,1,0,1,1,0,0] => [2,3,1,4] => {{1,2,3},{4}} => 4
[1,0,1,1,0,0,1,0] => [2,1,4,3] => {{1,2},{3,4}} => 4
[1,0,1,1,0,1,0,0] => [2,4,1,3] => {{1,2,4},{3}} => 5
[1,0,1,1,1,0,0,0] => [2,1,3,4] => {{1,2},{3},{4}} => 4
[1,1,0,0,1,0,1,0] => [1,3,4,2] => {{1},{2,3,4}} => 4
[1,1,0,0,1,1,0,0] => [1,3,2,4] => {{1},{2,3},{4}} => 4
[1,1,0,1,0,0,1,0] => [3,1,4,2] => {{1,3,4},{2}} => 5
[1,1,0,1,0,1,0,0] => [3,4,1,2] => {{1,3},{2,4}} => 6
[1,1,0,1,1,0,0,0] => [3,1,2,4] => {{1,3},{2},{4}} => 5
[1,1,1,0,0,0,1,0] => [1,2,4,3] => {{1},{2},{3,4}} => 4
[1,1,1,0,0,1,0,0] => [1,4,2,3] => {{1},{2,4},{3}} => 5
[1,1,1,0,1,0,0,0] => [4,1,2,3] => {{1,4},{2},{3}} => 6
[1,1,1,1,0,0,0,0] => [1,2,3,4] => {{1},{2},{3},{4}} => 4
[1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => {{1,2,3,4,5}} => 5
[1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => {{1,2,3,4},{5}} => 5
[1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => {{1,2,3},{4,5}} => 5
[1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => {{1,2,3,5},{4}} => 6
[1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => {{1,2,3},{4},{5}} => 5
[1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => {{1,2},{3,4,5}} => 5
[1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => {{1,2},{3,4},{5}} => 5
[1,0,1,1,0,1,0,0,1,0] => [2,4,1,5,3] => {{1,2,4,5},{3}} => 6
[1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => {{1,2,4},{3,5}} => 7
[1,0,1,1,0,1,1,0,0,0] => [2,4,1,3,5] => {{1,2,4},{3},{5}} => 6
[1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => {{1,2},{3},{4,5}} => 5
[1,0,1,1,1,0,0,1,0,0] => [2,1,5,3,4] => {{1,2},{3,5},{4}} => 6
[1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => {{1,2,5},{3},{4}} => 7
[1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => {{1,2},{3},{4},{5}} => 5
[1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => {{1},{2,3,4,5}} => 5
[1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => {{1},{2,3,4},{5}} => 5
[1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => {{1},{2,3},{4,5}} => 5
[1,1,0,0,1,1,0,1,0,0] => [1,3,5,2,4] => {{1},{2,3,5},{4}} => 6
[1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => {{1},{2,3},{4},{5}} => 5
[1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2] => {{1,3,4,5},{2}} => 6
[1,1,0,1,0,0,1,1,0,0] => [3,1,4,2,5] => {{1,3,4},{2},{5}} => 6
[1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2] => {{1,3},{2,4,5}} => 7
[1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => {{1,3,5},{2,4}} => 8
[1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => {{1,3},{2,4},{5}} => 7
[1,1,0,1,1,0,0,0,1,0] => [3,1,2,5,4] => {{1,3},{2},{4,5}} => 6
[1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => {{1,3,5},{2},{4}} => 7
[1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => {{1,3},{2,5},{4}} => 8
[1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => {{1,3},{2},{4},{5}} => 6
[1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => {{1},{2},{3,4,5}} => 5
[1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => {{1},{2},{3,4},{5}} => 5
[1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => {{1},{2,4,5},{3}} => 6
[1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => {{1},{2,4},{3,5}} => 7
[1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => {{1},{2,4},{3},{5}} => 6
[1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3] => {{1,4,5},{2},{3}} => 7
[1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => {{1,4},{2},{3,5}} => 8
[1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => {{1,4},{2,5},{3}} => 9
[1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => {{1,4},{2},{3},{5}} => 7
[1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => {{1},{2},{3},{4,5}} => 5
