- FindStat

Identifier

Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000577: Set partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{2} = 1 + q$

$F_{3} = 1 + 3\ q + q^{2} + q^{3}$

$F_{4} = 1 + 7\ q + 3\ q^{2} + 8\ q^{3} + 3\ q^{4} + q^{5} + q^{6}$

$F_{5} = 1 + 15\ q + 7\ q^{2} + 34\ q^{3} + 18\ q^{4} + 14\ q^{5} + 18\ q^{6} + 8\ q^{7} + 3\ q^{8} + q^{9} + q^{10}$

$F_{6} = 1 + 31\ q + 15\ q^{2} + 122\ q^{3} + 72\ q^{4} + 69\ q^{5} + 147\ q^{6} + 83\ q^{7} + 71\ q^{8} + 33\ q^{9} + 45\ q^{10} + 18\ q^{11} + 8\ q^{12} + 3\ q^{13} + q^{14} + q^{15}$

$F_{7} = 1 + 63\ q + 31\ q^{2} + 406\ q^{3} + 252\ q^{4} + 263\ q^{5} + 822\ q^{6} + 544\ q^{7} + 554\ q^{8} + 399\ q^{9} + 613\ q^{10} + 351\ q^{11} + 307\ q^{12} + 160\ q^{13} + 102\ q^{14} + 96\ q^{15} + 45\ q^{16} + 18\ q^{17} + 8\ q^{18} + 3\ q^{19} + q^{20} + q^{21}$

Description

The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element.
This is the number of pairs

$i\lt j$ in different blocks such that

$i$ is the maximal element of a block.

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

$i\leq\pi(i)$ .