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Identifier

Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St001062: Set partitions ⟶ ℤ

Values

[1,2] => [1,2] => {{1},{2}} => 1

[2,1] => [2,1] => {{1,2}} => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{2} = q + q^{2}$

$F_{3} = q + 4\ q^{2} + q^{3}$

$F_{4} = q + 16\ q^{2} + 6\ q^{3} + q^{4}$

$F_{5} = q + 69\ q^{2} + 41\ q^{3} + 8\ q^{4} + q^{5}$

$F_{6} = q + 348\ q^{2} + 293\ q^{3} + 67\ q^{4} + 10\ q^{5} + q^{6}$

$F_{7} = q + 2016\ q^{2} + 2309\ q^{3} + 602\ q^{4} + 99\ q^{5} + 12\ q^{6} + q^{7}$

Description

The maximal size of a block of a set partition.

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

$i\leq\pi(i)$ .

Map

invert Laguerre heap

Description

The permutation obtained by inverting the corresponding Laguerre heap, according to Viennot.
Let

$\pi$ be a permutation. Following Viennot [1], we associate to

$\pi$ a heap of pieces, by considering each decreasing run

$(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of

$\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.