[1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => {{1},{2},{3,5},{4}} => 6
[1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4] => {{1},{2,5},{3},{4}} => 7
[1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => {{1,5},{2},{3},{4}} => 8
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}} => 5
[1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => {{1,2,3,4,5,6}} => 6
[1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6] => {{1,2,3,4,5},{6}} => 6
[1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,1,6,5] => {{1,2,3,4},{5,6}} => 6
[1,0,1,0,1,0,1,1,0,1,0,0] => [2,3,4,6,1,5] => {{1,2,3,4,6},{5}} => 7
[1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,6] => {{1,2,3,4},{5},{6}} => 6
[1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,1,5,6,4] => {{1,2,3},{4,5,6}} => 6
[1,0,1,0,1,1,0,0,1,1,0,0] => [2,3,1,5,4,6] => {{1,2,3},{4,5},{6}} => 6
[1,0,1,0,1,1,0,1,0,0,1,0] => [2,3,5,1,6,4] => {{1,2,3,5,6},{4}} => 7
[1,0,1,0,1,1,0,1,0,1,0,0] => [2,3,5,6,1,4] => {{1,2,3,5},{4,6}} => 8
[1,0,1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => {{1,2,3,5},{4},{6}} => 7
[1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,1,4,6,5] => {{1,2,3},{4},{5,6}} => 6
[1,0,1,0,1,1,1,0,0,1,0,0] => [2,3,1,6,4,5] => {{1,2,3},{4,6},{5}} => 7
[1,0,1,0,1,1,1,0,1,0,0,0] => [2,3,6,1,4,5] => {{1,2,3,6},{4},{5}} => 8
[1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6] => {{1,2,3},{4},{5},{6}} => 6
[1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => {{1,2},{3,4,5,6}} => 6
[1,0,1,1,0,0,1,0,1,1,0,0] => [2,1,4,5,3,6] => {{1,2},{3,4,5},{6}} => 6
[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => {{1,2},{3,4},{5,6}} => 6
[1,0,1,1,0,0,1,1,0,1,0,0] => [2,1,4,6,3,5] => {{1,2},{3,4,6},{5}} => 7
[1,0,1,1,0,0,1,1,1,0,0,0] => [2,1,4,3,5,6] => {{1,2},{3,4},{5},{6}} => 6
[1,0,1,1,0,1,0,0,1,0,1,0] => [2,4,1,5,6,3] => {{1,2,4,5,6},{3}} => 7
[1,0,1,1,0,1,0,0,1,1,0,0] => [2,4,1,5,3,6] => {{1,2,4,5},{3},{6}} => 7
[1,0,1,1,0,1,0,1,0,0,1,0] => [2,4,5,1,6,3] => {{1,2,4},{3,5,6}} => 8
[1,0,1,1,0,1,0,1,0,1,0,0] => [2,4,5,6,1,3] => {{1,2,4,6},{3,5}} => 9
[1,0,1,1,0,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => {{1,2,4},{3,5},{6}} => 8
[1,0,1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,6,5] => {{1,2,4},{3},{5,6}} => 7
[1,0,1,1,0,1,1,0,0,1,0,0] => [2,4,1,6,3,5] => {{1,2,4,6},{3},{5}} => 8
[1,0,1,1,0,1,1,0,1,0,0,0] => [2,4,6,1,3,5] => {{1,2,4},{3,6},{5}} => 9
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,4,1,3,5,6] => {{1,2,4},{3},{5},{6}} => 7
[1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,3,5,6,4] => {{1,2},{3},{4,5,6}} => 6
[1,0,1,1,1,0,0,0,1,1,0,0] => [2,1,3,5,4,6] => {{1,2},{3},{4,5},{6}} => 6
[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,5,3,6,4] => {{1,2},{3,5,6},{4}} => 7
[1,0,1,1,1,0,0,1,0,1,0,0] => [2,1,5,6,3,4] => {{1,2},{3,5},{4,6}} => 8
[1,0,1,1,1,0,0,1,1,0,0,0] => [2,1,5,3,4,6] => {{1,2},{3,5},{4},{6}} => 7
[1,0,1,1,1,0,1,0,0,0,1,0] => [2,5,1,3,6,4] => {{1,2,5,6},{3},{4}} => 8
[1,0,1,1,1,0,1,0,0,1,0,0] => [2,5,1,6,3,4] => {{1,2,5},{3},{4,6}} => 9
[1,0,1,1,1,0,1,0,1,0,0,0] => [2,5,6,1,3,4] => {{1,2,5},{3,6},{4}} => 10
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,5,1,3,4,6] => {{1,2,5},{3},{4},{6}} => 8
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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 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
Map
weak exceedance partition
Description
The set partition induced by the weak exceedances of a permutation.
This is the coarsest set partition that contains all arcs $(i, \pi(i))$ with $i\leq\pi(i)$.
This is the coarsest set partition that contains all arcs $(i, \pi(i))$ with $i\leq\pi(i)$.
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